diff --git "a/OE_MM_maths_zh_COMP.tsv" "b/OE_MM_maths_zh_COMP.tsv" new file mode 100644--- /dev/null +++ "b/OE_MM_maths_zh_COMP.tsv" @@ -0,0 +1,259 @@ +id question solution final_answer context image modality difficulty is_multiple_answer unit answer_type error question_type subfield subject language +0 "在平面直角坐标系 $x O y$ 中, 曲线 $C_{1}: y^{2}=4 x$, 曲线 $C_{2}:(x-4)^{2}+y^{2}=8$. 经过 $C_{1}$上一点 $P$ 作一条倾斜角为 $45^{\circ}$ 的直线 $l$, 与 $C_{2}$ 交于两个不同的点 $Q, R$, 求 $|P Q| \cdot|P R|$的取值范围. + +" ['设 $P\\left(4 t^{2}, 4 t\\right)$, 则直线\n\n$$\nl: x-y-4 t^{2}+4 t=0\n$$\n\n该直线与圆相交, 因此\n\n$$\n\\frac{\\left|4-4 t^{2}+4 t\\right|}{\\sqrt{2}}<2 \\sqrt{2}\n$$\n\n即\n\n$$\n-1" ['如下图, 设抛物线方程为 $C: y^{2}=2 p x$, 焦点 $F\\left(\\frac{p}{2}, 0\\right)$, 连接 $C_{1} C_{2}, P F$.\n\n\n\n\n\n设 $P\\left(2 p a^{2}, 2 p a\\right)$, 则根据抛物线的光学性质, $|F Q|=|P F|$, 于是 $Q\\left(-2 p a^{2}, 0\\right)$, 进而由 $|P Q|=2$, 可得\n\n$$\n4 p^{2} a^{4}+p^{2} a^{2}=1,\n$$\n\n即\n\n$$\np^{2} a^{2}=\\frac{1}{4 a^{2}+1}\n$$\n\n圆心 $C_{1}, C_{2}$ 都在过点 $P$ 且与 $O P$ 垂直的直线 $l$ 上, 设直线 $l$ 的参数方程为\n\n$$\n\\left\\{\\begin{array}{l}\nx=2 p a^{2}+t \\\\\ny=2 p a-a t\n\\end{array}\\right.\n$$\n\n根据题意, $C_{1}, C_{2}$ 对应的参数满足\n\n$$\n\\sqrt{1+a^{2}} \\cdot|t|=2 p a-a t\n$$\n\n即\n\n$$\nt^{2}+4 p a^{2} t-4 p^{2} a^{2}=0\n$$\n\n因此圆 $C_{1}$ 的面积 $S_{1}$ 与圆 $C_{2}$ 的面积 $S_{2}$ 之和\n\n$$\n\\begin{aligned}\nS_{1}+S_{2} & =\\pi\\left[\\left(1+a^{2}\\right) t_{1}^{2}+\\left(1+a^{2}\\right) t_{2}^{2}\\right] \\\\\n& =\\pi\\left(1+a^{2}\\right)\\left[\\left(-4 p a^{2}\\right)^{2}+2 \\cdot 4 p^{2} a^{2}\\right] \\\\\n& =\\pi\\left(1+a^{2}\\right)\\left(16 a^{2}+8\\right) \\cdot p^{2} a^{2} \\\\\n& =\\frac{\\pi\\left(16 a^{4}+24 a^{2}+8\\right)}{4 a^{2}+1} \\\\\n& =\\pi\\left(4 a^{2}+1+\\frac{3}{4 a^{2}+1}+4\\right) \\\\\n& \\geqslant \\pi(4+2 \\sqrt{3})\n\\end{aligned}\n$$\n\n\n\n等号当 $4 a^{2}+1=\\sqrt{3}$, 即 $a^{2}=\\frac{\\sqrt{3}-1}{4}$ 时取得. 此时\n\n$$\np^{2}=\\frac{1}{a^{2}\\left(4 a^{2}+1\\right)}=\\frac{4}{(\\sqrt{3}-1) \\cdot \\sqrt{3}}\n$$\n\n于是 $F$ 点的坐标为 $\\left(\\frac{1}{\\sqrt{3-\\sqrt{3}}}, 0\\right)$.'] ['$\\left(\\frac{1}{\\sqrt{3-\\sqrt{3}}}, 0\\right)$'] 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Multimodal Competition False Tuple Open-ended Plane Geometry Math Chinese +2 "平面直角坐标系中 $x O y$ 中, $P$ 是不在 $x$ 轴上的一个动点, 过 $P$ 作抛物线 $y^{2}=4 x$ 的两条切线, 切点设为 $A, B$, 且直线 $P O \perp A B$ 于 $Q, R$ 为直线 $A B$ 与 $x$ 轴的交点. + + +求 $\frac{P Q}{Q R}$ 的最小值." ['设直线 $A B$ 的倾斜角为 $\\theta$, 根据对称性, 不妨设 $\\theta$ 为锐角. 过 $P$ 作 $x$ 轴的垂线, 设垂足为 $H$, 则 $\\triangle O P H$ 与 $\\triangle O R Q$ 相似, 于是\n\n$$\n\\frac{P Q}{Q R}=\\frac{P O+O Q}{Q R}=\\frac{\\frac{2}{\\sin \\theta}+2 \\sin \\theta}{2 \\cos \\theta}=\\frac{3-\\cos 2 \\theta}{\\sin 2 \\theta}\n$$\n\n令右侧代数式为 $t$, 则\n\n$$\nt \\sin 2 \\theta+\\cos 2 \\theta=3\n$$\n\n于是\n\n$$\n\\sqrt{1+t^{2}} \\geqslant 3\n$$\n\n解得 $t \\geqslant 2 \\sqrt{2}$, 等号当 $\\theta=\\frac{1}{2} \\arctan 2 \\sqrt{2}$ 时取得. 因此所求的最小值为 $2 \\sqrt{2}$.\n所以最终答案是 $2 \\sqrt{2}$。'] ['$2 \\sqrt{2}$'] 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Multimodal Competition False Numerical Open-ended Plane Geometry Math Chinese +3 "作斜率为 $\frac{1}{3}$ 的直线 $l$ 与椭圆 $C: \frac{x^{2}}{36}+\frac{y^{2}}{4}=1$ 交于 $A, B$ 两点 (如图所示), 且 $P(3 \sqrt{2}, \sqrt{2}$ )在直线 $l$ 的左上方. + + + +若 $\triangle A P B=60^{\circ}$, 求 $\triangle P A B$ 的面积." ['当 $\\angle A P B=60^{\\circ}$ 时, 结合 (1) 的结论可知 $k_{P A}=\\sqrt{3}, k_{P B}=-\\sqrt{3}$.\n\n直线 $P A$ 的方程为\n\n$$\ny-\\sqrt{2}=\\sqrt{3}(x-3 \\sqrt{2})\n$$\n\n代入 $\\frac{x^{2}}{36}+\\frac{y^{2}}{4}=1$ 中, 消去 $y$ 得\n\n$$\n14 x^{2}+9 \\sqrt{6}(1-3 \\sqrt{3}) x+18(13-3 \\sqrt{3})=0\n$$\n\n它的两根分别是 $x_{1}$ 和 $3 \\sqrt{2}$, 所以\n\n$$\nx_{1} \\cdot 3 \\sqrt{2}=\\frac{18(13-3 \\sqrt{3})}{14}\n$$\n\n即\n\n$$\nx_{1}=\\frac{3 \\sqrt{2}(13-3 \\sqrt{3})}{14}\n$$\n\n所以\n\n$$\n|P A|=\\sqrt{1+(\\sqrt{3})^{2}} \\cdot\\left|x_{1}-3 \\sqrt{2}\\right|=\\frac{3 \\sqrt{2}(3 \\sqrt{3}+1)}{7}\n$$\n\n同理可求得\n\n$$\n|P B|=\\frac{3 \\sqrt{2}(3 \\sqrt{3}-1)}{7}\n$$\n\n因此\n\n$$\n\\begin{aligned}\nS_{\\triangle P A B} & =\\frac{1}{2} \\cdot|P A| \\cdot|P B| \\cdot \\sin 60^{\\circ} \\\\\n& =\\frac{1}{2} \\cdot \\frac{3 \\sqrt{2}(3 \\sqrt{3}+1)}{7} \\cdot \\frac{3 \\sqrt{2}(3 \\sqrt{3}-1)}{7} \\cdot \\frac{\\sqrt{3}}{2} \\\\\n& =\\frac{117 \\sqrt{3}}{49} .\n\\end{aligned}\n$$'] ['$\\frac{117 \\sqrt{3}}{49}$'] 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Multimodal Competition False Numerical Open-ended Plane Geometry Math Chinese +4 "如图, $P$ 是揁物线 $y^{2}=2 x$ 上的动点, 点 $B 、 C$ 在 $y$ 轴上, 圆 $(x-1)^{2}+y^{2}=1$ 内切于 $\triangle P B C$,求 $\triangle P B C$ 面积的最小值. + +" ['设 $P\\left(x_{0}, y_{0}\\right), B(0, b), C(0, c)$, 不妨设 $b>c$. 直线 $P B$ 的方程: $y-b=\\frac{y_{0}-b}{x_{0}} x$,化简得 $\\left(y_{0}-b\\right) x-x_{0} y+x_{0} b=0$. 又圆心 $(1,0)$ 到 $P B$ 的距离为 $1, \\frac{\\left|y_{0}-b+x_{0} b\\right|}{\\sqrt{\\left(y_{0}-b\\right)^{2}+x_{0}^{2}}}=1$, 故 $\\left(y_{0}-b\\right)^{2}+x_{0}^{2}=\\left(y_{0}-b\\right)^{2}+2 x_{0} b\\left(y_{0}-b\\right)+x_{0}^{2} b^{2}$ ，易 知 $x_{0}>2$ ，上式化简得 $\\left(x_{0}-2\\right) b^{2}+2 y_{0} b-x_{0}=0$, 同理有 $\\left(x_{0}-2\\right) c^{2}+2 y_{0} c-x_{0}=0$. 所以 $b+c=\\frac{-2 y_{0}}{x_{0}-2}, b c=\\frac{-x_{0}}{x_{0}-2}$,\n\n则 $(b-c)^{2}=\\frac{4 x_{0}^{2}+4 y_{0}^{2}-8 x_{0}}{\\left(x_{0}-2\\right)^{2}}$. 因 $P\\left(x_{0}, y_{0}\\right)$ 是拋物线上的点, 有 $y_{0}^{2}=2 x_{0}$, 则 $(b-c)^{2}=\\frac{4 x_{0}^{2}}{\\left(x_{0}-2\\right)^{2}}$, $b-c=\\frac{2 x_{0}}{x_{0}-2}$. 所以 $S_{\\triangle P B C}=\\frac{1}{2}(b-c) \\cdot x_{0}=\\frac{x_{0}}{x_{0}-2} \\cdot x_{0}=\\left(x_{0}-2\\right)+\\frac{4}{x_{0}-2}+4 \\geq 2 \\sqrt{4}+4=8$. 当 $\\left(x_{0}-2\\right)^{2}=4$ 时, 上式取等号, 此时 $x_{0}=4, y_{0}= \\pm 2 \\sqrt{2}$. 因此 $S_{\\triangle P B C}$ 的最小值为 8 .'] ['$8$'] 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Multimodal Competition False Numerical Open-ended Geometry Math Chinese +5 " + +如图, 给定凸四边形 $A B C D, \angle B+\angle D<180^{\circ}, P$ 是平面上的动点, 令 $f(P)=P A \cdot B C+P D \cdot C A+P C \cdot A B$. +设 $E$ 是 $\triangle A B C$ 外接圆 $O$ 的 $\overparen{A B}$ 上一点, 满足: $\frac{A E}{A B}=\frac{\sqrt{3}}{2}, \frac{B C}{E C}=\sqrt{3}-1$, $\angle E C B=\frac{1}{2} \angle E C A$, 又 $D A, D C$ 是 $\odot O$ 的切线, $A C=\sqrt{2}$, 求 $f(P)$ 的最小值." ['记 $\\angle E C B=\\alpha$, 则 $\\angle E C A=2 \\alpha$, 由正弦定理有 $\\frac{A E}{A B}=\\frac{\\sin 2 \\alpha}{\\sin 3 \\alpha}=\\frac{\\sqrt{3}}{2}$, 从而\n\n$\\sqrt{3} \\sin 3 \\alpha=2 \\sin 2 \\alpha$, 即 $\\sqrt{3}\\left(3 \\sin \\alpha-4 \\sin ^{3} \\alpha\\right)=4 \\sin \\alpha \\cos \\alpha$, 所以\n\n$3 \\sqrt{3}-4 \\sqrt{3}\\left(1-\\cos ^{2} \\alpha\\right)-4 \\cos \\alpha=0$, 整理得 $4 \\sqrt{3} \\cos ^{2} \\alpha-4 \\cos \\alpha-\\sqrt{3}=0$,\n\n解得 $\\cos \\alpha=\\frac{\\sqrt{3}}{2}$ 或 $\\cos \\alpha=-\\frac{1}{2 \\sqrt{3}}$ （舍去）, 故 $\\alpha=30^{\\circ}, \\angle A C E=60^{\\circ}$. 由已知 $\\frac{B C}{E C}=\\sqrt{3}-1=\\frac{\\sin \\left(\\angle E A C-30^{\\circ}\\right)}{\\sin \\angle E A C}$, 有 $\\sin \\left(\\angle E A C-30^{\\circ}\\right)=(\\sqrt{3}-1) \\sin \\angle E A C$, 即 $\\frac{\\sqrt{3}}{2} \\sin \\angle E A C-\\frac{1}{2} \\cos \\angle E A C=(\\sqrt{3}-1) \\sin \\angle E A C$, 整理得 $\\frac{2-\\sqrt{3}}{2} \\sin \\angle E A C=\\frac{1}{2} \\cos \\angle E A C$,故 $\\tan \\angle E A C=\\frac{1}{2-\\sqrt{3}}=2+\\sqrt{3}$, 可得 $\\angle E A C=75^{\\circ}$, 从 而 $\\angle E=45^{\\circ}$, $\\angle D A C=\\angle D C A=\\angle E=45^{\\circ}, \\triangle A D C$ 为等腰直角三角形. 因 $A C=\\sqrt{2}$, 则 $C D=1$. 又 $\\triangle A B C$也是等腰直角三角形, 故 $B C=\\sqrt{2}, \\quad B D^{2}=1+2-2 \\cdot 1 \\cdot \\sqrt{2} \\cos 135^{\\circ}=5, B D=\\sqrt{5}$. 故 $f(P)_{\\min }=B D \\cdot A C=\\sqrt{5} \\cdot \\sqrt{2}=\\sqrt{10}$.'] ['$\\sqrt{10}$'] 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Multimodal Competition False Numerical Open-ended Geometry Math Chinese +6 " + +已知正三棱雉 $S-A B C$ 的高 $S O=3$, 底面边长为 6 , 过点 $A$ 向其所对侧面 $S B C$ 作垂线, 垂足为 $O^{\prime}$, 在 $A O^{\prime}$ 上取一点 $P$, 使 $\frac{A P}{P O^{\prime}}=8$, 求经过点 $P$ 且平行于底面的截面的面积." ['正三棱锥 $S-A B C$ 的高为 $S O$, 故 $A O \\perp B C$, 设 $A O$ 交 $B C$ 于 $E$, 连 SE. 则可证 $B C \\perp$ 面 $A E S$. 故面 $A E S \\perp$ 面 $S B C$.\n\n由 $A O^{\\prime} \\perp$ 面 $S B C$ 于 $O^{\\prime}$, 则 $A O^{\\prime}$ 在面 $A E S$ 内, $O^{\\prime}$ 在 $S E$ 上. $A O^{\\prime}$ 与 $S O$ 相交于点 $F$.\n\n$\\because A B C$ 为正三角形, $A B=6$, 故 $A E=3 \\sqrt{3}, O E=\\sqrt{3}$.\n\n$\\because S O=3, \\therefore \\tan \\angle O E S=\\sqrt{3}, \\angle E=60^{\\circ}$.\n\n$\\therefore O^{\\prime} E=A E \\cos 60^{\\circ}=\\frac{3}{2}\\sqrt{3}$.\n\n\n\n作 $O^{\\prime} G \\perp$ 平面 $A B C$, 则垂足 $G$ 在 $A E$ 上. $O^{\\prime} G=O^{\\prime}E \\sin 60^{\\circ} =\\frac{9}{4}$\n\n$\\because \\frac{A P}{P O^{\\prime}}=8, \\therefore \\frac{P H}{O^{\\prime} G}=\\frac{8}{9}, \\Rightarrow P H=2$.\n\n设过 $p$ 与底面平行的截面面积为 s, 截面与顶点 $S$ 的距离 $=3-2=1$.\n\n\n\n$\\therefore S_{\\triangle ABC}=\\frac{\\sqrt{3}}{4}\\cdot 6^{2}=9 \\sqrt{3}$.\n\n$\\therefore \\frac{s}{S_{\\triangle ABC}}=\\left(\\frac{1}{3}\\right)^{2}$, 故 $s=\\sqrt{3}$.'] ['$\\sqrt{3}$'] 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Multimodal Competition False Numerical Open-ended Geometry Math Chinese +7 "设双曲线 $x y=1$ 的两支为 $C_{1}, C_{2}$ (如图), 正三角形 $P Q R$ 的三顶点位于此双曲线上.设 $P(-1,-1)$ 在 $C_{2}$ 上， $Q 、 R$ 在 $C_{1}$ 上, 求顶点 $Q 、 R$ 的坐标. + + +" ['$p(-1,-1)$, 设 $Q\\left(x_{2}, \\frac{1}{x_{2}}\\right)$, 点 $p$ 在直线 $y=x$ 上. 以 $p$ 为圆心, $\\mid P Q\\mid$ 为半径作圆,此圆与双曲线第一象限内的另一交点 $R$ 满足 $|P Q|=|P R|$, 由圆与双曲线都是 $y=x$ 对称, 知 $Q$与 $R$ 关于 $y=x$ 对称. 且在第一象限内此二曲线没有其他交点 (二次曲线的交点个数). 于是 $R\\left(\\frac{1}{x_{2}}, x_{2}\\right)$.\n\n$\\therefore P Q$ 与 $y=\\mathrm{x}$ 的夹角 $=30^{\\circ}, P Q$ 所在直线的㑔斜角 $=75^{\\circ} \\cdot \\tan 75^{\\circ}=\\frac{1+\\frac{\\sqrt{3}}{3}}{1-\\frac{\\sqrt{3}}{3}}=2+\\sqrt{3}$.\n\n$P Q$ 所在直线方程为 $y+1=(2+\\sqrt{3})(x+1)$, 代入 $xy=1$, 解得 $Q(2-\\sqrt{3}, 2+\\sqrt{3})$, 于是 $R(2+\\sqrt{3}$, $2-\\sqrt{3})$.'] ['$(2-\\sqrt{3}, 2+\\sqrt{3}), (2+\\sqrt{3}, 2-\\sqrt{3})$'] 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Multimodal Competition True Tuple Open-ended Geometry Math Chinese +8 "在锐角三角形 $A B C$ 中, $A B$ 上的高 $C E$ 与 $A C$ 上的高 $B D$ 相交于点 $H$,以 $D E$ 为直径的圆分别交 $A B 、 A C$ 于 $F 、 G$ 两点, $F G$ 与 $A H$ 相交于点 $K$, 已知 $B C=25, B D=20$, $B E=7$, 求 $A K$ 的长. + +" ['$\\because B C=25, B D=20, B E=7$,\n\n$\\therefore C E=24, C D=15$.\n\n$\\because A C \\cdot B D = C E \\cdot A B \\Rightarrow A C=\\frac{6}{5} A B$\n\n$\\because B D \\perp A C, C E \\perp A B, \\Rightarrow B、E、D、C$ 共圆,\n\n$\\Rightarrow A C(A C-15)=A B(A B-7), \\Rightarrow \\frac{6}{5} A B\\left(\\frac{6}{5} A B-15\\right)=A B(A B$\n\n$-18)$,\n\n$\\therefore A B=25, A C=30 . \\Rightarrow A E=18, A D=15$.\n\n$\\therefore D E=\\frac{1}{2} A C=15$.\n\n延长 $A H$ 交 $B C$ 于 $P$, 则 $A P \\perp B C$.\n\n$\\therefore A P \\cdot B C=A C \\cdot B D \\Rightarrow A P=24$.\n\n\n\n连 $D F$, 则 $D F \\perp A B$,\n\n$\\because A E=D E, D F \\perp A B . \\Rightarrow AF = \\frac{1}{2} AE =9$.\n\n$\\because D 、 E 、 F 、 G$ 共圆, $\\Rightarrow \\angle A F G=\\angle A D E=\\angle A B C, \\Rightarrow \\triangle A F G \\sim \\triangle A B C$,\n\n$\\therefore \\frac{A K}{A P} = \\frac{A F}{A B} \\Rightarrow A K = \\frac{9 \\times 24}{25}=\\frac{216}{25}$.'] ['$\\frac{216}{25}$'] 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Multimodal Competition False Numerical Open-ended Geometry Math Chinese +9 "如图, 有一列曲线 $\mathrm{P}_{0}, \mathrm{P}_{1}, \mathrm{P}_{2}, \cdots \cdots$, 已知 $\mathrm{P}_{0}$ 所围成的图形是面积为 1 的等边三角形, $P_{k+1}$ 是对 $P_{k}$ 进行如下操作得到的: 将 $P_{k}$ 的每条边三等分, 以每边中间部分的线段为边,向外作等边三角形，再将中间部分的线段去掉 $(\mathrm{k}=0,1,2,3, \cdots)$ ，记 $\mathrm{S}_{\mathrm{n}}$ 为曲线 $\mathrm{P}_{\mathrm{k}}$ 所围成图形面积. + + + +$\mathrm{P}_{0}$ + + + + +求数列 $\left\{\mathrm{S}_{\mathrm{n}}\right\}$ 的通项公式;" ['对 $\\mathrm{P}_{0}$ 进行操作, 容易看出 $\\mathrm{P}_{0}$ 的每条边变成 $\\mathrm{P}_{1}$ 的 4 条边, 故 $\\mathrm{P}_{1}$ 的边数为 $3 \\times 4$;同样, 对 $P_{1}$ 进行操作, $P_{1}$ 的每条边变成 $P_{2}$ 的 4 条边, 故 $P_{2}$ 的边数为 $3 \\times 4^{2}$, 从而不难得到 $P_{n}$ 的边数为 $3 \\times 4^{n}$\n\n已知 $P_{0}$ 的面积为 $S_{0}=1$, 比较 $P_{1}$ 与 $P_{0}$, 容易看出 $P_{1}$ 在 $P_{0}$ 的每条边上増加了一个小等边三角形, 其面积为 $\\frac{1}{3^{2}}$, 而 $\\mathrm{P}_{0}$ 有 3 条边, 故 $\\mathrm{S}_{\\mathrm{1}}=\\mathrm{S}_{0}+3 \\times \\frac{1}{3^{2}}=\\mathbf{1}+\\frac{1}{3}$\n\n再比较 $P_{2}$ 与 $P_{1}$, 容易看出 $P_{2}$ 在 $P_{1}$ 的每条边上増加了一个小等边三角形, 其面积为 $\\frac{1}{3^{2}} \\times$ $\\frac{1}{3^{2}}$, 而 $\\mathrm{P}_{1}$ 有 $3 \\times 4$ 条边, 故 $\\mathrm{S}_{\\mathrm{2}}=\\mathrm{S}_{1}+3 \\times 4 \\times \\frac{1}{3^{4}}=1+\\frac{1}{3}+\\frac{4}{3^{3}}$\n\n类似地有: $\\mathrm{S}_{3}=\\mathrm{S}_{2}+3 \\times 4^{2} \\times \\frac{1}{3^{6}}=\\mathbf{1}+\\frac{1}{3}+\\frac{4}{3^{3}}+\\frac{4^{2}}{3^{5}}$\n\n$\\therefore \\mathrm{S}_{\\mathrm{n}}=1+\\frac{1}{3}+\\frac{4}{3^{3}}+\\frac{4^{2}}{3^{5}}+\\cdots+\\frac{4^{n-1}}{3^{2 n-1}}=1+\\frac{3}{4} \\sum_{k=1}^{n}\\left(\\frac{4}{9}\\right)^{k}=\\frac{8}{5}-\\frac{3}{5} \\cdot\\left(\\frac{4}{9}\\right)^{n}$ ( $※$ )\n\n下面用数学归纳法证明 ( $※$ ) 式\n\n当 $\\mathrm{n}=1$ 时, 由上面已知 $(※)$ 式成立,\n\n假设当 $\\mathrm{n}=\\mathrm{k}$ 时, 有 $\\mathrm{S}_{\\mathrm{k}}=\\frac{8}{5}-\\frac{3}{5} \\cdot\\left(\\frac{4}{9}\\right)^{k}$\n\n当 $n=k+1$ 时, 易知第 $k+1$ 次操作后, 比较 $P_{k+1}$ 与 $P_{k}, P_{k+1}$ 在 $P_{k}$ 的每条边上增加了一个小等边三角形, 其面积为 $\\frac{1}{3^{2(k+1)}}$, 而 $\\mathrm{P}_{\\mathrm{k}}$ 有 $3 \\times 4^{\\mathrm{k}}$ 条边。故\n\n$$\nS_{k+1}=S_{k}+3 \\times 4^{k} \\times \\frac{1}{3^{2(k+1)}}=\\frac{8}{5}-\\frac{3}{5} \\cdot\\left(\\frac{4}{9}\\right)^{k+1}\n$$\n\n综上所述, 对任何 $\\mathrm{n} \\in \\mathrm{N},(※)$ 式成立.'] ['$\\frac{8}{5}-\\frac{3}{5} \\cdot\\left(\\frac{4}{9}\\right)^{n}$'] 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Multimodal Competition False Expression Open-ended Geometry Math Chinese +10 "如图, 有一列曲线 $\mathrm{P}_{0}, \mathrm{P}_{1}, \mathrm{P}_{2}, \cdots \cdots$, 已知 $\mathrm{P}_{0}$ 所围成的图形是面积为 1 的等边三角形, $P_{k+1}$ 是对 $P_{k}$ 进行如下操作得到的: 将 $P_{k}$ 的每条边三等分, 以每边中间部分的线段为边,向外作等边三角形，再将中间部分的线段去掉 $(\mathrm{k}=0,1,2,3, \cdots)$ ，记 $\mathrm{S}_{\mathrm{n}}$ 为曲线 $\mathrm{P}_{\mathrm{k}}$ 所围成图形面积. + + + +$\mathrm{P}_{0}$ + + + + +求 $\lim _{n \rightarrow \infty} S_{n}$ 。" ['$\\lim _{n \\rightarrow \\infty} S_{n}=\\lim _{n \\rightarrow \\infty}\\left[\\frac{8}{5}-\\frac{3}{5} \\cdot\\left(\\frac{4}{9}\\right)^{n}\\right]=\\frac{8}{5}$.'] ['$\\frac{8}{5}$'] 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Multimodal Competition False Numerical Open-ended Geometry Math Chinese +11 "如图, 在 $\triangle A B C$ 中, $\angle A=60^{\circ}, A B>A C$, 点 0 是外心, 两条高 $B E 、 C F$ 交于 $H$ 点, 点 $M$ 、 $\mathrm{N}$ 分别在线段 $\mathrm{BH} 、 \mathrm{HF}$ 上, 且满足 $\mathrm{BM}=\mathrm{CN}$, 求 $\frac{M H+N H}{O H}$ 的值. + +" ['在 $\\mathrm{BE}$ 上取 $\\mathrm{BK}=\\mathrm{CH}$ ，连接 $\\mathrm{OB} 、 \\mathrm{OC} 、 \\mathrm{OK}$,由三角形外心的性质知 $\\angle \\mathrm{BOC}=2 \\angle \\mathrm{A}=120^{\\circ}$\n\n由三角形垂心的性质知\n\n$\\angle \\mathrm{BHC}=180^{\\circ}-\\angle \\mathrm{A}=120^{\\circ}$\n\n$\\therefore \\angle \\mathrm{BOC}=\\angle \\mathrm{BHC}$\n\n$\\therefore B 、 C 、 H O$ 四点共圆\n\n$\\therefore \\angle \\mathrm{OBH}=\\angle \\mathrm{OCH} \\quad \\mathrm{OB}=\\mathrm{OC} \\quad \\mathrm{BK}=\\mathrm{CH}$\n\n$\\therefore \\triangle \\mathrm{BOK} \\cong \\triangle \\mathrm{COH}$\n\n$\\because \\mathrm{BOK}=\\angle \\mathrm{BOC}=120^{\\circ}, \\quad \\angle O \\mathrm{KH}=\\angle \\mathrm{OHK}=30^{\\circ}$\n\n观察 $\\triangle \\mathrm{OKH}$\n\n$\\frac{\\mathrm{KH}}{\\sin 120^{\\circ}}=\\frac{\\mathrm{OH}}{\\sin 30^{\\circ}} \\Rightarrow \\mathrm{KH}=\\sqrt{3} \\mathrm{OH}$\n\n又 $\\because \\mathrm{BM}=\\mathrm{CN}, \\mathrm{BK}=\\mathrm{CH}$,\n\n$\\therefore \\mathrm{KM}=\\mathrm{NH}$\n\n$\\therefore \\mathrm{MH}+\\mathrm{NH}=\\mathrm{MH}+\\mathrm{KM}=\\mathrm{KH}=\\sqrt{3} \\mathrm{OH}$\n\n$\\therefore \\frac{M H+N H}{O H}=\\sqrt{3}$.'] ['$\\sqrt{3}$'] 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Multimodal Competition False Numerical Open-ended Geometry Math Chinese +12 "在直角坐标系 $xoy$ 中, 点 $A\left(x_{1}, y_{1}\right)$ 和点 $B\left(x_{2}, y_{2}\right)$ 的坐标均为一位的正整数. $O A$ 与 $x$ 轴正方向的夹角大于 $45^{\circ}, O B$ 与 $x$ 轴正方向的夹角小于 $45^{\circ}, B$ 在 $x$ 轴上的射影为 $B^{\prime}, A$在 $y$ 轴上的射影为 $A^{\prime}, \triangle O B B^{\prime}$ 的面积比 $\triangle O A A^{\prime}$ 的面积大 33.5, 由 $x_{1}, y_{1}, x_{2}, y_{2}$ 组成的四位数 + +$\overline{x_{1} x_{2} y_{2} y_{1}}=x_{1} \cdot 10^{3}+x_{2} \cdot 10^{2}+y_{2} \cdot 10+y_{1}$. 试求出所有这样的四位数. + +" ['$x_{2} y_{2}-x_{1} y_{1}=67 . x_{1}y_{2}$. 且 $x_{1}, y_{1}, x_{2}, y_{2}$ 都是不超过 10 的正整数.\n\n$\\therefore x_{2} y_{2}>67, \\Rightarrow x_{2} y_{2}=72$ 或 81. 但 $x_{2}>y_{2}$, 故 $x_{2} y_{2}=91$ 舍去. $\\therefore x_{2} y_{2}=72 . x_{2}=9$, $y_{2}=8$.\n\n$\\therefore x_{1} y_{1}=72-67=5 . \\Rightarrow x_{1}=1, \\quad y_{1}=5, \\quad \\therefore \\overline{x_{1} x_{2} y_{2} y_{1}}=1985$.'] ['1985'] 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Multimodal Competition False Numerical Open-ended Geometry Math Chinese +13 " + +如图, 封闭多面体 $A B C D F E$, 平行四边形 $A B C D$ 中, $A C$ 与 $B D$ 相交于点 $O, E$ 在平面 $A B C D$ 的射影为 $O, E F / /$ 平面 $A B C D$, 且 $E F=\frac{1}{2} B C$, 其中 $A B=B C=A C=A E=a$. +若平面 $A B C D \perp$ 平面 $C F D$，求多面体 $A C D F E$ 的体积;" ['过 $\\mathrm{O}$ 做 $\\mathrm{ON} \\perp \\mathrm{CD}$ 于 $\\mathrm{N}$, $\\because$ 平面 $A B C D \\perp$ 平面 $C F D$,\n\n平面 $A B C D \\cap$ 平面 $C F D=C D, O N \\subset$ 平面 $A B C D \\therefore O N \\perp$ 平面 $C F D$\n\n$\\because O E / / F M, O E \\not \\subset$ 平面 $C F D, F M \\subset$ 平面 $C F D, \\therefore O E / /$ 平面 $C F D$,\n\n$\\therefore \\mathrm{E}$ 到平面 $CFD$ 的距离等于 $\\mathrm{O}$ 到平面 $CFD$ 的距离\n\n$\\triangle C O N$ 中, $O N=\\frac{1}{2} a \\sin 60^{\\circ}=\\frac{\\sqrt{3}}{4} a, \\triangle O A E$ 中, $O E=\\frac{\\sqrt{3}}{2} a$\n\n$V_{C-A E F D}=V_{E-A C D}+V_{E-C D F}=\\frac{1}{8} a^{3}+\\frac{1}{16} a^{3}=\\frac{3}{16} a^{3}$.'] ['$\\frac{3}{16} a^{3}$'] 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Multimodal Competition False Expression Open-ended Solid Geometry Math Chinese +14 " + +如图, 封闭多面体 $A B C D F E$, 平行四边形 $A B C D$ 中, $A C$ 与 $B D$ 相交于点 $O, E$ 在平面 $A B C D$ 的射影为 $O, E F / /$ 平面 $A B C D$, 且 $E F=\frac{1}{2} B C$, 其中 $A B=B C=A C=A E=a$. +求二面角 $A-C E-B$ 余弦值." ['\n\n建系如图所示: $A\\left(0,-\\frac{1}{2} a, 0\\right) 、 B\\left(\\frac{\\sqrt{3}}{2} a, 0,0\\right)$ 、\n\n$C\\left(0, \\frac{1}{2} a, 0\\right) 、 E\\left(0,0, \\frac{\\sqrt{3}}{2} a\\right)$\n\n$\\overrightarrow{C E}=\\left(0,-\\frac{1}{2} a, \\frac{\\sqrt{3}}{2} a\\right), \\overrightarrow{B C}=\\left(-\\frac{\\sqrt{3}}{2} a, \\frac{1}{2} a, 0\\right)$\n\n平面 $\\mathrm{ACE}$ 的法向量为 $\\overrightarrow{O B}=\\left(\\frac{\\sqrt{3}}{2} a, 0,0\\right)$,\n\n设平面 $\\mathrm{BCE}$ 法向量为 $\\vec{n}=(x, y, z)$ 则\n\n$\\vec{n} \\cdot \\overrightarrow{C E}=0, \\vec{n} \\cdot \\overrightarrow{B C}=0$, 令 $y=2, \\vec{n}=\\left(\\frac{2 \\sqrt{3}}{3}, 2, \\frac{2 \\sqrt{3}}{3}\\right)$\n\n$\\therefore \\cos \\langle\\vec{n}, \\overrightarrow{O B}\\rangle=\\frac{1}{\\frac{\\sqrt{3}}{2} \\cdot \\sqrt{\\frac{4}{3}+\\frac{4}{3}+4}}=\\frac{\\sqrt{5}}{5}$.\n\n$\\therefore$ 二面角 $A-C E-B$ 的余弦值为 $\\frac{\\sqrt{5}}{5}$.'] ['$\\frac{\\sqrt{5}}{5}$'] 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Multimodal Competition False Numerical Open-ended Solid Geometry Math Chinese +15 " + +如图, 在四棱锥 $P-A B C D$ 中, $P A \perp$ 平面 $A B C D$, 四边形 $A B C D$为矩形, $E$ 是 $P D$ 的中点, $M$ 是 $E C$ 的中点, 点 $Q$ 在线段 $P C$ 上且 $P Q=3 Q C$. +当 $\angle P B A$ 为多大时, 在线段 $P C$ 上存在点 $F$ 使得 $E F \perp$ 平面 $P A D$ 且 $E F$与平面 $P B C$ 所成角为 $45^{\circ}$ 同时成立?" "['取 $P C$ 中点 $F, \\because E$ 是 $P D$ 的中点, $\\therefore E F / / C D$, 又由题意知 $Q$ 是 $F C$ 的中点, $M$ 是 $E C$ 的中点, $\\therefore E F / / Q M$,\n\n$\\therefore Q M / / C D / / A B$. 又 $Q M \\not \\subset$ 平面 $P A B, A B \\subset$ 平面 $P A B$,\n\n$\\therefore Q M / /$ 平面 $P A B$ .\n\n当 $F$ 为 $P C$ 中点时, $E F / / A B \\because P A \\perp$ 平面 $A B C D, \\therefore P A \\perp A B$. 又 $\\because$ 四边形 $A B C D$ 为矩形, $\\therefore A B \\perp A D, \\therefore A B \\perp$ 平面 $P A D, \\therefore E F \\perp$ 平面 $P A D$ .\n\n$\\because P A \\perp B C, A B \\perp B C, \\therefore B C \\perp$ 平面 $P A B, \\therefore$ 平面 $P B C \\perp$ 平面 $P A B, \\therefore \\angle P B A$ 为 $A B$ 与平面 $P B C$ 所成角, $\\therefore \\angle P B A=45^{\\circ}$.' + '取 $P C$ 中点 $F, \\because E$ 是 $P D$ 的中点, $\\therefore E F / / C D$, 又由题意知 $Q$ 是 $F C$ 的中点, $M$ 是 $E C$ 的中点, $\\therefore E F / / Q M$,\n\n$\\therefore Q M / / C D / / A B$. 又 $Q M \\not \\subset$ 平面 $P A B, A B \\subset$ 平面 $P A B$,\n\n$\\therefore Q M / /$ 平面 $P A B$ .\n\n当 $F$ 为 $P C$ 中点时, $E F / / C D$.\n\n$\\because P A \\perp$ 平面 $A B C D$ 且 $C D \\subset$ 平面 $A B C D$\n\n$\\therefore P A \\perp C D$\n\n$\\because A B C D$ 为矩形 $\\therefore C D \\perp A D$\n\n且 $P A \\cap A D=A$\n\n$\\therefore C D \\perp$ 平面 $P A D \\quad \\therefore E F \\perp$ 平面 $P A D $.\n\n另一方面: 过点 $A$ 作 $A G \\perp P B$ 于 $G$.\n\n由上理\n$\\left.\\begin{array}{c}P A \\perp B C \\\\ A B \\perp B C \\\\ P A \\cap A D=A\\end{array}\\right\\} \\therefore B C \\perp$ 平面 $P A B$\n\n$\\because A G \\subset$ 平面 $P A B \\quad \\therefore A G \\perp B C$\n\n$\\because P B \\cap B C=B$\n\n$\\therefore A G \\perp$ 平面 $P B C$\n\n$\\therefore A B$ 在平面 $P B C$ 内的射影为 $P B$\n\n$\\therefore \\angle A B P$ 就是直线 $A B$ 与平面 $P B C$ 所成的角\n\n$\\because \\angle P B A=45^{\\circ}$ 且 $E F / / C D, C D / / A B$\n\n$\\therefore E F / / A B$ 且 $E F$ 与平面 $P B C$ 成角 $45^{\\circ}$.']" ['$45$'] 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Multimodal Competition False ^{\circ} Numerical Open-ended Solid Geometry Math Chinese +16 " + +如图, $E A \perp$ 平面 $A B C, A E / / C D, A B=A C=C D=2 A E=4$, $B C=2 \sqrt{3}, M$ 为 $B D$ 的中点. +求三棱锥 $E-A B M$ 的体积." ['由题意知, $V_{E-A B M}=V_{M-A B E}, \\because M N / /$ 平面 $A B E$,\n\n$$\n\\therefore V_{M-A B E}=V_{N-A B E}=V_{E-A B N}=\\frac{1}{3} \\cdot S_{\\triangle A B N} \\cdot A E\n$$\n\n$$\nA N=\\sqrt{13}, S_{\\triangle A B N}=\\frac{1}{2} \\cdot B N \\cdot A N=\\frac{\\sqrt{39}}{2}, \\therefore V_{E-A B M}=\\frac{1}{3} \\times \\frac{\\sqrt{39}}{2} \\times 2=\\frac{\\sqrt{39}}{3} .\n$$'] ['$\\frac{\\sqrt{39}}{3}$'] 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Multimodal Competition False Numerical Open-ended Solid Geometry Math Chinese +17 " + +如图, 在四棱锥 $A-B C D E$ 中, 平面 $A B C \perp$ 平面 $B C D E ; \angle C D E=\angle B E D=90^{\circ}, A B=C D=2$, $D E=B E=1, \quad A C=\sqrt{2}$. +求点 $D$ 到面 $A E B$ 的距离." ['$\\because V_{D-A E B}=V_{A-D E B}$, 且 $A C \\perp$ 面 $B C D E$ .\n\n设 $D$ 到面 $A E B$ 的距离为 $d$, 则 $\\mathrm{S}_{\\triangle A E B} \\cdot d=S_{\\triangle D E B} \\cdot \\sqrt{2}$.\n\n在 $\\triangle A E B$ 中, $A E=\\sqrt{E C^{2}+A C^{2}}=\\sqrt{7}, A B=2, B E=1$,\n\n$\\cos \\angle A B E=\\frac{2^{2}+1^{2}-\\sqrt{7}^{2}}{2 \\times 2 \\times 1}=-\\frac{1}{2}, \\therefore \\angle A B E=120^{\\circ}$,\n\n则 $\\mathrm{S}_{\\triangle A \\mathrm{EB}}=\\frac{1}{2} \\times 1 \\times 2 \\times \\frac{\\sqrt{3}}{2}=\\frac{\\sqrt{3}}{2}$.\n\n$\\mathrm{S}_{\\triangle D E B}=\\frac{1}{2} \\times 1 \\times 1=\\frac{1}{2}, \\quad \\therefore \\frac{\\sqrt{3}}{2} \\cdot d=\\frac{1}{2} \\times \\sqrt{2}, \\quad \\therefore d=\\frac{\\sqrt{6}}{3}$\n\n所以点 $D$ 到面 $A E B$ 的距离为 $\\frac{\\sqrt{6}}{3}$.'] ['$\\frac{\\sqrt{6}}{3}$'] 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'] Multimodal Competition False Numerical Open-ended Solid Geometry Math Chinese +18 "2018 年初, 某市政府考绩办准备对 20 家政府机构部门进行年度考核评估, 并根据考核结果得分（最低分 60 分, 最高分 100 分) 将这些机构分别评定为 A, B, C, D 四个类型, 考核评估标准如下表: + +| 评估得分 | $[60,70)$ | $[70,80)$ | $[80,90)$ | $[90,100]$ | +| :------: | :---------: | :---------: | :---------: | :----------: | +| 评分类型 | D | C | B | A | + +考核评估后, 对 20 家政府机构的评估分数进行统计分析, 得到频率分布直方图如图: + + +评分类型为 $\mathrm{D}$ 的政府机构部门有多少家?" ['评分类型为 $\\mathrm{D}$ 的政府机构部门的频率为 $0.015 \\times 10=0.15$,\n\n所以评分类型为 $\\mathrm{D}$ 的政府机构部门共有 $0.15 \\times 20=3$ 家.'] ['$3$'] 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'] Multimodal Competition False Numerical Open-ended Probability and Statistics Math Chinese +19 "2018 年初, 某市政府考绩办���备对 20 家政府机构部门进行年度考核评估, 并根据考核结果得分（最低分 60 分, 最高分 100 分) 将这些机构分别评定为 A, B, C, D 四个类型, 考核评估标准如下表: + +| 评估得分 | $[60,70)$ | $[70,80)$ | $[80,90)$ | $[90,100]$ | +| :------: | :---------: | :---------: | :---------: | :----------: | +| 评分类型 | D | C | B | A | + +考核评估后, 对 20 家政府机构的评估分数进行统计分析, 得到频率分布直方图如图: + + +现从评分类型为 $\mathrm{A}$ 和 $\mathrm{D}$ 的所有政府机构部门中随机抽取两家进行分析, 求这两家来自同一评分类型的概率." ['评分类型为 $\\mathrm{A}$ 的政府机构部门有 $0.020 \\times 10 \\times 20=4$ 家,\n\n设评分类型为 $A$ 的 4 家政府部门为 $\\mathrm{a}_{1}, \\mathrm{a}_{2}, \\mathrm{a}_{3}, \\mathrm{a}_{4}$,\n\n评分类型为 $D$ 的 3 家政府部门为 $\\mathrm{b}_{1}, \\mathrm{~b}_{2}, \\mathrm{~b}_{3}$,\n\n从评分类型为 $\\mathrm{A}, \\mathrm{D}$ 的政府部门中随机抽取两家的所有可能情况有 $\\left(\\mathrm{a}_{1}, \\mathrm{a}_{2}\\right),\\left(\\mathrm{a}_{1}, \\mathrm{a}_{3}\\right),\\left(\\mathrm{a}_{1}, \\mathrm{a}_{4}\\right),\\left(\\mathrm{a}_{1}, \\mathrm{~b}_{1}\\right),\\left(\\mathrm{a}_{1}, \\mathrm{~b}_{2}\\right),\\left(\\mathrm{a}_{1}, \\mathrm{~b}_{3}\\right),\\left(\\mathrm{a}_{2}, \\mathrm{a}_{3}\\right),\\left(\\mathrm{a}_{2}, \\mathrm{a}_{4}\\right),\\left(\\mathrm{a}_{2}, \\mathrm{~b}_{1}\\right),\\left(\\mathrm{a}_{2}, \\mathrm{~b}_{2}\\right),\\left(\\mathrm{a}_{2}, \\mathrm{~b}_{3}\\right),\\left(\\mathrm{a}_{3}, \\mathrm{a}_{4}\\right),\\left(\\mathrm{a}_{3}, \\mathrm{~b}_{1}\\right),\\left(\\mathrm{a}_{3}\\right.$,$\\left.\\mathrm{b}_{2}\\right),\\left(\\mathrm{a}_{3}, \\mathrm{~b}_{3}\\right),\\left(\\mathrm{a}_{4}, \\mathrm{~b}_{1}\\right),\\left(\\mathrm{a}_{4}, \\mathrm{~b}_{2}\\right),\\left(\\mathrm{a}_{4}, \\mathrm{~b}_{3}\\right)$,$\\left(\\mathrm{b}_{1}, \\mathrm{~b}_{2}\\right),\\left(\\mathrm{b}_{1}, \\mathrm{~b}_{3}\\right),\\left(\\mathrm{b}_{2}, \\mathrm{~b}_{3}\\right)$, 共 21 种,其中满足条件的共有 9 种, 所以这两家来自同一评分类型的概率为 $\\frac{9}{21}=\\frac{3}{7}$.'] ['$\\frac{3}{7}$'] 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Df0VB/Y1l/1wT+QrRwKz9F/wCQLZf9cU/kK0MUAGBRRigCgBMGgjinUh6UAcr4a/5CV3/vt/OupXqa5fwz/wAhG7/32/nXUL3oAdRRRQAUUUUAcZ4+H+iWv/XWul0jjSrb/rmK5vx7j7Ja/wDXX+ldLpP/ACCrX/rmKALtFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFADD0P1qN0WQMpHDKRUpHNIMHHtQBw83gqez1ZL7S2UMc+YGYDrU9n4bvdItilnIHlckyFmAzmuvbjlR81HOfu9OlAHD3PhG9+13M1tIG+0wiORHYDp1q1L4Ulj8MyaZbY8yUfMGOBXX49qNu7qooA47SdD1jSNAjsLdkeRCMhnwMDtS/2DO99Lqd/FGjJGwRYjnqK7ArleOKhuxm0l/hJQ5/KgDifDvhfRNStWuL7TLeeYn5pJF5PNdnaWlvaRCK2jWKJeiL0rG8JnfYtuHGf610RXIw3SgDFvfCmiancNcXumW80zcF2HOKv2OnWunW4t7SFI4x/CtWQCD02n1pxUDGBk+tAGLf+FtG1S48+802CWRehZa0LSztbC2S3tYVjhQYREU8CrWOM7RmkwyjPTPUUAcF8RPD+mT6NcX0lnA1zkbpCPmPNXdK8FeG5dKhJ0a1eRkGWZOvH1qT4hrnwxOdoOCv863dGydHttvy4QfyFAGb/AMIL4YHB0W0yB2T/AOvS/wDCCeGP+gLa/wDfH/166FVw+d2eKfQBzf8Awgnhj/oC2v8A3x/9ej/hBPDH/QFtf++P/r10lFAHNN4E8L7T/wASW16f3P8A69Z3h3TLPTNZuLawtI4UwPugjvXan7p+lcvp+W8R3BLHOB2oA6fAPHcd6cOnrTAMDDc08dKAFooooAK4/wCI3/IrXX+4a7CuP+I3/IrXX+4aAOg0X/kC2X/XBP5CtCs/Rf8AkC2P/XBP5CtAUAFFFFABSN92lpG+7QBy3hj/AJCF3/vt/OupXvXLeGP+Qhd/77fzrqV70ALRRRQAUUUUAcZ4+/49bX/rrXTaT/yCrX/rmK5nx9/x62v/AF1rptJ/5BVr/wBcxQBdooooAKKKKACiiigAooooAKKKKACiiigAooooAKMUUUAFGBRRQAUUUUAIahuv+PSf/cb+VTGobr/j0n/3G/lQBgeDv+Qc31/rXTVzPg7/AJBzfX+tdNQAUUUUAIaTuaU0nc0Acn8Q/wDkV5vqv863NF/5BFv/ALg/kKw/iH/yK831X+dbui/8gi3/ANwfyFAGgOlLQOlFABRRRQAjfdP0rltMH/FRz/QV1LfdP0rltM/5GKf6CgDqR1paQdaWgAooooAK4/4jf8itdf7hrsK4/wCI3/IrXX+4aAOg0X/kC2P/AFwT+QrQFZ+i/wDIFsf+uCfyFaAoAKKKKACkPSlpD0oA5Xwx/wAhC8/32/nXUr3+tct4Y/5CF5/vt/OupXv9aAHUUUUAFFFFAHGePv8Aj1tf+utdNpP/ACCrX/rmK5nx9/x62v8A11rptJ/5BVr/ANcxQBdooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAQ1Ddf8ek/wDuN/KpjUN1/wAek/8AuN/KgDA8Hf8AIOb6/wBa6auZ8Hf8g5vr/WumoAKKKKACm9zSmkHU80Acn8Q/+RXm+o/nW7ov/IIt/wDcH8hWD8Qj/wAUxN9V/nW7ov8AyCLf/cH8hQBpDpRQOlFABRRRQAjfdP0rldN/5GK4+grqm+6fpXK6b/yMU/0FAHU96dTe9OoAKKKKACuP+I3/ACK11/uGuwrj/iN/yK11/uGgDoNF/wCQLY/9cE/kK0BWfo3/ACBbL/rin8hV8UALRRRQAUh6UtI3SgDlfDH/ACELz/fb+ddSvf61y3hj/kIXn++3866le/1oAdRRRQAUUUUAcZ4+/wCPW1/66102k/8AIKtf+uYrmfH3/Hra/wDXWum0n/kFWv8A1zFAF2iiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigBj5z1qvcsPsk33uI2/lVhh82f0qveZNpNlyo2HGPpQBheDmDabuBxknj8a35LqOHInniQf7TYrzvS/EstjaPY6dbvc3YOArqVA/GtOHwxdawwutXmZ2brAeVT8RQB2UV1DOuYZQ4HdTmns7cEc56151LHe+C9UjMZZ9OcjcD/CO9d7Y3aXtqs0TAhxkUAQ6vqaaVYyXUrAKowB6nsK4i38Ya15kd9LaKuny/dLZ3YPqKv+O5jc/Y9Li+aRp45SB6Bua6W6063nsTBKuFK7cYoA5nxzeJdeEHngcOj7evbmup0TnS7ck/8ALNePwrybxHJfeG7K60u6RntZWHkSYJzg/pXd6P4t0uLSIFkkYSKoGCh9KAOu5Ddadurm/wDhM9J34aVhxn7ho/4TXRv+er/98GgDpN1JlvQVzn/Ca6N/z1f/AL4NA8a6N/z1b/vg0AdES208DpXLaXn/AISS5XuFB/Wp/wDhM9GKn963/fBrnrDxTpsOuXFyzvtdQB8tAHoYORS5b0Fc4PGWkgEeY3TP3DSjxno2P9Y3/fBoA6LLegoy3oK53/hNNG/56t/3waP+E00b/nq3/fBoA6LLegrkPiJn/hF7jvlDV3/hNNG/56t/3wa5nxz4o0678N3Udu7vIyEKAtAHcaL/AMgay56Qp/IVfBNcfpXi7SodLtEeRgwhVWGw9hV7/hNdH/56t/3waAOjy3oKFJPWud/4TTRv+erf98GlHjPRz0lb8UNAHR/jTTzxXPHxlpH/AD1b/vg00+NNJHSVv++DQBB4YOb+8IPR2/nXUg5GRXnegeJtNtLu5Z3YCRyR8vqa6EeNNJ7yN/3waAOky3oKQE5rnv8AhNNG/wCerf8AfBo/4TTR/wDnq3/fBoA6PNITxXOf8JppP/PVv++DSjxnpBIBlbH+4aAM/wAegm2tfQS10mkEtpdt7IK4Hxh4msL20gWJ2Zllzwvaug0zxjpUemQI0rAqgB+XvQB1340fjXOf8JlpP/PVv++DR/wmWk/89W/74NAHQ5P+TRlvSud/4TPRv+erf98Gl/4TPRv+erf98GgDoct6UZb0rnv+Ez0b/nq3/fBo/wCEz0b/AJ6t/wB8GgDoct6UZb0rnv8AhM9G/wCerf8AfBo/4TPRv+erf98GgDoAzliMDFGc8ZwRXNv4z0hQSZnVexCGsib4oaHaat9kuHZdwG1whOc0Ad2GPQ07tVGw1C3v4FngbKNyDV00AFFFFADqKKKACiiigAooooAidhu2hvmxwKxNV8TWOmEQzyDzmB/dYzgd625VDDHccivLQfN+IN+t1Ew2pKI9/IxtoA9B0hdOuYBd2aIVk78c1p7Rjb2rz74YeYqX8RZmhVhsJr0PFAFHU7CDULGS2nAMbDB9a4zRNQl0HW30e/bbDgtA59Ow9K72UNgEEADrkVj634esNet1S6VvlOQ0bbWHtmgDmrUf2z41Nwp3R26spxyAetd27hE8w4K9cnjFZej6BaaJHstFZFPLeY25ifrVfxdPLb+G7hoMhsY3CgCG/wBS0TU51spAtzJ6Y5FY9+/hGwuvs86qpTBc5PyiuJ02/wBS8L6Xb3z7bpLndvCr8w+mansXa98L67NdlNxh3fMPmwTQB6ZF4a0S5jjmFsCrAbTu+9U//CJaN/z6f+PVlfDaeefwpGbh9zLIyjr90dOtdj+NAGGfCWi4/wCPQfnUS+EtHJP+i8dvmroe1RMFPDE4HTFAHG6hD4V0u48m6QKe/U4qCP8A4RCezlurcK8aZ5zjkfzrb1+OzsbC7v5Y0dyACHANeY3dtNbT6JM42ia6bKgYVlIzyKAO70q28Nau5jghHmgYZS2Djsa2R4R0fH/HqP8AvquHjdl+I0UNuPl+zxl/L4H6V6qOgoAwv+ER0f8A59f/AB6mnwnpG7H2MY/3q6Co2IB54oA5u78PaDYWzXFzCFhUZZs9KwreXwje3qWiqqtI22PPIc10HjJ4R4eulnnSNNvzAivGdOu9Wi1zTm1KMLapcEwMqhc8dz196APS7hPClrfC38pWkH3lDdK2rbwxoVxEJI7YFWGc7s15t4cSObw3q11PIHuDNKFJPIUHjGa9G8DtK+hJ5iMuMYLHORQBb/4RLRv+fUf99UHwlo3/AD6jH+9W4RRxwKAMBvCejqv/AB6bv+BVi6lF4U02fZKi5XBcZPArtpAcDnFeU+JEutI1HVNQu4CbRogNzAYGDQB0U1p4Xhs1u0iVo26ENU2kWPhjWomks0EoU7W5xg15ZoDXytY/bGZLeW93oH7qea7Lw5JJD8RLiKFT9nMTFiv3c0Adt/wiWjHn7L/49QfCOi7f+PUD/gVbUb5XkUrDI5z+FAGC/hPRkUn7Jn/gVZU1r4SttRjspgomkwFTJ612TBsfeAHvXlvi/K+PtKCMoPmrzj2NAG7qtv4V0pxHdxrGxAxzkgHvTNOTwjqFpNLZ7JI4jmQ7sbcfWueuZGuPiBcwXQ37bJeDyFH48VxN69xbeILmCwcm3dG8wJ0GetAHqmmHwpql59mgVWfkryRuHtW6nhTR3J/0cfTNee3YSDQ/DxtpEiuRGo3gcuN3t/WvV7N2e3V3U+Zjn3oAz/8AhEtH/wCfX/x6j/hEtH/59f8Ax6tyigDBPhPRwSfs3b1oHhPSNv8Ax7Dk+tbo+Yc013Cxsx6LzQBhSeGNGixm1BJP97FMHhrQ2dgsIO3gjJ4rl9a1Aa34jitluGt7WL/WPkgE5yBUB1618NeLZB9oL2VycsWbIXtQB2DeGNHjBP2cZb1JrKk8A+HpdSN88Ue4gAfPnGKw/iBrcq2cE2n3Qk3ZLCPIzx+led6ZrFzKYLZ470tNKQ3708A++aAPoexsLeyQJbJiIDIOau5+U964bwbr13qp2xxOtrFmIluSSPeu5XGSF6UALRSYPrRg+tAElFFFABRRRQAUUUUANZQ3sfWue1DwpZ3uoNeMZFmYMCVbHWuhI96BQBm6Jotrolp5FsDjuT1NalNUYJp1ACEZpCgJzTqKAIyoAPU5NV76xiv7RreQZRutWzSMMigDkx4G09ViUtI0Uf3ULZH5VDdeAdOuo5o988ccq7XVHI4rsAg9aCrDoaBMq2FnDp9pHbRIFRFCjA646k1cpAAO3NOxQMbSDaRginUYBoEjL1nRYNYtvImZxGeu04zVbUfC9lqFtDDIrARfcKnBBrd6dKOBQM57TfCVlp+pNfoZGnKKnztngV0WDScGnUAGKZgE9KfSGgDP1fSrfV9Pks7hT5cgwSvWuej8BaastoWaZhbPvj3OTz7+1dhn2o2hutAHInwLpw81Y/NVJfvANgV0djaR2FpFbxZCIuBmre0CjaB0oABzQfvAUuKRj+dADDwdvJz3rndT8OPrF0Bfvvt15VF6H2I710gzt+YUiqAeuaAOe1Dwhpt5DboUZfI5TacYp2h+FbLRLiWeEyPJIfmMjbv510ARaXbQAgz6U7HFGKWgBCMiua1TwhZapqkGozNKJYmypVsCulIpCOOaAOZ1HwbY6lcfaHeRZgnllkbBK+h9qq6Z8PtK05bjHmSNPu3FzkjPpXYcdQKQ4PagDlrTwRYWzQnMrrCMIHbIArqETaoHpT8GjFACUUuKMUAJUU0YaB0J4YYqXqKZMpaPAoA868T2VrDBFa2akytIrOw+9wfWqLabbX/jfSomCyrCrCQdQD2zXZweHEW+uLqWQyNJkqCfu5HGPxqloPhAaRrVzqD3Dy+c2R5h6UAc38RNK+zafLNEh2Y+UL/B9a5C0s7e40G7vIDtuUj/AHSfxbuxr13X/Dp1hUjM0ipn5h2NZVj4BistZ+2q+UCKoiP3eO9AB8NGt4fD/lCRHlLl3A67j1ruMlgQo21ylj4Pis9Za+tppIo2OWiTpn1rq1bC8Z49aAE3D+9RuHrS7R6D8qNq+g/KgCWkHQUtIOgoAWkpe9JSYCN1pe1I3WlPSkgDqaBQOooWqABRQvSigA7Ud/wo7Ud/woAWkPSlpD92gBp+7SZ+Y0p+7SfxGgBRThTRThQSFNPSnU09KCgH3qUCkH3qUUAKQMilpD94UooAKafvU6mn71AC0UUUAFItLQtAC009c06mmgBqk+tLikWloARaf/FTFp/8VAC0UUUAFJS0lADMn1p1Np1ACU+mU+gAooooASkpaSgBmAZM4puAQ4I4p/8AHTf79AB1Cg8g9qXA9BSD+CnUAHbFO7U2ndqAIcn1pcn1pKKAP//Z'] Multimodal Competition False Numerical Open-ended Probability and Statistics Math Chinese +20 " + +已知直线过定点 $P(1,1)$, 且与找物线 $x^{2}=4 y$ 交 $A, B$ 两点, $l_{1}, l_{2}$ 分别过 $A, B$ 两点且与抛物线相切, 设 $l_{1}, l_{2}$交点为 $C$. +求交点 $C$ 的轨迹方程." ['设 $A\\left(x_{1}, y_{1}\\right), B\\left(x_{2}, y_{2}\\right)$, 易知直线 $\\mathrm{AB}$ 的斜率存在设为 $k, A B$ 方程为$y=k(x-1)+1$,\n\n联立 $\\left\\{\\begin{array}{c}y=k(x-1)+1 \\\\ x^{2}=4 y\\end{array}\\right.$, 消去 $y$, 得 $x^{2}-4 k x-4+4 k=0$,\n\n$\\therefore x_{1}+x_{2}=4 k, x_{1} x_{2}=4 k-4$.\n\n$\\because y^{\\prime}=\\frac{x}{2}, \\therefore k_{A C}=\\frac{x_{1}}{2}, A C$ 直线方程为 $y-y_{1}=\\frac{x_{1}}{2}\\left(x-x_{1}\\right)$, 将 $y_{1}=\\frac{x_{1}^{2}}{4}$ 代入, 化简得 $y=\\frac{x_{1}}{2} x-\\frac{x_{1}^{2}}{4}$\n\n所以直线 $A C$ 方程为: $y=\\frac{x_{1}}{2} x-\\frac{x_{1}^{2}}{4}$\n\n同理, 直线 $B C$ 方程为: $y=\\frac{x_{2}}{2} x-\\frac{x_{2}^{2}}{4}$,\n\n联系 $A C, B C$ 方程可得交点坐标 $x_{C}=\\frac{x_{1}+x_{2}}{2}=2 k, y_{C}=\\frac{x_{1} x_{2}}{4}=k-1$,\n\n所以点 $C$ 的轨迹方程为: $x-2 y-2=0 $.'] ['$x-2 y-2=0$'] 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Multimodal Competition False Equation Open-ended Plane Geometry Math Chinese +21 "如图, 多面体 $A-P C B E$ 中, 四边形 $P C B E$ 是直角梯形, 且 $P C \perp B C, P E \| B C$, 平面 $P C B E \perp$平面 $A B C, A C \perp B E, M$ 是 $A E$ 的中点, $N$ 是 $P A$ 上的点. + + +若 $P E=\frac{1}{3} B C$ 且 $A C=B C=P C$, 求二面角 $E-A B-C$ 的余弦值." ['设 $P E=1, A C=B C=P C=3$, 在平面 $P C B E$ 中作 $E H \\perp B C$ 于 $H$, 连接 $E H, A H$, 如图.\n\n\n\n根据题意, $P C \\perp A B C$, 于是 $P C \\perp C A$, 由 $A C \\perp B E$, 于是 $A C \\perp P C B E$, 因此 $\\triangle A C B$ 是直角三角形, 且 $C$ 为直角顶点. 因此\n\n$$\nS_{\\triangle A B H}=\\frac{2}{3} S_{\\triangle A B C}=\\frac{2}{3} \\cdot \\frac{9}{2}=3,\n$$\n\n又\n\n$$\n\\left.\\begin{array}{l}\nB E=\\sqrt{B H^{2}+E H^{2}}=\\sqrt{13}, \\\\\nA E=\\sqrt{A H^{2}+E H^{2}}=\\sqrt{19}, \\\\\nA B=\\sqrt{A C^{2}+B C^{2}}=\\sqrt{18},\n\\end{array}\\right\\} \\Rightarrow S_{\\triangle A B E}=3 \\cdot \\sqrt{\\frac{11}{2}},\n$$\n\n于是所求二面角的余弦值为\n\n$$\n\\frac{3}{3 \\cdot \\sqrt{\\frac{11}{2}}}=\\frac{\\sqrt{22}}{11}.\n$$'] ['$\\frac{\\sqrt{22}}{11}$'] 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Multimodal Competition False Numerical Open-ended Solid Geometry Math Chinese +22 " + +如图, 在三棱柱 $A B C-A_{1} B_{1} C_{1}$ 中, 侧棱 $A A_{1} \perp$ 底面 $A B C$, $A B=A C=2 A A_{1}, \angle B A C=120^{\circ}, D 、 D_{1}$ 分别是线段 $B C 、 B_{1} C_{1}$ 的中点, 过线段 $A D$ 的中点 $F$ 作 $B C$ 的平行线, 分别交 $A B 、 A C$ 于点 $M 、 N$. +求锐二面角 $A-A_{1} M-N$ 的余弦值." "['设 $A A_{1}=1$ ，\n\n如图: 过 $A_{1}$ 作 $A E_{1} / / B C$, 建立以 $A_{1}$ 为坐标原点, $A_{1} E, A_{1} D_{1}, A_{1} A$ 分别为 $x, y, z$ 轴的空间直角坐标系.\n\n\n\n则$A_{1}(0,0,0),A(0,0,1),B(\\sqrt{3},1,1),C(-\\sqrt{3},1,1)$.\n\n因为 $F$ 是 $A D$ 的中点, $M N / / B C$, 所以 $M, N$ 分别为 $A B, A C$ 的\n\n中点, 则 $M\\left(\\frac{\\sqrt{3}}{2}, \\frac{1}{2}, 1\\right), N\\left(-\\frac{\\sqrt{3}}{2}, \\frac{1}{2}, 1\\right)$,\n\n则 $\\overrightarrow{A_{1}M}=\\left(\\frac{\\sqrt{3}}{2}, \\frac{1}{2}, 1\\right), \\overrightarrow{A_{1} A}=(0,0,1), \\overrightarrow{N M}=(\\sqrt{3}, 0,0),$\n\n设平面 $A A_{1} M$ 的法向量为 $\\vec{m}=(x, y, z)$,\n\n则 $\\left\\{\\begin{array}{l}\\vec{m} \\cdot \\overrightarrow{A_{1} M}=0, \\\\ \\vec{m} \\cdot \\overrightarrow{A_{1} A}=0,\\end{array}\\right.$ 得 $\\left\\{\\begin{array}{l}\\frac{\\sqrt{3}}{2} x+\\frac{1}{2} y+z=0, \\\\ z=0,\\end{array}\\right.$\n\n令 $x=1$, 则 $y=-\\sqrt{3}$, 则 $\\vec{m}=(1,-\\sqrt{3}, 0)$\n\n同理设平面 $A_{1} M N$ 的法向量为 $\\vec n=(a, b, c)$,\n\n则 $\\left\\{\\begin{array}{l}\\vec{n} \\cdot \\overrightarrow{A_{1} M}=0, \\\\ \\vec{n} \\cdot \\overrightarrow{N M}=0,\\end{array}\\right.$ 得 $\\left\\{\\begin{array}{l}\\frac{\\sqrt{3}}{2} a+\\frac{1}{2} b+c=0, \\\\ \\sqrt{3} a=0,\\end{array}\\right.$\n\n令 $b=2$, 则 $\\mathrm{c}=-1$, 则 $\\vec n=(0,2,-1)$,\n\n则\n\n$$\ncos(\\vec m, \\vec n)=\\frac{\\vec m \\cdot \\vec n}{|\\vec m| \\cdot |\\vec n|}=-\\frac{\\sqrt{15}}{5},\n$$\n\n因为二面角 $A-A_{1} M-N$ 是锐二面角,\n\n所以二面角 $A-A_{1} M-N$ 的余弦值是 $\\frac{\\sqrt{15}}{5}$.' + '连接 $A_{1} F$, 过点 $A$ 作 $A E \\perp A_{1} F$ 于点 $E$, 过 $E$ 作 $E H \\perp A_{1} M$\n\n\n\n因为 $A B=A C, D$ 是 $B C$ 的中点, 所以 $B C \\perp A D$,\n\n因为 $M N / / B C$, 所以 $M N \\perp A D$,\n\n因为 $A A_{1} \\perp$ 平面 $A B C, M N \\subset$ 平面 $A B C$,\n\n所以 $A A_{1} \\perp M N$,\n\n因为 $A D, A A_{1} \\subset$ 平面 $A D D_{1} A_{1}$, 且 $A D$ I $A A_{1}=A$,\n\n所以 $M N \\perp$ 平面 $A D D_{1} A_{1}$.\n\n$N M \\perp A E$ ，又 $A E \\perp A_{1} F ， A_{1} F \\cap N M=F$ ，\n\n$\\therefore A E \\perp$ 面 $A_{1} N M, \\therefore A E \\perp A_{1} M$\n\n$A_{1} M \\perp E H, A_{1} M \\perp A E, E H \\cap A E=E$,\n\n$\\therefore A_{1} M \\perp$ 面 $A H E, \\therefore A H \\perp A_{1} M$\n\n故 $\\angle A H E$ 是二面角 $A-A_{1} M-N$ 的平面角\n\n设 $A A_{1}=1$, 则 $A M=A N=1 ， A_{1} M=A_{1} N=\\sqrt{2}$ ，\n\n故 $A H=\\frac{\\sqrt{2}}{2}, A_{1} F=\\frac{\\sqrt{5}}{2}, A E=\\frac{\\sqrt{5}}{5}$, 故 $\\sin \\angle A H E=\\frac{A E}{A H}=\\frac{\\frac{\\sqrt{5}}{5}}{\\frac{\\sqrt{2}}{2}}=\\frac{\\sqrt{10}}{5}$,\n\n则 $\\cos \\angle A H E=\\sqrt{1-\\sin ^{2} \\angle A H E}=\\frac{\\sqrt{15}}{5}$\n\n因为二面角 $A-A_{1} M-N$ 是锐二面角,\n\n所以二面角 $A-A_{1} M-N$ 的余弦值是 $\\frac{\\sqrt{15}}{5}$.']" ['$\\frac{\\sqrt{15}}{5}$'] 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Multimodal Competition False Numerical Open-ended Solid Geometry Math Chinese +23 " + +如图, 正方形 $A D E F$ 与 $\square A B C D$ 所在的平面互相垂直, 且 $A B=2 A D=2 a$, $\angle B A D=60^{\circ}, G$ 为 $B D$ 的中点. +求平面 $C G E$ 与平面 $A D E F$ 所成锐二面角的余弦值." ['平面 $A D E F \\perp$ 平面 $A B C D, E D \\perp A D, \\therefore E D \\perp$ 平面 $A B C D, E D \\perp D B$,\n\n即 $D A 、 D B 、 D E$ 两两垂直，\n\n以 $D A 、 D B 、 D E$ 分别为 $x, y, z$ 轴, 建立如图所示空间直角坐标系,\n\n\n\n则 $A(a, 0,0), G\\left(0, \\frac{\\sqrt{3}}{2} a, 0\\right), C(-a, \\sqrt{3} a, 0), E(0,0, a)$,\n\n$\\overrightarrow{E G}=\\left(0, \\frac{\\sqrt{3}}{2} a,-a\\right), \\quad \\overrightarrow{G C}=\\left(-a, \\frac{\\sqrt{3}}{2} a, 0\\right)$,\n\n取平面 $A D E F$ 的法向量为 $\\boldsymbol{m}=(0,1,0)$, 设平面 $C G E$ 的法向量为 $\\boldsymbol{n}=(x, y, z)$,\n\n则 $\\left\\{\\begin{array}{l}\\boldsymbol{n} \\cdot \\overrightarrow{E G}=0 \\\\ n \\cdot \\overrightarrow{G C}=0\\end{array}\\right.$, 即 $\\left\\{\\begin{array}{l}\\frac{\\sqrt{3}}{2} a y-a z=0 \\\\ -a x+\\frac{\\sqrt{3}}{2} a y=0\\end{array}\\right.$, 令 $y=1$, 则 $x=\\frac{\\sqrt{3}}{2}, z=\\frac{\\sqrt{3}}{2}$,\n\n故 $\\boldsymbol{n}=\\left(\\frac{\\sqrt{3}}{2}, 1, \\frac{\\sqrt{3}}{2}\\right)$,\n\n设平面 $C G E$ 与平面 $A D E F$ 所成锐二面角为 $\\theta$, 则 $\\cos \\theta=\\frac{|\\boldsymbol{m} \\cdot \\boldsymbol{n}|}{|\\boldsymbol{m}||\\boldsymbol{n}|}=\\frac{\\sqrt{10}}{5}$.'] ['$\\frac{\\sqrt{10}}{5}$'] 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'] Multimodal Competition False Numerical Open-ended Solid Geometry Math Chinese +24 "为加强对企业产品质量的管理, 市质监局到区机械厂抽查机器零件的质量, 共抽取了 600 件螺帽, 将它们的直径和螺纹距之比 $Z$ 作为一项质量指标，由测量结果得如下频率分布直方图: + + + +附: $\sqrt{224} \approx 14.97$. + +若 $Z \sim N\left(\mu, \sigma^{2}\right)$, 则 $P(\mu-\sigma + +附: $\sqrt{224} \approx 14.97$. + +若 $Z \sim N\left(\mu, \sigma^{2}\right)$, 则 $P(\mu-\sigma + +附: $\sqrt{224} \approx 14.97$. + +若 $Z \sim N\left(\mu, \sigma^{2}\right)$, 则 $P(\mu-\sigma + +如图, 正三棱柱 $\mathrm{ABC}-\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime}$ 的所有棱长都为2 , 点 $M, N$ 分别是 $B A^{\prime}$ 与 $B^{\prime} C^{\prime}$ 的中点. +求三棱雉 $\mathrm{A}^{\prime}-\mathrm{MNC}$ 的体积." ['$\\because A^{\\prime} B^{\\prime}=A^{\\prime} C^{\\prime}, N$ 为 $B^{\\prime} C^{\\prime}$ 的中点, $\\therefore A^{\\prime} N \\perp B^{\\prime} C^{\\prime}$,\n\n$\\because B B^{\\prime} \\perp$ 平面 $A^{\\prime} B^{\\prime} C^{\\prime}, \\therefore B B^{\\prime} \\perp A^{\\prime} N \\therefore A^{\\prime} N \\perp$ 平面 $B C C^{\\prime} B^{\\prime}$,\n\n$\\therefore V_{A^{\\prime}-M N C}=V_{B-M N C}=V_{M-B C N}=\\frac{1}{3} \\cdot S_{\\square B C N} \\cdot \\frac{1}{2} A^{\\prime} N=\\frac{1}{3} \\cdot 2 \\cdot \\frac{\\sqrt{3}}{2}=\\frac{\\sqrt{3}}{3}$.'] ['$\\frac{\\sqrt{3}}{3}$'] 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Multimodal Competition False Numerical Open-ended Solid Geometry Math Chinese +28 " + +如图,过抛物线 $y^{2}=2 p x(p>0)$ 上一点 $P(1,1)$, 作两条直线分别交抛物线于 $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$, 若 $P A$ 与 $P B$ 的斜率满足 $k_{P A}+k_{P B}=0$. +求直线 $A B$ 的斜率." ['由抛物线 $y^{2}=2 p x(p>0)$ 过点 $P(1,1)$, 得 $2 p=1$, 即 $y^{2}=x$,\n\n将条件 $k_{P A}+k_{P B}=0$ 写为 $\\frac{y_{1}-1}{x_{1}-1}+\\frac{y_{2}-1}{x_{2}-1}=0$,\n\n注意到 $y_{1}^{2}=x_{1}, y_{2}^{2}=x_{2}$, 代入上式得到 $\\frac{1}{y_{1}+1}+\\frac{1}{y_{2}+1}=0$,\n\n通分整理得 $y_{1}+y_{2}=-2$.\n\n设直线 $A B$ 的斜率为 $k_{A B}$, 由 $y_{1}^{2}=x_{1}, y_{2}^{2}=x_{2}$,\n\n得 $k_{A B}=\\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\\frac{1}{y_{1}+y_{2}}\\left(x_{1} \\neq x_{2}\\right),$\n\n由于 $y_{1}+y_{2}=-2$,\n\n将其代入上式得 $k_{A B}=\\frac{1}{y_{1}+y_{2}}=-\\frac{1}{2}$.'] ['$-\\frac{1}{2}$'] 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Multimodal Competition False Numerical Open-ended Plane Geometry Math Chinese +29 " + +如图,过抛物线 $y^{2}=2 p x(p>0)$ 上一点 $P(1,1)$, 作两条直线分别交抛物线于 $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$, 若 $P A$ 与 $P B$ 的斜率满足 $k_{P A}+k_{P B}=0$. +直线 $A B$的斜率$-\frac{1}{2}$.若直线 $A B$ 在 $y$ 轴上的截距 $b \in[0,1]$, 求 $\triangle A B P$ 面积的最大值." ['设直线 $A B$ 的方程为 $y=-\\frac{1}{2} x+b$,\n\n由 $\\left\\{\\begin{array}{l}y^{2}=x \\\\ y=-\\frac{1}{2} x+b\\end{array}\\right.$, 消去 $y$ 整理, 得 $\\frac{1}{4} x^{2}-(b+1) x+b^{2}=0,$\n\n$\\because \\mathrm{b} \\in[0,1]$\n\n$\\therefore \\Delta=(b+1)^{2}-b^{2}>0$,\n\n且 $x_{1}+x_{2}=4(b+1), x_{1} x_{2}=4 b^{2} ，$\n\n$|A B|=\\sqrt{\\frac{5}{4}} \\cdot \\sqrt{\\left(x_{1}+x_{2}\\right)^{2}-4 x_{1} x_{2}}=2 \\sqrt{5} \\sqrt{2 b+1}$\n\n又点 $P$ 到直线 $A B$ 的距离为 $d=\\frac{|3-2 b|}{\\sqrt{5}}$,\n\n所以 $S_{\\triangle A B P}=\\frac{1}{2} \\cdot|A B| \\cdot d=\\frac{1}{2} \\cdot 2 \\sqrt{5} \\sqrt{2 b+1} \\cdot \\frac{|3-2 b|}{\\sqrt{5}}=\\sqrt{1+2 b}|3-2 b|$\n\n令 $f(x)=(1+2 x)(2 x-3)^{2}$, 其中 $x \\in[0,1] ，$\n\n则由 $f^{\\prime}(x)=2(2 x-3)(6 x-1)=0$, 得 $x=\\frac{1}{6}$ 或 $x=\\frac{3}{2}$,\n\n当 $x \\in\\left(0, \\frac{1}{6}\\right)$ 时, $f^{\\prime}(x)>0$, 所以 $f(x)$ 单调递增,\n\n当 $x \\in\\left(\\frac{1}{6}, 1\\right)$ 时, $f^{\\prime}(x)<0$,所以 $f(x)$ 单调递减,\n\n故 $f(x)$ 的最大值为 $f\\left(\\frac{1}{6}\\right)=\\frac{256}{27}$.\n\n故 $\\triangle A B P$ 面积 $S_{\\triangle A B P}$ 的最大值为 $\\sqrt{f\\left(\\frac{1}{6}\\right)}=\\frac{16 \\sqrt{3}}{9}$.'] ['$\\frac{16 \\sqrt{3}}{9}$'] 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Multimodal Competition False Numerical Open-ended Plane Geometry Math Chinese +30 "在三棱锥 $A-P C B$ 中, 其中 $\angle B P C=90^{\circ}, \angle A P C=\angle A P B=60^{\circ}$, 且 $P A=P B=P C=2$. + + + +$D$ 为线段 $A C$ 上一点, 且 $A D=\frac{2}{3} A C$, 求三棱锥 $D-A P B$ 的体积." ['$\\because P A=P B=P C=2$, 又 $\\angle A P C=\\angle A P B=60^{\\circ}$,\n\n$\\therefore \\triangle A P B$ 和 $\\triangle A P C$ 都是等边三角形, $A B=A C=2$.\n\n取 $B C$ 中点 $H$, 连接 $A H, \\therefore A H \\perp B C$.\n\n\n\n$\\therefore A H^{2}=A C^{2}-C H^{2}=2$\n\n$\\because \\angle B P C=90^{\\circ}, \\quad \\therefore B C=2 \\sqrt{2}, \\quad P H=\\sqrt{2}$\n\n在 $\\triangle P H A$ 中, $A H^{2}=2, P H^{2}=2, P A^{2}=4$,\n\n$\\therefore P A^{2}=P H^{2}+A H^{2}, \\quad \\therefore A H \\perp P H, \\quad P H \\cap B C=H$\n\n$\\therefore A H \\perp$ 平面 $P B C$.\n\n$\\because A H \\subset$ 平面 $A B C, \\therefore$ 平面 $A B C \\perp$ 平面 $B P C$ .\n\n$V_{D-A P B}=\\frac{2}{3} V_{C-A P B}=\\frac{2}{3} V_{A-P B C}=\\frac{2}{3} \\times \\frac{1}{3} \\times S_{\\triangle P B C} \\times A H=\\frac{4 \\sqrt{2}}{9}$.'] ['$\\frac{4 \\sqrt{2}}{9}$'] 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Multimodal Competition False Numerical Open-ended Solid Geometry Math Chinese +31 "如图，已知长方形 $A B C D$ 中, $A B=4, A D=2, M$ 为 $D C$的中点, 将 $\triangle A D M$ 沿 $A M$ 折起, 使得平面 $A D M \perp$ 平面 $A B C M$. + + +若 $\overrightarrow{D E}=\lambda \overrightarrow{E B}(\lambda>0)$, 当二面角 $E-A M-D$ 大小为 $\frac{\pi}{3}$ 时, 求 $\lambda$ 的值." ['以 $M$ 为原点, $M A$, $M B$ 所在直线为 $x$ 轴, $y$ 轴, 建立如图所示的空间直角坐标系则 $M(0,0,0), A(2 \\sqrt{2}, 0,0), B(0,2 \\sqrt{2}, 0), D(\\sqrt{2}, 0, \\sqrt{2})$\n\n$$\n\\begin{aligned}\n& \\text { 设 } E(x, y, z) \\because \\overrightarrow{D E}=\\lambda \\overrightarrow{E B} \\\\\n& \\therefore(x-\\sqrt{2}, y, z-\\sqrt{2})=\\lambda(-x, 2 \\sqrt{2}-y,-z) \\\\\n\n\\end{aligned}\n$$\n\n$$\n\\therefore \\left\\{\\begin{array}{l}\nx=\\frac{\\sqrt{2}}{1+\\lambda} \\\\\ny=\\frac{2 \\sqrt{2} \\lambda}{1+\\lambda} \\\\\nz=\\frac{\\sqrt{2}}{1+\\lambda}\n\\end{array}\\right.\n$$\n\n$\\therefore E\\left(\\frac{\\sqrt{2}}{1+\\lambda}, \\frac{2 \\sqrt{2} \\lambda}{1+\\lambda}, \\frac{\\sqrt{2}}{1+\\lambda}\\right)$\n\n设平面 $E A M$ 的法向量为 $\\vec{m}=(x, y, z)$, 则 $\\left\\{\\begin{array}{l}\\vec{m} \\bullet \\overrightarrow{M A}=0 \\\\ \\vec{m} \\bullet \\overrightarrow{M E}=0\\end{array}\\right.$ 即 $\\left\\{\\begin{array}{l}2 \\sqrt{2} x=0 \\\\ \\frac{\\sqrt{2}}{1+\\lambda} x+\\frac{2 \\sqrt{2} \\lambda}{1+\\lambda} y+\\frac{\\sqrt{2}}{1+\\lambda} z=0\\end{array}\\right.$\n\n$\\therefore\\left\\{\\begin{array}{l}x=0 \\\\ y=-1 \\therefore \\vec{m}=(0,-1,2 \\lambda) \\\\ z=2 \\lambda\\end{array}\\right.$\n\n又 $\\because$ 平面 $D A M$ 的一个法向量为 $\\vec{n}=(0,1,0)$\n\n$\\therefore|\\cos \\langle\\vec{m}, \\vec{n}\\rangle|=\\frac{|-1|}{\\sqrt{1+4 \\lambda^{2}}}=\\frac{1}{2}$\n\n$\\because \\lambda>0 \\therefore \\lambda=\\frac{\\sqrt{3}}{2}$.'] ['$\\frac{\\sqrt{3}}{2}$'] 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Multimodal Competition False Numerical Open-ended Solid Geometry Math Chinese +32 " + +如图, 平面四边形 $A B C D$ 中, $\triangle A B D$ 为等边三角形, $B D=2$, $B C=C D=\sqrt{2}$, 沿直线 $B D$ 将 $\triangle A B D$ 折成 $A^{\prime} B D$. +当 $A^{\prime} C=\sqrt{3}$ 时, 求三棱锥 $A^{\prime}-B C D$ 的体积." ['取 $B D$ 中点 $M$, 连接 $A^{\\prime} M, M C$.\n\n\n\n$\\because \\triangle A B D$ 为等边三角形, $B D=2, \\therefore A^{\\prime} M \\perp D B$,\n\n$A^{\\prime} M=\\sqrt{3}$,\n\n$\\because B C=C D=\\sqrt{2}, \\therefore M C=1$,\n\n当 $A^{\\prime} C=2$ 时, $A^{\\prime} M^{2}+M C^{2}=A^{\\prime} C^{2}$,\n\n$\\therefore A^{\\prime} M \\perp M C$,\n\n$\\therefore A^{\\prime} M \\cap C M=M$, $A^{\\prime} M, C M \\subset$ 平面 $A^{\\prime} M C, C M \\perp B D$,\n\n$\\therefore B D \\perp$ 平面 $A^{\\prime} M C$, 取 $M C$ 的中点 $N$, 连接 $A^{\\prime} N$,\n\n$\\because A^{\\prime} C=\\sqrt{3}, \\therefore A^{\\prime} N=\\frac{\\sqrt{11}}{2}, \\therefore S_{\\triangle A^{\\prime} M C}=\\frac{1}{2} \\cdot M C \\cdot A^{\\prime} N=\\frac{1}{2} \\times 1 \\times \\frac{\\sqrt{11}}{2}=\\frac{\\sqrt{11}}{4}$\n\n$\\therefore V_{A^{\\prime}-B C D}=\\frac{1}{3} \\cdot S_{A^{\\prime} M C} \\cdot B D=\\frac{1}{3} \\times \\frac{\\sqrt{11}}{4} \\times 2=\\frac{\\sqrt{11}}{6}$.'] ['$\\frac{\\sqrt{11}}{6}$'] 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Multimodal Competition False Numerical Open-ended Solid Geometry Math Chinese +33 " + +如图, 在四棱锥 $\mathrm{P}-\mathrm{ABCD}$ 中, $\mathrm{PD} \perp$ 平面 $\mathrm{ABCD}$, 底面 $\mathrm{ABCD}$ 是菱形, $\angle \mathrm{BAD}=60^{\circ}$ $A B=2 a, P D=2\left(1-a^{2}\right)$, 其中 $0\n\n$\\because P D / /$ 平面 $E A C$, 平面 $E A C \\cap$ 平面 $P B D=O E, \\mathrm{PD} \\subset$ 平面 $\\mathrm{PBD}$\n\n$\\therefore P D / / O E$.\n\n$\\because O$ 是 $B D$ 的中点, $\\therefore E$ 是 $P B$ 的中点\n\n$\\because$ 四边形 $A B C D$ 是菱形, $\\angle B A D=60^{\\circ}, \\therefore B H \\perp A D$\n\n又 $B H \\perp P D, A D \\cap P D=D, \\therefore B H \\perp$ 平面 $P A D$, 且 $B H=\\frac{\\sqrt{3}}{2} A B=\\sqrt{3} a$,\n\n故 $V_{P-E A D}=V_{E-P A D}=\\frac{1}{2} V_{B-P A D}=\\frac{1}{2} \\times \\frac{1}{3} \\times S_{\\triangle P A D} \\times B H=\\frac{\\sqrt{3}}{3} \\cdot a^{2}\\left(1-a^{2}\\right)$.\n\n由基本不等式得 $V_{P-E A D}=\\frac{\\sqrt{3}}{3} \\cdot a^{2}\\left(1-a^{2}\\right) \\leq \\frac{\\sqrt{3}}{3} \\cdot\\left(\\frac{a^{2}+1-a^{2}}{2}\\right)^{2}=\\frac{\\sqrt{3}}{12}$,\n\n当且仅当 $a=\\frac{\\sqrt{2}}{2}$ 时取等号, 即最大体积为 $\\frac{\\sqrt{3}}{12}$.'] ['$\\frac{\\sqrt{3}}{12}$'] 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4QS1g+E2kWN9dSaYbyDYZQMEeoZiMDI9aALXwn1S41rww2o6jBFHfyzNuZYVXcAxA+v1r0PqeeDXkOl+JtNl8Zab4Y8NCGaOyQiS6fPIBAwMYB5PX2r1wZxtY/NjnbQAMATgjP1rx74u+BBfeV4i0y4MOrIwCpnG/ByNvvkmvZgox05qOWNCnKKQDkZGaAPJfhP8RNV8SXkmhatbqs9rGd0hY72II6ivXxwBXh/wATPBt14fvz4y8Om5F4sweVEI2BQOuAPpXeeBfH+neLtOgSO5RtSWJTPFgghuhxn6UAdrRUaMS3XIqSgAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiijtQAh6Vk69qUWkaJe3zzpD5MLOrv0zjj9a1WztOOteRfHnXBYeEE01BE/258MCwyoU5Bx6UAc78C9Rj1DxHrd5dWpe8uZDIs6x/KoJyRn3OK99VAvTqa4f4WeHrXQvBFj9n+czxid3ZcMS3OPwzXboSDk96AH81yvxHGPAmqHOP3R7ZrqzzWH4s0mbXfDl7p0Dqks6bVLdAaAPN/2eM/8IhfZHHnjH617IOlef/CzwXe+CdEuLK9milaWQOCi9Ov+NegZ7d6AI3wW5rxj4/Xminw/b21zMW1dWV4EVuBnqSPoDXqOseJdL0JHbULnyVSMy5I7D/PSvANKtV+LfxC1e8uJk+zQ27rBFMwzjDBMY64bBoA4iNLqf+xb7xJcXE+jOxRSshZkRTggDtz717J4T0Twh4z0qS0i13VLpovlkWa5Zc+4XdjFQ/ChtNvrDUvBOswQTSWUroihSd6FjuP0zis7xf8AB+80m8n1jwxIwtABI9qGO44OSBjHFAHo/hz4faV4O1ZZNH08PHMuZLmWUs6H2yK7fkFsN26V4x4D+Kst1qEWlavHbabDGBGqSRsHc+uSa9ft7u3uo91vPFMBx+7YMfxxQBdX7ooIyKFPyiloAgeASQtE4BVhhgeQa+cviH4Zu/APjWDxLo9u6aa0ivIInIw2fmBx2I4r6UrJ1rS4Nb0y6sLnBjmjZMsOhIIyPpQBT8IeLNP8XaPHqFjITwBJGesbY6V0AdSSM9K+adLvdZ+CXit7O+i8/SLtss+3AI7EH1r3vQtUXWbM6hDLDLaSMfs5jHIXg8+9AG7RSA8UtABRRRQAUUUh6GgBvmDOOaUMD0quZENz5ZkXdtzsB+YjpUyDH0oAkopO9LQAUUUUAFFFFABRRRQAUhpaax7dqAEzwa+ffiXdjxl8TdM8PafCkxtZV88sPQ5YfTFe+3EwhheUsFCqTljgcV4J8LbmfxL8V9a1sgwpy5UrkHquAf1oA93sbeO1tobeKNUjRAAB2wMVbwKYi85GakoASsHxfqtxonhi91G2C+fDHuXcOMit6uU+Ipz4F1QDkmE0AY/wo8Y6j4z0G4vNT8vzIpdo2ADjn29q7tnWNSzchQSee1eSfs+xSReEL4SRuuZgRuGM9fWk+MPj+fRrSPRdHmT7bdfLIUOWUHt+NAHJeOtZf4k+N4NF03zVsLXcs0vQZBGWP05H41f+CXhVJvEeoeIYUlisrZmhtgXOWJGGz68EGun8DfDOWx8DXkV1deXqGqoHa4QfNGG5xXX+BvB8fgnQ306K4e4DSmUuT1zgf0oA8P0a9l8BfG28Oo2jObmR0CxcnDsGBFfS0ZEsasBgHsew9K8O+OMVzpfiHQ/EFlbqpgyJJyM8gjGfwzXsmhajDq+i2l9DKsqSxhgwGASRQBz/AIx+Hej+K7KUSQJBeH7lzGuGXivHLm18ZfB28e6jn+06dLJtG8lsj6dq+lCMjBqpeaXaahAIby2injB3BZBkZoA4XwR8WdJ8UW5junWxvUwCjnhj7V6CpIPHQjOa8n8Y/CbR7y9ifSI5rO9k+ZXjztB965TTfih4m8FamdH8S2rzxRsqeftOVGeue9AH0MGJ70/aMYIyPesPRPE2k+IIFn0++hlBA+UOM5PbFbIfjkd8UAYHjLwvB4r8OXOmyhFmdcxSsufLb1rw/wAJ6te/C7x83h3UtTD6XIxB/iCE9Dz098dq+kGOQMGvOviN4AtPFGlk29nGtyA8vnD7+4DIH4mgDv4J454Y5onDxuNwYHIIIqZSSxBNeCfCXx9caffR+EdWUxbCyrLMTkndwOfY17yrADIIIOMfSgCWim7xn6U6gAoPSiigCobSD7eLvZ++2lM+1Wl+mKTbzmlGaAFooooAKKKKACiiigAooooAKjfpjrnrUlRsM8UAcf8AE25S3+HmtGSfymMBCkHBY+g96wfgjoT6X4FivJGR2vm88HHzAEYwT+FYPxtk1LVdX0DwvaYCXzGTaepZTgD/AMeNeo+ErK307wtp9naQyRQxRBVSQfMAD3/GgDcjzin01cdKdQAlZHiFtOj0e4bVAPsSjMufQVr1yfxG+bwLqgyARESKAMifx/4e07wVc6xo3kS29uPLCIAAzAcfyrzb4e+G7vxDr93411+0H9nhmlCSJkkgcYB7CuK+HPhjUfF+rx6cs5TT4m82VW+727d+tfUWrRTab4UuI9LgHmxQkQoi5BOPSgDylfF+reJPibDa+Eb13sIlBuEkGEAAAIA9q9tVmCKJCC2Pnx0PHavGvgtoupafNqup6tFHZvPIf3cqFGJJJJXJ6ZzXsP2u3yP38OMdd4xQBwvxn0aXV/h7dlJNn2Nhcn/axkY/WnfBrUrW8+Hun28D7pLZfKkHow5/qK7S/jt7/T57eTZNBIjB0PIYV4n8CtSuo9f1vRQqrbRt5u3H3WJxn9KAPeUILZzUlRRnIBIGR6VLQBCx/eDrjvWJ4l8LaX4m0yW01C3RhIMeaB8646GuhxSEDHI4oA+b/EXwx17wDL/bnhu8mnt4JFYRKx3YxzkdxXb+E/jNpWoNHY62DY34CRsJBgM5OK9XZVZWBUEH171534u+FGheIJbvUYrYx6i0TCPY21S+OCR9cUAegRujoJEIKkZBHIxRjtk4HcV82WviHx58ML2KDVVluNPV0DFvmG3B4B6V7b4V8daN4xt3l024AkjbDRSHBz9DQBwvxi8Bpf2X/CQaVat/adu37wwjG9cdfwrR+EvxAtdc0eLSL64b+1LcbWEh++B6fSvTyMqVIBHQgfrmvnr4geFtS8D+Ll8YaQitaGTcY0H3cnofrQB9DZOccZ9KlFcl4I8b6f4z0hbq2YC4QATxd1P+FdT0GeR7UAS96KYpJ5p9ABRRRQAUUUUAFFFFABRRRQAUUUUAFVruUwWssqxtKVUtsUZLYHSrBI9aw/Ft8uneF9RuftIt2WB9jlwvzbTjHvmgDxOx1a6+Ifxusp4w8FtpnzCKUBWTaQHAx6mvoUBvlPSvBv2f9Ea5udS8SXQm88v5ayMDtkDZLEevNe+L0FAABijNLTSeOuKAG+YN3avEPjR48kjC+HdKmjdpuJyOWUdMV1nxX8Y/8Ir4ZYW0hjv7r5YQBnjvXnng3wtLp/hK98d6iouNT2mSFbhDgd8kHrzQB3vwb8Lt4c8Ms8rQyyXRDh0HQc8Hv3q54j+J+n+HvE0WgyWVzcXcoUJ5agg5OKq/DHxte+J/Ct5qWqpDELZgALdCBtwecc+leaeDL2HxP8ZrjU7rfNaB28maX5fLODtHPfPagD0v4rWNtP4JutZkkuba+gtswBLho9pODggHk1yXwm8LWnivwZc3Wp3l/JdmZ4lmW7kyo2jHGcdaX9oHWCYtN0FIX8y4bzQ6tgYHGD+del+ANCGgeDdPtDBHFceXunK8hnPfjrxigDxlNT8R/CHxn9l1Kea/0m8bI8xy+VzjPzHqM1avtTl8DfGWO/VVSw1YK3kxoOFc4UH0wTmq37QGp2t14m0uwR2862Q+bxjGSCP0rU+J9hb2vhDwxrFskl81rKjNc/fJQbTgsO2QQKAPdoZFkVHXIDYIz6EZqeuZ8KXkGs2cOrwXDETwRjyPMDLGQoz0+ldNQAUUUUAJjimMmR7+oqSkoAztW0ez1vTprHUIEmgl4YN+h9q8N8W/B/V9F1GTVvBlxLHDuUi2jkYODnp7gV9AuMrjpTCM54IyMZBoA8R8J/GEabDFovi2O5XUkl8uWZkAVeTjJ/KvVJxo/i3TriyZhdWrHEgVuvHqKx/GXw50fxbYz+bbRw6hIMpdYywP+1jqK8eubjx18I9Q+zea9/pi4G8xkxlfwPBx60ATXaav8GvGrXVtbbtFuG/hyV2dgT6ivcvC2ur4h0CHUt8RWYbh5ZyB9feud03X/DPxM8LfZpzDvlTBtnYB0bHJAPNeX+H/ABBqPws8bvo+sPcjR2YiKPd8qg9Gz0OKAPpBCenpT6p6feW9/ax3NtIJIZFDK4Oc1czQAUUmRijI9aAFopMj1oyPWgBaKKKACiiigAoooJwKAGNxyOtePfH3V3svDNrpojUpfTbWcsTt2nPSvX2YbgD3r558ezS+Nfi3Y6JZubq1tWUzQvwoIJLYP+7QB6n8M9Gn0Lwla2krI8JRZ43B+Zt43MD24J4xXbD1xVa2gitIYreJNkaIFRB0AFWc460AMcndWPf+ILDT7K+uZbmPbaKS5Y4AbHC/yrYY8Ent74rwj4j+KtQ8TeJz4N0KE+S8oF3LENxk555HAAHegDB0mx1X4yeMprrVJymmW7fcUnaFzxt+or2XxvZR2Pw5vrW2jO2K3KKpOcYFbXh/RLfQtFtrOGPAjiUMT1Y47nFaF1LBBbySXTIsKqS5fGMfjQB4p8FdW0zQPBN5JrU8drDJOFXzxgP97tXrNlY6E1ktzZWlkLVwJFkSFQGx0Y8frXhXjzVbXxz430vRdAdbiyjfEkO3YhYH3wD3r3d9DtZfDx0dMwQNB5QEZwVyCDj86APn/wCIOt2vjH4r2FhExNraOIvNtzknOCevoRivXNf8daV4K0+f+0NSW4uyu63tgAHC4G0EDjGetc4vwA0BZBKl/qCuGJDCUDHPX7tauk/BbwtYTySXkcmomQDat024LQB5N8O9DvPHnxHk1nUIpjapK07mRdwJJ+VDntg/pXtHxJ0qGX4cX9nbiG2jChlTG1RtOcAAe1dhaWFpY2yW9tBHFEAAFUAZA9QKL6zivrKa0mXMUsZjbHuMZFAHlH7P+p28vhq40yMN5tvLl2HK8kkYr2Wvnf4X3UnhX4n6n4fIEdtM7kNMME4bAxnrxX0G0m0dOc0AS0U0Nlc0m7BOOcdaAH0VHuO7tTt1ADqKjMhwSFyBTRI3cdsn2oAkIB7Vm6volnrUKQ30ZkiXPyluDnjkd6mm1GKG9itH3+bKCykISMDrk4wKtAl+v5UAeBeO/hK2jTJq/hW6e0OTvi8xgSf9nFckfFc/iPT5PDni6JftaLizuXXa6OOgJ6kE/Wvqh4UcAOgYA55HtXL+KfAmh+K7V0u7NBOAQk6DDKaAPG/hT41fwpqVx4f125kFuf8AUqSCA3sT/nmvoSzuheWkVxtZRIu4Buor5R8b/DLXvCNytwd99bucJPChJXHTPXtXrnwj+JVprljBoWoSLBqFum1C7cSj6n+KgD1pSWAOcY4rC1XxjoOhz/ZtU1aC1n2b9khwcHOD+lQeNfF9l4N0N9QuwWcjbDF/eevDPBehXfxb8aXGta6zG1gwWAGA4ydqj265+tAHp8Hxr8MS6iLVhdxIXKfaXjxEAO+c9Peul0HxppviO6u000SSxW7bWuMARuf9k1dufDGlz6PJphsoDA0XlYKDoBxWH8OfBk/gvR57CeeOUSTtKojBwoOOOvtQB3FFFFABRRRQAUh5GBS031oAytYvktLZg8UkglGPlOAvHUntXhvwc0SfVfHeoeJAssVtCzhQxyHJyMZ9s16P8X9Raw8AagsatvmUx7lbBXOeap/A6z+y+A1bzhL50xkyEK4yBx70Aek/MOeTn9KXdg88jGeaeTjFcr4u8Y2PhXS7m6nJaeJAwQ9y3AoA534p/EG08PaVLpdvKX1S6TYgib5kz3/lWb8IPAt1oIuNY1PD3l0MLIHLfIecf41yHwx8LXHjbxbc+LdXhWWw8xiiynPzEkgfQDIr32wsrfTbVLeBSsS9AecfSgC2OBj/ACa4T4seI9O0fwhd29xOPtFyhWOMdfyrsTf2SymL7ZBv/u+au4fhmvML74S3Wt+OX1bWdS+2aeG3xxE/pQB5p8OfhdrOv/8AE0W6l0xIuYpl4dj6ivSR8K/EMbNIfGd+cHOQea9PeH+zdKZLKJdsEeI4wMZwK4rwR8Sotf1K50bVEWz1SByvlE4Dc9uxNAG/NY3Gt+DW0+3uri0uPK8gXDDD7lwN3XvjNcN/wqbxGQB/wmuofXef8a9dTGOmOemOnen4oA8/8JeBNX8P60by98R3V/E0ZTypTkAkg/0qbxn4L1bxHqcVxYeIbnTo1j2GOM8E5Jz+td0RTCOQfSgD5T8YeFtS0Hx/p9vf6rO4lZSt/IcFfofavpqyhZ9EjgjumcmDYJ+5JHWvO/jrp1pL4OTUHhBuIJAI2zyOv+FdL8NdcbXPBNhcyNEHEYTYp5GPWgDlp/hP4hluHlXxlfqCxYKHwBntW94R8B6x4e1oXt54iur6IKw8mQ5XJGK75fuinUAcJ4y8Far4k1OC5sdfutOjjQqyRH7xzXNj4T+Iw/8AyOd7t6+9evYoxQBhappN5d+EpNKgvZIro24iW5H3gwA5z+FeHt4A+KayErq7n5uv2hvWvotyMdSPpTM55BI/HigDxvwP4W8e6HrU19q10LuPySFjedjlsj29ql8e+DfGmo6wt74fv5YI5UBkj85gFbnpXd+KvGmj+E7UvqUxDvGWjiUcuB1xXn938fdPntpF0jSbqW8HCrtJGPXGKAOaTwF8VRIpbVm27hn/AEh8V3vjvUINP+Hq215rH2bU4YAwEcpUu/161r2viA+LfCKxRGS1vr+EqqsjLsbPPWvF/E/hXR/CqwL4o1O7vL+TkCFxgex4NAHLLrPiW80Oa9fXy0SYQwPMxdvoK6T4VfDvWdS8Q2uqXEUtnaW7LMJHXl8HOB9a2Ph74BsNV1e31rULcW9jM/8AodqPnD44yx7dK+hYI0t4UiiRY0QYVQeMelAHi/7QWowyafpmkcrcyT+bkj5AoBHJr0b4faXDpPgnS4YxBvMKmR4R8rse9cH8edH1XUNEt7i0s0ltreTdK6rmTv8Ap3rP+Gvxh0i00Kw0PVVe3ltoyizAEqQBxmgD3YHI+lABz1Arx+7+I2oeM7qDS/B8FzDMLjMl3IDs8sZB56elen6ZczzeZFPE6vC2xmPRz6igDUooooAKKKKACmEkDP5089DUMsywxPM/CopY/SgDwr4r6zc+JfGGm+ENOxKA4aZCcBm4I/nXtemWVvpmmwWtrCkUcaABUGOfpXgmh2a+NfjPfalplw9otodyvtOCUIU/ng19CIAoCNyVAyTxk+tAFG71m2s7oW905izGXaX+BQPf15rwb4hapf8AxK8Zp4a0S2V7a1kw1wnIbjkk+gHSuh+NfjGGGNPDEMJnmuVJYxn5o24x785Jrq/h/wCDz4T0LTo4bWP7VcLuvp9wLDjIxQB0XhnQoPDXh610uBQpjQByvduMn8TWd438cWHgzSnuJiHumH7qLPJPYkemam8WeNdI8G2yS6m/MhwkaHLflWd4h8DaP46uNM1a6D/ugHAJ4cdQCP8APWgDynwX4P8AEHjXXm8T3VxLZ27T+aqEkbhnpivowJtiVSMkYqK2torS3SCCNY441ARV4AqwASBnmgCKdJDDKIW2yEfKTXknxA+GuoalDa6poe2PV0l3zSI20vn6V7CRxnoaZtI6du+etAGB4YhvNE8PWVprmoCe/Y4MjHkk8gfh/St8SbmwG4B5PFeP/GPQdfkkttf02YraaZGXeNT8zNnrjOfWt/4W+MLbxRp1wWHk3iykyQM+WxgYOPTNAHo1NZN3el3DOM0bh60Ac7400Vdd8KX1kYg8jRkxg9mA4rz34G3mzTL3SrhEiuIJSuwN83GOtewSENE/cYI45r5y8Jatp/g74wakl680cU5MavICBuOev6UAfRynAoLfLkfga5vWPHGgaKmbrUYS20OEDgkg1PLrbXHhdtW0qP7Q0kReGPHJ9OKANt50iXdI6quM5PFU7PWrHUmZbG6juCn3vLOdv1rwZ/DnxP8AHJlup7k2cauVSIts4J9M13fw4+Gd74IuvtkupmVp49txDjgntg+xoAqeN/i3daFrt3oGm6Y89+uBDJjIJIrn/D+p/FDX/EFpq7W3kWisI5InAClc8nGPQn8q9ml0XTXvRfNYwtdocrKV+arzLgAL8vPXPAoA43VvDlj44ubGe+t/3FoSGR1+YsDjH0rY0rwjoWkO8llptvHIRgkKDxWuhTfjcocjO0dfc4+teZeMvi5a+Frq804IlxegYtvKbIXP97FAGj8RvGemeE9Ie3gWMalMpWFY1GU9D7Vw3gj4Yah4h1Ma/wCLZWnjcbo492c/X2qj4Q8HX/iG/Txd4qhmuI5Jf3MBByeeCR6V9CQRJHEscabIgvyqP4R6UARWemWljbxW9vAkcUX+rXH3f85q3sAPHWhAe9O70ARSwJMhSRFZGGGB6GvPdS+Efh/UtWjmktVhtYdrrHEMbmzyD7dK9IprLk8UAYsHhuxtIbaHT4ltIoJPMCxDG7g8H862BH8m0nNKFIp1AC0UUUAFJS0UAMZsZrmfHs4tvBepuZjHiFgGzjn0rpZOBnB/CvDfjzrzn+z9EhmjMcrgzRjlh+XvQBp/APTLi18LXl5c24zcS5jc8swBOf1r0DxZ4mtPC+gXN9dXEMbqhEKufvNjgfnVTQobHwv4DtPLnEEEVqJd8pyFJAJ/U14RZDVPiz4/js769kl022c/vI42VdgyefTOMZoA6T4WaHfeMPF934u1+ESeXgIkqYyTyu31AAxXrPjLxPD4R0KS+NtJPJgrFFEvftn2rlviL48t/h3oVvpunxg6g8QW3BGQqqAMn9K3PAGvzeLvBttc6lb5uHUrMHiKrJnglc9RigDyrwf4Q1b4m64fE/id3FkrnyoWGMjPQD0r6AgijtoEhjUBIwFVfQUttbxW8IhhQJGgwqqMAVMq45oAXANKMUwk/X3pglGGAYMR6dqAJ6TFR7jwG6etRT3cMETyTzJGicl2YAAfjQA+aKOWPy5I1kRuGVhkGuBu/hrat41tvEWlXb6cwz9ojiGBJ+FdXrWu2mjaBPrM4aW3ii8z92Ccrj2zXiV98V/HOtytceHNFmjsV+TBgaRtw9xx0oA9xOt6emrx6W1yn2wxmURk87RgZP5iuU8Q/Frw1oLXEJu1mu4G2mCM8mvGINP8c6ZLN421O2uHjuMxXJ5EoQ9dq9VHHHtXTfDPwr4E1a5+0SXjX1+ZGxDN8uRjnII5+tAHQ+Hfi9ceIPF8FkunS22mSqNski/MGOPT15rkvj74e+zanba3b2xEcoHmyBurV7jpGix6dvkaO3aRsqjRxbSE7Lz7Y/KuT+L3hbUfFPhdYNMCb4HMz+Y4XgYPf6UAef8AgD4axaitlqGsQ3GoRTruEoclFHo3Oa920jR7PRLBbLT42jtlOVXcTt/EmvNvgVqDzeF5rGe78y4t5iojZwSgBIPH5V6uvUDpn9aAHAED+tO7dKMU0vjrQAP0HFMJJU5AH6U52wuSRjNeW/Ev4lWujWtxo2kXDvrj4SPy0LbcnBHHegCj8VvGNnp94mmWMl0NbKYie35AzggHn3Fc98PfhHf3mrf2z4rV+vmRwynJc9cmt34e/DfVDqsPinxNdrdXbx5WCUbmTkYPX0FewhM/w8Z49RQAyJVhRY402IowFXoAKsDpSbMdKfQAgGKWiigAooooAKKKKACiiigApCaWkNAEF1OkEDyyHCICzHHQV883V+PiL8YbaKzMQtLN8nzExv2kk9PYd6918TXq2Xh2+mM6w4iJV2OMGvmHw14nh8PafrGpXEEp1W+JjtZ1BGM4Bww9iaAPS/iz4nOrvD4K8PN517cELKsQBULjG39a6fwJ4UXw34Lu9O025WTUyG3y7cbZCMAepA615N4cl0nwHosfivVYhqGvTSfurdpAHizn5yp59K7H4XXvi3xFr7a7q2ptFpszv5Voxx5h24+Rc9AOc0AM8GfCOfUbqfU/Gkst44lYRwPKSM5zvz/T3r2S1tYrSBLeGNYoo12qq9AP6VOQSeM4x3qvc3UFnE09xNHDCvVpHCgZ9zQBYU7QOtOZ8fyryfxF8dPD+jXK29pFNqLgkSNG2xVx7kHNb3hPxdd+PNM+12Nm+mwq7LJ5oLFwRwVPHvQB1l5qljY20lxcXUUUcf3mLDiuU0Txr4Z1LxK1lpV61xc3C7mcSEoMc4AJx+Qrz6b4ReI9a8QXcN/qssWkfaDJHuk3lgSTwO3WvQND+FPhjQLu3vLS2f7VbjiRiMscY54oAy/iTqHjuC7t7TwnYOYWG5541UnP41xHh/4WeL/EN2+oeI9WurOOViWh847iR0yOmK+glGMAfKPQdqz9bvWsNFvLtJRE0UTMHK5C4B5x3oA878ceLU8J+Cl0WWSGTWHjEMEKAN8qjClgfYfrRoHhLxffeE2a910WFzOjFYLeBFUZAHJA/UV5f8PVtfGXxLabxBJcXzsS0RUHBIPGeDgY+navqJV2qMYwF27RzQB4T8O/HOvSeKpfCGvw/b4gxtw7KAYwvGeeoNehaX4G8OeCrrUdchiAMgaQs4H7pQM7V9M14vDcPq/xvkvrdbmRLO5LzeYfMbahKkDGMj2r6Pv7Cy1zSDa3kDSW86AMjAg8j07GgDH8HeN9J8Z2002nh1kicpJHJ1Hoa37yBp7OeBOGljZAfTIxXzlcW2q/CHxWb/TZhe6JLLiTyyCMf3SR0I6fhXu3hXxZZeLtMW9sFmCkAMHQjB7896APFPhlFceHfitfaTdTPG00jY2JlXyeOTzX0WpUOcZ+teEfFaaTRfiVo2sW088chAVkjjOSOnXv9PevbbKdbu0iuAW2yKGG5SucjuKAL24AZ9KhZhkdfWh5FjQyOQqgckngV4X4y+Lt7Zz3+laPeQ3t1NJ5UDQwkGJc+uSGPoR0oA3fij8UP7DU6FomJdWmGwmPnyv/AK9UfhN8OpbeWXxB4ltZDqTSHy0n54xkMPzqL4Z+A7XTHtvEWvM19qd3LiEKC6xMecuRkZyDyele08/Xp0/xoAUJwMgY9PQVIgOcmnDkUoFABRRRQAUUUUAFFFFABRRRQAUUUnNADQcnI/Gm7t+cHA9+9DuqDkgDrXh978StS1D4sWuj6feFdOV9kipkB+KANr47621h4PWxWEMLxtu4/wAI/wAmvFfhta2+seLtPtNTvBFbQvvjifkMw7Y/Kuo+PWuSXniaCwSVzbQJzGVwN2BmvKbS+n0+8S6tJGimjOUZeMUAfSs3gTQ774jLrF1eY2yIiWcyqY5TjouOfzre8f8Ai218B6XZzW+lC4uC2yCMJhYx36dPwrx7wHqXhyxni8ReIddnm1KOUyxwAHb3yD+lem/8Lk8F3rWguiT5oJbfHkRYH070AcInxN+IviO8kn0TTpI7RSoMaxAhfxIzXq/iDRtV8XeC7bSr7ZaTXSg3Tpz5eOflHrVGD4reBbVSLe+EQJBIWLGajufjL4OiuLaNLsusrMC4jPyDHU0AaPhv4YeGfD1sY47COeR1Xe8yhixA5PPSuvtrW3tIBDbwpFH2VBgflXDn4v8AgsKW/tFyQOnlnmoLL41eDrq28yW7eE5K7DHnpQB6KMYwc0Ngde1eczfGrwfDdwQpdO6SfekCkBPwoh+NPg+a8nha6eNI/uyFOGoA9GBBrnvG2D4K1bcP+Xd8f98mubvPjT4PtbYyw3bztnhQmKS6+KngbULFrS41EtDMm1wUPQj6UAcZ+zxplo9vqeqOp+1qwiDE/wAB5PH4CvQ/iX4sj8LeELmRGU3k6eTDGxwfm4JGPTNeL6e+l+GvEs2oeGPFkdtbuTiGSEkY98EfyqW1fQdT1+PU/Fvi9794GVoVjjKgYbOD1zQB0HwM8IXIuZfFt3MQJw8axtncfm+Yt+Ir3dOh+bjsa830v4xeC2tDi5e1VHZRHsPIBwD+I5rQ/wCFw+C+n9pvn/rmaAOsm0jT5rZoJ7OGSJzuKMgwT9PWuE8feP7T4cwW1lZ6WkjynIQDYgH4Veb4w+DCMf2mf+/RrnNf+IPw58SSLpuqD7RBsJWZkzsPtQBzfjPxJZfEPwKutWsj22oaawd7ZOSPx9K9N+GXiB9c8DWt7c4jeFdjsT1wByfzr5yu9ag8H69er4bvlvNNu1ZWR48Daex+nSqWheJNct9MudD0ovtvT82wEsBnt6CgD1f4jfEa+8Q6j/wifhPfMztslnjOCx6YHoK2/hp8IU8O3Laprax3N7n92vUKPXnvWv8ADDwDp3hjT1u2MdzqkqhpZeCU3DO32/8ArV6JjnnHU9B2oAitbWKBdscaouSQijABJyT+dWdo9BSKMY9fWn0AFFFFABRRRQAUUUUAFFFFABRRRQAGmMSKcxqGeVYomlkYqiKWY9gByaAOG+JvigaJohtLW6kg1K6UrbmMAknpjB968F8L6be6L8VNOg1PatyZA8hz3I711FxqGteNviedV03TxeW1jJ5UW5sxgg4DfpmsrxXBqGn/ABUtdQ8TWm1JZVIEB9yB0oA+jNQ8M6Lq9x595p1tPNj7zrk4qsfAvhnJ/wCJNa/98CtmydJLOBo87GRSueuMd6tYoA5s+BfDIA/4k9t/3wKUeBvDXT+x7XH+4K6SigDnf+EE8M/9Aa1/74FNbwP4aGf+JPa8nJ/diuko/CgDmz4G8NHkaRbDP+wKB4G8Nnro9rnt+7FdJRQBzQ8C+Gen9j23H+wKX/hBfDJ/5hFtz/0zFdJRQBzn/CC+GccaRaj/AIAKT/hBfDJGP7GtMd/3YrpKKAOc/wCEG8M4x/Y9rj/rmKQeBvDP/QGtv++BXSUUAc3/AMIP4ab72j2ufaMUo8C+GTydHtf++BXR4ooA5w+BfDIGf7Gtf++BSHwJ4ZxxpFtj/cFdJRQBzR8D+Gcf8ge1/wC+BU9l4Y0LS7j7TaadbQuBtDhBke1bWcsBXkPxluNZ0TS01ez1uWFVkTy7YMcMcjPegD1S3sobV5GhjVGlbLle+BxVnvjvXgOkX/xJ1bwTda9b6kd6HKRMp+ZfUc113wc8c3firTLy31N3kvrZt7yY4KngD8CCfxoA9TXrTqjTg8HOakoAKKKKACiiigAooooAKKKKACiiigBr9ODXA/Eg+LLqwOneG7PPnjEs5PIHTArv2GRyM03nGMjH0oA86+GelXfhjw+9hqGmm0MZ3vM5H7xjnPT61yHxD8O+KfGmuWV5aaOy29mwKtuH7wA59a9vmt47iBoZkDxvwynoRSJEsMSxRRqqLwAOwoAy/C11qFxpMa6lYGzliAQKSDkDvW7UeMsOmB7VJQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAETE5I79q+bvjnq7ap4stNEjik/0fBIBzuySOn4V9G3EqRqzswXAOGz0r5xg+1+NfjoZ7aNI/skvJJHRTnr+dAEmu+P49F8I2XhLw55qSugWaWUYdCw5Htk133wh8B3PhS1uL64vIpzfxIdkY+6cn2rmv2gU0mOwtFiSBdRllDOV4dlwRz+ld78MLLUrfwhYTXmo/abeW3XyYwhHl8nryaAO6Xr0FPqNA27k8VJQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAhFJsFOooAQDFLRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFB5FFFAHLeNNI1XXdJbTtNe3jEv33mZgV+mBXlWk/BTxZoWonUNN1+0guSD84BJwRyOVr36igDxXSfgxd3PiL+1PF2pLqPfy0J2sfQ8DFevafYQafYw2drH5cEK7UjB4Aq3S0AIBg0tFFABRRRQAUUUUAFFFFABRRRQAUUUUAf//Z'] Multimodal Competition False Numerical Open-ended Solid Geometry Math Chinese +34 " + +如图, 在斜三棱柱 $A B C-A_{1} B_{1} C_{1}$ 中, $A C=B C=A A_{1}=a, \angle A C B=90^{\circ}$,又点 $B_{1}$ 在底面 $A B C$ 上的射影 $D$ 落在 $B C$ 上, 且 $B C=3 B D$. +若三棱锥 $B_{1}-A A_{1} C_{1}$ 的体积为 $24 \sqrt{2}$, 求实数 $a$ 的值." ['$\\because B_{1} D \\perp$ 平面 $A B C, \\therefore B_{1} D \\perp B C$, 又 $B D=\\frac{a}{3}, B_{1} B=A A_{1}=a$,\n\n$\\therefore B_{1} D=\\sqrt{B B_{1}^{2}-B D^{2}}=\\frac{2 \\sqrt{2}}{3} a$,\n\n$\\therefore$ 四边形 $B_{1} B C C_{1}$ 的面积 $S_{\\text {四边形 } B_{1} B C C_{1}}=a \\times \\frac{2 \\sqrt{2}}{3} a=\\frac{2 \\sqrt{2}}{3} a^{2}$,\n\n$\\therefore S_{\\Delta B_{1} B C_{1}}=\\frac{1}{2} S_{\\text {四边形 } B_{1} B C C_{1}}=\\frac{\\sqrt{2}}{3} a^{2}$\n\n$\\because V_{B_{1}-A A_{1} C_{1}}=V_{B_{1}-A C C_{1}}=V_{A-B_{1} C C_{1}}=V_{A-B B_{1} C_{1}}$,\n\n$\\because$ 点 $B_{1}$ 在底面 $A B C$ 上的射影 $D$ 落在 $B C$ 上,\n\n$\\therefore B_{1} D \\perp$ 平面 $A B C, \\because A C \\subset$ 平面 $A B C, \\therefore B_{1} D \\perp A C$,\n\n又 $\\angle A C B=90^{\\circ}, \\therefore B C \\perp A C$,\n\n又 $B_{1} D \\cap B C=D, B_{1} D, B C \\subset$ 平面 $B B_{1} C_{1} C \\therefore A C \\perp$ 平面 $B B_{1} C_{1} C$.\n\n故三棱锥 $A-B_{1} B C_{1}$ 的高为 $A C=a$,\n\n$\\therefore V_{B_{1}-A A_{1} C_{1}}=V_{A-B B_{1} C_{1}}=\\frac{1}{3} \\times \\frac{\\sqrt{2}}{3} a^{2} \\times a=24 \\sqrt{2}$.\n\n$\\therefore a=6$.'] ['$6$'] 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Multimodal Competition False Numerical Open-ended Solid Geometry Math Chinese +35 "如图, 已知抛物线 $C$ 的顶点在原点, 焦点为 $F(0,1)$. + + +求抛物线 $C$ 的方程." ['设抛物线 $\\mathrm{C}$ 的方程是 $x^{2}=a y$, 则 $\\frac{a}{4}=1$, 即 $a=4$.故所求抛物线 $C$ 的方程为 $x^{2}=4y$.'] ['$x^{2}=4y$'] 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Multimodal Competition False Equation Open-ended Plane Geometry Math Chinese +36 "如图, 已知抛物线 $C$ 的顶点在原点, 焦点为 $F(0,1)$. + + +已知抛物线 $C$ 与直线 $y=-\frac{1}{2} x+b$ 相交于 $P 、 Q$ 两点（点 $P$ 在第一象限内), 且 $P F \perp Q F$, 求抛物线 $C$ 在点 $P$ 处的切线方程." ['设抛物线 $\\mathrm{C}$ 的方程是 $x^{2}=a y$, 则 $\\frac{a}{4}=1$, 即 $a=4$.故所求抛物线 $C$ 的方程为 $x^{2}=4y$.\n\n设 $P\\left(x_{1}, y_{1}\\right), Q\\left(x_{2}, y_{2}\\right)$,\n\n直线 $P Q$ 的方程是 $y=-\\frac{1}{2} x+b$.\n\n将上式代入抛物线 $\\mathrm{C}$ 的方程, 得\n\n$x^{2}+2 x-4 b=0$,\n\n故 $x_{1}+x_{2}=-2, x_{1} x_{2}=-4 b$\n\n所以 $y_{1} y_{2}=b^{2}, y_{1}+y_{2}=1+2 b$,\n\n而 $\\overrightarrow{F P}=\\left(x_{1}, y_{1}-1\\right), \\overrightarrow{F Q}=\\left(x_{2}, y_{2}-1\\right)$,\n\n$\\overrightarrow{F P} \\cdot \\overrightarrow{F Q}=x_{1} x_{2}+\\left(y_{1}-1\\right)\\left(y_{2}-1\\right)$\n\n$=x_{1} x_{2}+y_{1} y_{2}-\\left(y_{1}+y_{2}\\right)+1$\n\n$=-4 b+b^{2}-(1+2 b)+1$\n\n$=b^{2}-6 b=0$\n\n$\\therefore b=6$ 或 $b=0$ (舍去)\n\n$\\therefore x_{1}=4, y_{1}=4$\n\n即点 $P$ 的坐标是 $(4,4)$\n\n所以抛物线 $C$ 在点 $P$ 处的切线方程为 $y-4=2(x-4)$, 即 $2 x-y-4=0$.'] ['$2 x-y-4=0$'] 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Multimodal Competition False Equation Open-ended Plane Geometry Math Chinese +37 "为响应国家“精准扶贫，产业扶贫的战略，哈市面向全市征召《扶贫政策》义务宣传志愿者, 从年龄在 $[20,45]$ 的 500 名志愿者中随机抽取 100 名, 其年龄频率分布直方图如图所示. + + +求图中 $x$ 的值." ['根据频率分布直方图可得 $(0.01+0.02+0.04+x+0.07) \\times 5=1$, 解得 $x=0.06.$'] ['$0.06$'] 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Multimodal Competition False Numerical 1e-2 Open-ended Probability and Statistics Math Chinese +38 "为响应国家“精准扶贫，产业扶贫的战略，哈市面向全市征召《扶贫政策》义务宣传志愿者, 从年龄在 $[20,45]$ 的 500 名志愿者中随机抽取 100 名, 其年龄频率分布直方图如图所示. + + +在抽出的 100 名志愿者中按年龄采用分层抽样的方法抽取10名参加中心广场的宣传活动, 再从这 10 名志愿者中选取 3 名担任主要负责人。记这 3 名志愿者中 “年龄低于 35 岁”的人数为 $X$, 求 $X$ 的数学期望." ['根据频率分布直方图可得 $(0.01+0.02+0.04+x+0.07) \\times 5=1$, 解得 $x=0.06$.\n\n用分层抽样的方法, 从 100 名志愿者中选取 10 名, 则其中年龄 “低于 35 岁” 的人有 6 名, “年龄不低于 35 岁” 的人有 4 名,\n\n故 $X$ 的可能取值为 $0,1,2,3$.\n\n$P(X=0)=\\frac{C_{4}^{3}}{C_{10}^{3}}=\\frac{1}{30}, P(X=1)=\\frac{C_{6}^{1} C_{4}^{2}}{C_{10}^{3}}=\\frac{3}{10}, P(X=2)=\\frac{C_{6}^{2} C_{4}^{1}}{C_{10}^{3}}=\\frac{1}{2}, P(X=3)=\\frac{C_{6}^{3}}{C_{10}^{3}}=\\frac{1}{6}$. 故 $X$ 的分布列为\n\n| $Y$ | 0 | 1 | 2 | 3 |\n| :---- | :--------------- | :--------------- | :-------------- | :-------------- |\n| $P$ | $\\frac{1}{30}$ | $\\frac{3}{10}$ | $\\frac{1}{2}$ | $\\frac{1}{6}$ |\n\n$E(Y)=0 \\times \\frac{1}{30}+1 \\times \\frac{3}{10}+2 \\times \\frac{1}{2}+3 \\times \\frac{1}{6}=1.8$'] ['$1.8$'] 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iigAzRRiiglhRRRQIKKKKACiiloASilooASlxRRQMXFAHFFKKTKExRinUUiSOiiikIKKKKACiiigAJxRSGkoAdRTaKAHUU2igB1FNooAdRTaKAHUU2igB696iJ+an5wKjPWgDnvFC5sJK5zwpdLFqRUmuq8QR77CTIridCTbq7cUmJ7nqiNujUjuKepqCA4gT6VMvSmMdRRRQAUUUUAFFFFABRRRQAUUUUAKKdTRTqACiiigAooooAKKKKACiiigAooooAKKKKAA0lKaSqRSCiiimMKKKKACiiipAQ0lKaSkAUY96KSmhMcBRSDmlzVEiUUUVLAKKKKQBSGlpDQAlFFFABRRRQAVwHjPov+9Xf1wHjPov+9QwY/wAI/db613KdK4bwj91vrXcp0oQIa33qee1Mb71PPagAFOFNFOFABRRRQAUUUUAYWv8AMD/SuY0Q/wDExFdTrwBtn+lcvogH9oigOp3tv9wU9utMtx+7FPbrVl2HJSSUsdEgosAg6UtAHFFFgsFFFFFgsFRv0NSVG/3TSYmee+Kf+Qnaf9dBXcWv+qi/3R/KuH8U/wDITtP+ugruLX/VRf7o/lUIhFo9acfuimnrTj90UxjAeadmmiigB2aM02igB2aM02igBx5pCc0lFABSUtJQAUh6UtIelADDSYpaKAGkYpKcRSEYoAaRikpxGaQjFACUopKUUAOFOpop1ADqelMp6UwH0UUUgFFLSCloAKWkpapFIKKKKZQUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUlLSUAFFFFAgooooAKKKKBMKKKKCQooooAKWkpaACiiigAooooGKOaUUgpRQULRRRUiI6KDxSZFIkWiiigAooxQeKAENJSsajaRF6mgLj6Ki+0RnjdUgYHpQAtFITg0A56UALRSFgO9M8+PPWgLklFNV1bGDmnY6UAFFBBFMaVFOCaAHk8UwnNIZUYcGlwTQBma0pbTpK4bRwRrDV3msEDTpM1wukkf2w9JiZ6VD/AKhPpUqHIqCA/uE+lTIQBzTGSUUgYUAigBaKM0hIoAWikyKM0ALRRmigAooooAUU6minZoAKKM0UAFFFB4oAKKTIoyKAFopMijNAC0UZooAKKKKAA0lKaQc1SKQUUUmRTGLRSBgaU8CgAopAQaWpC4hpKU80lILhSUE0U0JiiikBxS1VyQoozRmpYBRSZFGaQC0hpc0h5oASiiigAooooAK4Dxn0X/erv64Dxn91f96hiY/wj91vrXcp0rhfCByrfWu6XgUJghrfep57Uw8tTz2oGApwpgODTgaYC0UUUDCiikzSsKxja7/x7P8ASuW0XjURXU67/wAez/SuV0U51EUdQ6nf2/8AqxTm6023/wBWPpTm+9V3LHR0SUJxRJRcLgOlFIDxRmi4C0UmRRkUALUb/dNPyKY/3TSYmee+Kf8AkJ2n/XQV3Fr/AKqL/dH8q4fxT/yE7T/roK7i1/1MX+6P5VCIRaPWnH7opp604/dFMYwcUZopKAFzRmkooCwuaM0lFAWFzRSUtAWCkpaSgLBSGlzSHmgLCU2nUfhTCw08008048UlILDTSGnE000BYSlFJSigLDhTqaOtOoCw6npTKenFMLD6KKKQWFFLSCloCwUtJS1SKQUUUUygooooAKKKKACiijNABRRmjNABRQOaKACiiigApKWkoAKKKKBBRRRQAUUUUCYUUUUEhRRRQAUtJS0AFFFFABRRRQAopRSClFBYtFFFSIjNMp55pPwpEgtPFNUUZxQA4kelMJoLcVEXGaAEnkCKWPQVyWseIFglKI3Nbuqz7LGZh2XNeXRebqetqoJIP+NS9yXudEdVu/KEqucGtfw5r5upHjnfJB4zV+DQ1XTVRkGa4DUZzo2twovyh5MHFSSesMSeRTo+GzUFlKJ7aNwf4R/KppDsXJ9KBWMrV9SSyQlm69K5pdXuZtxjc4zWZ421F5rqCKNiPnwcV0fh7TkNkGdckr3FA7ley16WK5CTvXY204mhVweDXmXi0ixuHkQbQK6TwvrAvLVI85KrQB1sj4ri/F/iFdKtpHDYYZ6Vt6lqa20ZJboK8q8byS32ny3OSEoA7Xwrrz6ntZn3Aiu6Q5jFeUfDwgRR/SvVYz+7H0q1sWjK14/6A+K4XRznWGrt9eP/ABL5K4TR2A1dsn0oYM9Rt/8AUJ9KkI9qr2zgwJjHSpWkC9SBTGSBakC1XFwmfvCphIPWgB+2mlaDKo6kD61GbhD0YGgB22nbaj80UecB1OKAJdtKF5qEXKH+IU4TDigCbbRtqL7QvcgUC4Q4ww/OgCQiikDBulIXUdWAoAdRmovMXPBB/GgtigCXNBOTUBmVerAfjQkykjDA0AThaCtKHprTKOrAfU0ALto200ToejA/jTTMAKAJdtB4qD7SgPLD86UTK54YH6UAS0UzdkUhkVepA/GgB5NKvWovMU9CDT1PNO47jm6UwKTnintImOWFNSRBn5gaOYOYYoweae/K8UrDPIpnmIDgsKd0HMKi04jHelDL2INNZgO4pCEp2KYZYx/EKBIp6GkAEGjBpS49aTzY/wC8v50AGKXFAdTjBBpS6jqRQAmKXFMMydnH50gmX+8Pzp3Afto2H1pnmr/eH50vmJ/fFIB+2jbTPOX++KPOX+8PzoAfto203zV/vijzV/vigB22jbTfNX++KPNX++KAHba4Dxpwq/Wu981f74rgfGbKUXDDrSYmHg7v9a71+BxXBeDe/wBa7xjxSQhVHApXGMUKPlFJIeRVFDO9OFNHPNOHFMYtB4ooNNDQUmPelopjMTXf+PZ/pXLaF/yFBXU67/x7P9K5bQv+QoKzlozOWjPQoR+7H0pG+9Tof9UPpTW+9TAePu01uacPu000DQdqSlpKZQHikpTRiqGJimt90/Sn01hwfpSYnsef+KR/xNLP/roK7i1H7mL/AHR/KuH8U/8AIUs/+ugruLb/AFMX+6P5VCM0WT1px+6KaetOP3RTKGDmigUUxijmigUUDQUUUUhhQRiig0AJSUtJQAlFFFABRRRTAaRxTacelNpANPWkPNKetJQAU4U2nCgBQOaeBmminigBaVaSlWmA+iiikAop2KaKdQAAZpcUClqkAUUUUwCiiigBCMUlKaSgYHikpTSUCCiiigBRRQKKBhRRRQAUYoooAMUYoooAMUYoooAMUYoooJYYoxRRQSGKMUUUAGKKKKACiiigAooooAUUopBSigsWiiipER0UHim0iRwpDSqaCM0AMbpVdutWWHHWoSooAxtaONMuP9yuG8HgHWkyAfmPX613utqP7MuOP4a848MXAg1tATj5j/Ope5L3PaHA8g8DpXiXj5iut2eOP3navYFuTJbkg8YryDxxG02uWQXqZRSEemeGZC+mKSf4RWldn9wfpWfodu9ppyK+ASoxV6Y7oz9KBnk2uNu1dcnP7wV6XovFgv8AuivMvEREOroSOsgr0bQpC1gD/sigRw3xHJWKQ1V+H12UaUuxwF9farfxBjadJVAyc1neGNOube2lcLjKHH5UgNvU7x9Q1GKJDlC2GIqv4x05IvCslvFkn/61YFnLqFteO0nZ8il8Va5cnSZNrc0wNr4fYhCI3Uda9Wi5jX6V4D4Jv783oZjwa9509i9opbriqiOJma9/x4SV5THqEltrDbQO1esa+B9gkrzrTtOt7nWG8xc8ihjZ6FoF211aqW7Cl8QXT2dlJKmMir2nWcVrboI1xxT72xhvIzHMMqetMZ59ZeJrqWfaV/UV3tjKZbZGPeqMPhTTIX3JFz9K3IbWOFAiDAAoV+otTlvEeqy2D/J6Vgaf4muZ7kIy8fUV317otpf/AOuUniqMPhPTLaTfHHg/SlqGpatnMlsjHqRVLV7p7W0kkTqBWxHAkaBFHAqK5sILqMpKMqRzTGef6d4luZrkKRx9RXcWkrSWyueuKqweFNMgfekeD9K147VI4wijgUK/UFfqcfr2sz2T4QGqmkeILi6mVWHX6V1934fsb05mTNMtvDGn2rbo48H6UtRal+1ctb7q5PxDr89jMqoOprs0iWNAq9MVm3vh6wv23TR5Oaeozl9D8QXF7fxxMvB9K7R+M89qzrXw5YWMokhjww6VqEA9fShAjiNd12ezlIVe/emaJr893KquO9dJeeH7G8YtKmTmiy8O2Fo4aOPBFGotTUiYtFurjfEOvT2MmEHfvXdJEqrtHSsu+8OWF8czR5OaGM5HRvEVxd3KIw6nsRXZyHC5qra+GNPs5A8SYIPpWm8SnqKF5gjzTUfE9zBfyxKMAe4re8LatNqCOZP0rTn8J6ZcStJJFlmPNW7DRbTTRi3UqM0tRal2RxGm48CuJ1vxLLbSlYxnHuK7iSJZF2t0rIuPC+nXTFpI8kmmwZiaJ4iku5ArYyTXYl8R7qybbw1p9nIHijIINbCICNp6UgOA1fxVcWuoSRKOF9xVnQNfn1GVgR0POK3r3wlp11M00keXbrxSab4ctdPdjBHjJ5qbMRq+eEtDIT0rh9Y8US2twwT19RXem2R4TG44NYl54T065cs0eSTT1CzK2ha4b1MOea0dQu2gt2dMVHZ6Ba2R/dJir09kk8e1xkYqijzy78V3MV0UC8fUV1OgapJfJ8+PwpJvCenSybmj5+laen6RDY/6pcUai1H6hM0FjJKOoFee3Pi+5jlK4H5ivTLi3S5gMbjg9a52TwVp8r7jFzQwdybw/qUl/CrNjp2qxrN81lAzjsM1YsNLi09QsS4AFM1uyW6spN47UD1POpfHEwuNikfmKkHjK69B+YqlZeHrSTxCkcifKf8AGu4TwbpR/wCWVSrkq5yy+Mrn0H5in/8ACZXOOg/MV1a+C9J/55fpUh8F6SR/qqYWZxx8Z3PoPzFJ/wAJnc+g/MV1zeCtJ/55fpTR4M0nP+qoGcr/AMJnc+g/MUf8Jnc+g/MV1v8Awhekf88qP+EL0j/nl+lKzA5L/hM7n0H5ij/hM7n0H5iut/4QvS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Multimodal Competition False Numerical Open-ended Probability and Statistics Math Chinese +39 "如图, 在 $\triangle \mathrm{ABC}$ 中, $\angle \mathrm{C}=90^{\circ}, \angle \mathrm{ABC}$ 的平分线交 $\mathrm{AC}$ 于点 $\mathrm{E}$, 过点 $\mathrm{E}$ 作 $\mathrm{BE}$的垂线交 $A B$ 于点 $F, \odot O$ 是 $\triangle B E F$ 的外接圆, $\odot O$ 交 $B C$ 于点 $D$. + + +过点 $\mathrm{E}$ 作 $\mathrm{EH} \perp \mathrm{AB}$, 垂足为 $\mathrm{H}$,若 $\mathrm{CD}=1, \mathrm{EH}=3$, 求 $\mathrm{BF}$ 及 $\mathrm{AF}$ 长." ['连结 DE.\n\n$\\because \\angle C B E=\\angle O B E, E C \\perp B C$ 于 $C, E H \\perp A B$ 于 $H$,\n\n$\\therefore \\mathrm{EC}=\\mathrm{EH}$.\n\n$\\because \\angle \\mathrm{CDE}+\\angle \\mathrm{BDE}=180^{\\circ}, \\angle \\mathrm{HFE}+\\angle \\mathrm{BDE}=180^{\\circ}$,\n\n$\\therefore \\angle \\mathrm{CDE}=\\angle \\mathrm{HFE}$.\n\n在 $\\triangle \\mathrm{CDE}$ 与 $\\triangle \\mathrm{HFE}$ 中,\n\n$$\n\\left\\{\\begin{array}{c}\n\\angle C D E=\\angle H F E \\\\\n\\angle C=\\angle E H F=90^{\\circ}, \\\\\nE C=E H\n\\end{array}\\right.\n$$\n\n$\\therefore \\triangle \\mathrm{CDE} \\cong \\triangle \\mathrm{HFE}$ (AAS),\n\n$\\therefore \\mathrm{CD}=\\mathrm{HF}$.\n\n $C D=H F$. 又 $C D=1$\n\n$\\therefore H F=1$\n\n在 Rt $\\triangle H F E$ 中, $E F=\\sqrt{3^{2}+1^{2}}=\\sqrt{10}$\n\n$\\because E F \\perp B E$\n\n$\\therefore \\angle B E F=90^{\\circ}$\n\n$\\therefore \\angle E H F=\\angle B E F=90^{\\circ}$\n\n$\\because \\angle E F H=\\angle B F E$\n\n$\\therefore \\triangle E H F \\sim \\triangle B E F$\n\n$\\therefore \\frac{E F}{B F}=\\frac{H F}{E F}$, 即 $\\frac{\\sqrt{10}}{B F}=\\frac{1}{\\sqrt{10}}$\n\n$\\therefore B F=10$\n\n$\\therefore O E=\\frac{1}{2} B F=5, O H=5-1=4$,\n\n$\\therefore$ 在 Rt $\\triangle O H E$ 中, $\\cos \\angle E O A=\\frac{4}{5}$,\n\n$\\therefore$ 在 Rt $\\triangle E O A$ 中, $\\cos \\angle E O A=\\frac{O E}{O A}=\\frac{4}{5}$,\n\n$\\therefore \\frac{5}{O A}=\\frac{4}{5}$\n\n$\\therefore O A=\\frac{25}{4}$\n\n$\\therefore A F=\\frac{25}{4}-5=\\frac{5}{4}$.'] ['$10,\\frac{5}{4}$'] 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Multimodal Competition True Numerical Open-ended Plane Geometry Math Chinese +40 "已知椭圆 $\mathrm{C}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(\mathrm{a}>\mathrm{b}>0)$ 的离心率为 $\frac{\sqrt{3}}{2}$,并且过点 $P(2,-1)$ + + +求椭圆 $\mathrm{C}$ 的方程." ['由 $\\frac{c}{a}=\\frac{\\sqrt{3}}{2}$, 得 $\\frac{c^{2}}{a^{2}}=\\frac{a^{2}-b^{2}}{a^{2}}=\\frac{3}{4}$, 即 $a^{2}=4 b^{2}, \\therefore$ 椭圆 $c$ 的方程可化为 $x^{2}+4 y^{2}=4 b^{2}$. 又椭圆 $C$ 过点 $P(2,-1), \\therefore 4+4=4 b^{2}$, 得 $b^{2}=2$, 则 $a^{2}=8 . \\therefore$ 椭圆 $C$ 的方程为 $\\frac{x^{2}}{8}+\\frac{y^{2}}{2}=1$;'] ['$\\frac{x^{2}}{8}+\\frac{y^{2}}{2}=1$'] 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x44/KklqJLU9ABzRTY/uU6mMKKKKACiiimgCiiiqAKKKKAClFJSigBaKKKAH0UYoqBhRRijFABRRRQA09aQUp60lABiiim5oERSHGSegrzHx1qxbUls42yrjsa9Hvn22sjegrxrVEk1DxTEyfMoJFJiZ3vgvThDaxyc5xXZgVmaDaiDTUBGCK1RTGgFLSZooAU0iilpRQMdRRRQAUUUUAIabTjTcUDCijFGKACijFGKACijFFABRRRigAooxRigAooxRQAUUUYoAKKMUYoAKKMUYoAKKMUUAFFFFABRRRQAUUUUAFFFFNAFFFFMAooooAbRQ1NoEPFNelFMkNAyrcdDXk3jY/6Zb/9dl/mK9XuOhryfxt/x+W//XZf5ipkRI9Y0r/kFWv/AFzFWJolkQqRwar6RzpVt/1zFXWHFSM8j8V+H10m+OpRMQ0jjgGu18JanHcWgVpASFqz4k0xNSsCjLnHNee6HfjSdTeGRtql8CgnY9gZgeho7VVs7hbmEOhyMVazVFBS0nelplISiiikIDTKeelMoEFKOKSnLy+KAI5JMNwKcYo5oTlFJ9xSzwjtTYyVGKLDscB4l0idZhLAHGD/AA0nh3WJ7crDcArnu1d1dQrKpBFc1q2ioI2mjU7h0qWQ2dJBdrMgKEHNWVI9a8zstfuLC78mR8IK72wvkuoVdDnI5oQR3NLNGahyafmqNQoooxQAUUUYoAKQmlppoASiiigBKKKKACm06m0wFFLSClpAKKdTRTqAHCnU0GnUAFFFFAC0UUUARvUatzU5TcOKiERzQSyZZOOlSL8/BqIRsB0pQ5Q80xGT4m01LrTZXPVE4rzfwjevFeNFIMfvD/OvUtWuB/Z86+qGvF4LsW3iSOPONzE/rUyEz2+Bg0YI6VLVLS332StV2mMKKKMUAFFGKKaAKKKKoAooooAKUUlKKAFooooAUtik8ymvTAOagZPmkLYpoPFI3SgB3mUeZ7VERzSgGgB5ozSjmn7RigCEtSE1IQKdtGKBGNqs2zT7g46Ka8z0GA3+rCVR0cj9a9I1kD7BcAkcqRXI+CrZVZj1/eH+dJ7ie56FaJ5VuF9KmqL+VKCaYyTFAHNMyaUMaAJMGjOOtJuNJnPWgY7dRvpMD1pKAH5pc1GDRmmBIRUZbFBY03HNILi7qN1GBRgUBcN1G6jAowKAuLmjIoAGaXaKAuJkUm6jAowKAuG6jdSYFJigLj80bhQOlGKAuG4Um6kwKMCgLi7hRuppAoxQFx26jcKbigAUBcdupdwpuBTsUBcM0m+nYFIVGaAuJ5lHmUbBRsFAXDzKPMo2CjYKAuG+l3UBBml2imFxN1G6nYoxRcLjd1G6nYpMUXC5Gx5xS4oIy4qRhgUAMBprqSM0zJzTmY7aAKdxwDXk/jb/AI+7f/rsv8xXqV25ANeVeMjm6t/+uy/zFS9iWet6P/yCrb/rmKunpWfpTf8AErtv+uYq6GzRYQkib0KnuDXmHjHR2tLqO4iGDuycCvUgRWdrOnR3tm4IGQOM0hPc5bwbrYkgELnnI4rugQwyDXjyh9G1kDJCjmvUdJvVu7GOQMCT6U0ykzR70ZFMJNJTuWiSio91G6gCQ03FJmk5oCyHYoHBBpvNGaBaD9xPU0YFMFLmncAOCOahliV0KkcGpQuaGT3pWFY5PVPD0crGRYwGzWXDqbaLIElyVz0xXdNEGBBxXOa9oP2yMsmQQO1LYVrGrpWrQ6lEHQgH0zWmDk4rxo3174av1Qq7R9ST0r0PQfEcWpW6Esu89s0rsfMdJmk3UisGAIPWl4p3HzBv9qXdScUZFMLi5pCaM00mgLi5ozTc0ZoC4tFNJ9KQ0BcfkU2ozQGphckBpc0xTThSHceDTs0zNOBoGOFPpmaM0APopnNAzmmBJRTc0m6nYCYdKYDzSqeKafrSJJPNX0qCQ5OaTFPUZHNAjO1GJprdlB5IxXlWoaJLB4kimPGMmvZHUEYxXC+LCsN1v2jgelSxM6nRZx9hRa1Ac1yXhe88+BBmuqQ00NElN3ilJ4qEkUAS7qDIB1qMGq15L5ZHOKYFzzRTgwNYq3yMcCQVeil3Dg0cwy7RUKP61JuouIdSimZpQ1O4D6XFNDU/NAFWaXYM1SbUkXvVq4jMi4FZUunttJANSBMdXT+8KeurKe9cHq+ptY37QlHIHsapReIyXxsk/I0riuel/wBpJ6ig6mg715nc+JzB1ST8jVePxd5hx5cmf9007oLnqY1QdmFP/tMY615gPE5B+5J/3yae/ikqygxyc+xpXC56R/ag/vCnjUxt+9XnH/CScfcf8jSXHigww7tkn5Gi4XOg8SaqywlVbqKzPB13sVtxx85/nXI6n4k+1J91/wAjRoWuCBT8rfepX1F1Paft6+opRfr615wvidm6Ryf98mn/APCSP/zzl/75NO47nov29fWj7evrXnX/AAkj/wDPOX/vk0f8JJJ/zyl/75NFwuejf2inqKT7eOxFedf8JHJ/zyl/75NA8SOP+Wcv/fJouFz0b7cPWk+3j1Feef8ACTP/AM85f++TTv8AhJH/AOecv/fJouO6PQxfj1oF+vqK89/4SRh1jk/75NNHif8A2H/I0+ZCuei/2gnrUonyMg15v/wkvP3H/I1OvjFVAG2T8jRdBc9B86jzq8//AOEzX+7J+Ro/4TNf7sn5Gi6C56B51HnV5/8A8Jmv92T8jR/wma/3ZPyNF0Fz0ETc07zq88HjJf7r/kad/wAJkv8Ack/I0XQXO+86jzq8/wD+EyX+6/5Gg+M1/uyf98mi6C535m96Tzq4A+MV/uv+RpP+ExH9x/yNF0O56EJuKUzV56PGS/3H/I0p8Zr/AHX/ACNF0Fzv/OpPPrgP+ExX+6/5Gk/4TEf3H/I0rgd/59Hn15+fGS/3X/I0n/CYj+4/5Gi4XPQfPpfOrz4eMl/uv+Ro/wCEyX+6/wCRouFz0ITe9L59efDxiv8Adk/I0Dxkv9x/yNFwPQvPpRN7157/AMJmv91/yNKPGa/3X/I07oLnofnD1o84eteef8Jov91/yNH/AAmi/wB1/wAjSuFz0Pzh60ecPWvPP+E0X+6/5Gj/AITRf7r/AJGi4XPQvPHrTftB9a8+PjRcfcf8jR/wmK/3X/I07hc9B88+tHn+9ednxsg/gf8AI0v/AAmaYztk/I0riPQ/P96PtB9a89HjJD/DJ+Rpj+NkU4Kv+RouB6Ks43DmpXmG3rXmn/CapuHyv+RqU+Nl6bH/AO+TRcLnfmUZqTIK15yPGik/dk/75NTf8JqoXlZP++TRcLnZXQBU15T42G24tz/02X+Yrek8aIwPyv8Aka4HxRr63c8WA3Ein9RQwZ7hpT/8Su2/65iryscda830/wAXrHYQptfhQOhq2vjNf7r/AJGlck9ADHNPbDoQe4rz0+NUHVH/ACNH/CfQqOQaQD/GWklt88a8gVR8Ia80U4s5WxtHSo9Q8YQXtu6bWJNcH/aMlrqrTxhlGRzigD6JhlWSMMD1p7soHWvPdM8UBrSIGT5sc81ffxBuXG8fnTuUmdd5i+tODr61xn9uMOr0DxB/tVVx3O23rSb19a4w+IsL979ajHiLJ+8fzouFzt960b1ri/7fP9/9ab/wkB/vUXA7betLvWuKPiLHV/1pP+Ej/wBv9aLhc7TeBQZK4weIx/f/AFpx8Q4/iP50XC515enh4ypDdxXGDxGvdv1o/wCEjXHLfrQwuaGu6BZ6pE+4fORwAK4B7a58M3pYArEO5NdcPECZzu5HTms3WLu21G3ZHAJqCDV0PxVDdqEaQZHFdVFOsgB3da+cXvrrS9UJjZgm/tXouj+LQ0CiRyDgdTQFz08MvrTfNWuLHiMMPlkB/Gga/n+L9aaZSZ2nmp600yL61xh8QAHG4/nSjxACfvH86dx3Ox81fWjzV9a5Ea5no1L/AG3/ALdFxXOu8xfWgutcedcI/joOuN/fp3Hc64utQvKoPWuWOunH36ibV3JzuNFwudesynvUolX1rihrTJyTxThr4/vUXC52nmr60olU9640a8D0b9aU66AM7j+dLmQXO0Ei+tSb19a4geIh/f8A1p3/AAkP+3+tO4XO13r60b19a4r/AISH/b/WkbxD/t/rS5gudtvX1pd6+tcP/wAJEf7360f8JF/t/rQFzsJbsRjrUBvx61yra6jfebP400a1Ef8A9dO4XOq+3D1qVb0Y61yP9sRgZP8AOpE1uHHX9aLhc6p7wetcN4zn3sQD2rQbW4j3/WuT8V6kjgkelJiZ0ngpv3MddyjGvJfCXiCOCJASQRXWv4sgjXINCYJnXluKjOc1wUvj2FWxg1GvjyJmxhqd0O56GprP1iN3tWKDkCuYn8aQwwI+Dk1Sfx7bupVg2CKVwuc5qniK60i7IlbaB71p6X8RYSih5hn61xnjLVLXUWkeNMVwHnPG+UOKlshs+pdL8TWl6mRKDW2l5HIflbNfKdh4kv7NxtmOK7DSviQ9s6+c7MO9CkNSPoDzDSiQ15bYfEiG7cALJz7Guus9eW52bc81aZSZ06ualDVVhbcuasrVIBTQVUqcinOMVAG+bFSIzb7Q7e9JYom49yKqQ+FrWMk7UP4V0arkUx+OlAWMKfwvZzYzHGPwqGPwjZI4YIn5VvljmnrzQFjC/wCEZtM58pPypJ/C1rPgiJBj2rfLbacJDjtQM5ceEYM9F/KrUnhe1lthGYkPuRW75h9qkD/J0oA8w8ReGILaPIRR+FQeFvD0Fyh6fexXW+KUaWzkYD7q5rA8BXIcMuc/Of51NtRPc7C00CytwN0MbH6Vc/sux/59o/yq95YNJ5YqhlH+zbD/AJ9Y/wAqP7MsP+fWP8qubBTSlAFT+zLD/n1j/KmNpdiTxax/lV3ZShAKAKA0qy/59Y/yp39l2X/PrH+VXcCjFAjNn0azmQqIIx+FZv8Awiltn+D8q6Sm0BY5/wD4RO2/2fypn/CJwei/lXR0u40BY5r/AIRK39F/Kj/hErf0X8q6XcaNxoCxzX/CJW/ov5Uf8Ilb+i/lXS7jRuNAWObHhK39F/Knf8InB6L+VdGG5pdxoCxzP/CJW/ov5Un/AAiVv6LXTkmm7qAsc0fCVv6L+VJ/wiUH+z+VdNmjNAWOZHhKD/Z/Kg+Erf8A2Pyrpc0hY0BY5n/hE4P9j8qP+ETg/wBn8q6XdTd1AHN/8Inb+i/lR/wikH+z+VdJuo3UBY5v/hFIPRPyoHhO39F/Kuk3UoaiwWOb/wCETg/2fyp3/CJW/wDs/lXR7qd+NMDmv+ESg9E/Kk/4ROD/AGa6bPvThzSsFjl/+ESg/wBn9aP+ESg/2f1rqsCjAosFjlf+ESg/2f1o/wCESg/2f1rqsCjAosFjlf8AhErf/Z/Kl/4RO3x/D+VdOeKbupNBY5BvA8LPu3L19KlHg2DGMr+VdVuNG6lYLHMr4Rt1XHy/lVW78Cw3JUhwuD6Guw3UbveiwWORTwVAhXkGp38JW4Xov5VvT3LRyACqF9r8FkhM7hcUiTLTwxBG33VP4VNJ4at5F4VR+FVf7duJ23QAFOuavQeI7Y4SWQBz2FAGVceE4iDjaPwrzzxXoi2c8GDnMoH617L9qWVdy4IrzPxuc3Fv/wBdl/mKGDOr0zwvA2nwPhclAelXR4Vg9F/KtnSkH9l23/XMVdAxQBy8vg2CZdu5R+FZ8vw5jZiRcj/P4V3XPamMz00ikjh4vh/FCcmcN/n6VQ1zwVElmWRhu9hXobbvSopbVLhCsh4wadgseE6BpskmqTQT3JjAfC5r0WDwISqt9tPIz1rA8VaHNp9/Fc2aHG/LGuz8K6yNUsMyEblGOPapsSU38EMyY+1Gof8AhAmz/wAfhrtVxjinYqki1E4p/A7Ff+Po01PArA/8fRruAOKUKOuKdh2RxP8AwgbZz9rbFOTwQYz/AMfJNdqOlIV70WDlOJl8CtI+4XZFM/4QFv8An8NdxijbRYVjiB4DdWB+1mpz4MJTb9orsQKXbRYLHDHwE5P/AB+Gm/8ACBP/AM/hrucU00mhNHFDwC//AD+Gmt4BcdLtq7tMUrfSpsFjz5/hpDIpZpwT71yHibwZdaeC1tK5AH8Jr2eUyIeBxUVxZreW7JIoO4elFgseGeHbGSWRYrm8KN3DGvQrPworxhhdbge9YniXwpNazvc2aNuHSk8P+KbmzmFtfEIBwKaQkjfbwOZDn7URR/wgLhc/azXVWV3HcQq6NkGrgYEU7DscengxlUD7SaX/AIQ5v+fk12G1aNq0WCyONbwc3/Pyab/wh7f8/JrsmVabtSnYdjkP+EPP/PyalTw0I12mQmuq2pTWTmiwWOVuPCv2iLaJipqr/wAII/8Az9mu0C4pwosFjjE8Curg/azU0nggyDH2oiuwAp4osPlOH/4QB/8An8al/wCEBb/n8NdzxRRYLI4b/hAW/wCfw0h8At/z+Gu6pCKLIXKjhv8AhAX/AOfs0f8ACAP/AM/h/Ou5xTwKOUOU4T/hAJP+fxqRfAjoebs13+eOKiaiwWOKbwWzIV+0mo18BSf8/jV2uacDxRYLHEnwFIpH+mNXM+J/Dhs0KmbdxXrbsa4TxiDJLt9RUtEtHMeGvChvEQi4KZ9DXVDwCzDBuzU3hG3aOKOu0iFNIaRwjfDlTybinRfD5I2B8/Nd6QMUzbTHY4c/D4St81ySOwqlfeAEt0LfaAAK9JQCsnxBFJJZyCJcsRxSaCx87+L7VLC4kiSQNj0NcjFbT3Dfu42b6V63N4FutWvd13EwB611ui/DXTrMKxzu75FTYix4jpfhS/vZVBtpVB74rvtM+EzXIUySsn1r2Ox0W1tFASNePVa0PKVfugD6CqUSuU81s/hdHagYuASPrXU6X4eWwAy2/HSui2ijbTsOxFGgUAVMBShcU/bVICNiSKjCCnkUlSAoOBTSM0tFADCnNOAxS0UANIzSgcU7FKBQA3YKd2xTsU2gDJ1yMHSbo452GvOvAU4indScHzD1+temamu/T519VIry3TIvsOrhNwXLk4/GkxM9dWYsMipBITVGxbfbhs5q2opjJAc0hNKtI3SgBu40KxpKFoAfRRRQAhFBFLRQAmKMUtFACYoxS0UAJijFLRQAgHNGKWigBMUEc0tFACYoxS0UANpCKfTSKAE203YKfmkoAbsFGwU6imA3YKUIKdRVAIEFPCikpwNKwCbBSdKfmmHrTAM0ZpKKAFzRmkooADyKZg0+jFFgG4pcU7FIelKwEdNZsUpY1Uu5vKgd89BmhoRh+ItcisW8snDkcc1zlnZXWvTiSX5oW6gisTVrt9Y8YWyKSYwcEdq9c0zTorS3VVUDHpUWEjPs9GitrcIqY4rivFtk9hMJovlwM16gy46iuB+IUipbsBg/LxSYMq+Edba7iSF3yTWT44H+kW3/AF2X/wBCFZ/gAyNfxsc4rQ8bn/SLb/rsv/oQpEnqWlD/AIlVr/1zFXKq6V/yCrX/AK5irlXY0sIOtONA60HpQA3AqJhipaaRmgRk6xZLc2UgK87eK820q+bQNXFo5wruTgccZr1uVQYyvrXmXjrRnUveQqQUHAFTJCaPRbO5W5hDqRg1cVRXnvgPWy+nQwTn9533HmvQ4WDrkGmmNMCMU7HFOYUg6Uykw20Y4paQ1QyPFJTiKbipJAU6mgU6gBCKQ06m0AAOKUsTSUUAKfmxmgHAxQKSgCC4gWZCrAEGvP8AxH4TLt5sCbWzuz716LjimyQrIpBUHik0FjyXTNeu9GuBDduduenSvRtN1WC/thLEevvWL4l8NJcwO8ahXA4OK4bS7688PagsUpkeJfUcUtSdT2qPDDmmScdKxdL12G8iVgwB9K2UYScg5ouUIKMU8rim1SAbSikpRQAtLikp1ACg04UmKUCgY7tRiilFAXFprcU4ChxxTQXGCpBUYFSCqEIx4ptK1LUsBmwUEcU/IpppARPXm3jS9MWsxwg8EV6LcnbA59BXjviaV7rxVAucg0mJnovhYb7NGrqFGKw/Ddv5OnR8Vu0xgelJilooAAcUyTD8Gn00gk0ARiGMEYWpRgAUgBo7UgFL7ajM4HVgKbJ92uY167mtEZ1LYHpVAzp/tB7HNMW6yetcnpWtpNFhn+bHSq76hPPeqqBgN3alcLnexS7h1zVis+xDeUuetaHaqQERFNxTyKTaakBuKMU7aaNpoAbijFO2mjaaAFQVJtpg4pd1ACkU0080w0AUrobonX2ryPxGZNP8SREcLya9idc159460kyyG5VfujrQwZ03hy+F1p8fPNdAtefeBpyIo42NegrSTEmKBSEU4DNBFMZERTloIoUUAPooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACmYp+KKAGYpuKkIpuKAG0UuKMUwDFGKdtNJiqAKXNJRmgApCKKUDNACYoxTtpo2mgBuKMU7aaNpoAQDmpAKZjBpwNAwqNhxUhpGHFICs1Y2vOY9LnYdozW4y1k63btNps6AdUIpMR5T4Bxfao8snJSUgH8a9xQhYxXinhO2bRtRdJRjdKx/U16sNQj8vqKmOhKLk8wSN2J6V5T4yvft94sKHIPGBXV+INbEVuVjbk5FcXo2mXOpaiJpBkbyenakBu+EdFFnapKVORWN425uLb/rsv/oQr0mG1Frb7AMCvN/G3/Hxbf8AXZf/AEIUmJnqmlf8gq1/65irlVNKH/Eqtf8ArmKuYqywHWg0AU7FAyM05AO9LikbjpQBFIOao6lYR3tq8bgfMPStA8mg8jFMDx2W2fQfELOMrCOlem6LfrdWyENnIrm/HGkPPZPJEvz5rK8G6z5Uxtpm5TgZqNmRsz1DJ70oNRLMsqKVOakUcVSKQ6kNLSGqKA00inUmKkkZilp2KMUANptSYo20AR0VJto20AMBpKdRQAynK3rRijA7UxjJFWQEHnIrmNe8NQ30bOB83bArqMYpkg+Q4osKx5C8V5oNxuCv5YPc12WieJobhQjyDd0rW1fSIL6zIdMtivLL3S77SNQ8yPhA1Q0JnsyTLKu4HinVxnh3xEs0SJI3Pua7BZFkjBU5zVIaYtKKWLBPNPIoAZThSgU4CgAFOFApwoASgU6gUAAoPNAopoBKUUuKWqAjIpaMUuKlgGKY1SYprCkBn6nIItPnYnGFryrT4P7W1xZ8Z2uRXf8Ai68+zWDRk/6xSK5XwHZMpZ2HVyR+dLqLqej2EIhtQvTFWsUqjA9qXaaYxuKMU7aaNpoAbinqM0m001m20AS7RTSBUfm00y0gEkHFYeuWqT2bg1p3FysYyxrG1G8E1q4Q/lTuM810CZm8Wz2ZY7FHrXrFlpUCosncjPSvHo459O8TyXWdu8gH9a9j02582yhYHkrSRKNeIBVwKlPSoI845qU1YwIooJoqQCiiigAooooAYWxQG5prDmhQc0ATZpcU0U7NAEbLWN4htRPpEoK844rbNRTxrLEUboaAPKvD87WGqCNuAB3r1S1bfbxt6jNeUeIlOl6y8q/cHeu+8N6vHfWcahwSF5pLclbnQKKG6UgdfWgkYplDTQozQaVeKAH0YpM0u4UAGKMUbhRuFABijFG4UbhQAYoxRuFG4UAGKMUbhRuFABijFG4UbhQAYoxRuFG4UAGKMUbhRuFACYpMUuaCaAEpp606igBmKAOaWjNMB1NNLmkNUAmaSg0UAFOWm0oOKAH4oxSbqM0ALRSZpaAEY4GabupXHy1HigpEgOaU01admkAwrUMsW9Sp6GrORSZpMRx+oeFHur1ZoHEeM9KqNpd7HdfZ/MbHrXeLtzk0SCMgnaufXFKwrHFw+GpGcNO+8Z6GuhtNLt7VBsiUH2q3s5HHH0qbotJisULgYFeVeNj/AKTb/wDXZf8A0IV6tdj5TivKPGv/AB9W/wD12X/0IUhM9Z0r/kFWv/XMVczVTSv+QVa/9cxVyqKEpaTFLQUgppGadSUAJto20tFMCrdWiTwlWAIPrXkfiLTZtG1MTwkgSSdhXsjnK4rnfEOkre2rNt5QbhU2IaGeGb03FoN7ZYAV0y9K8l0TXJbC/wDs0o2rvxzXqNpdR3EQdWyDTQ0WqQ0bhRnNUUFFFFSIKKKKACiijNABijFJmjNADaKKKACkxS0mapDQhFNK54p+aTPNAxhXPX0rPv8ASIL2FwyLn1xWiRSQuSG3CloKx5D4g0y50S6MsG4qOy1reGfFZkCxTgg+rV3WoafFeKVdVOfUV57r3hh7OQ3FsGznoKnYjY9EtbpZlBBBzVwHNeW6H4muLaYQXQ2ANjJr0myu47yEPCwYU0x3LYpwpvQZNKpB6UxjhTqaKXNAh1FGaKBhQKKBTsA7NFJS5pgNIpRRmgCpYhQKa5A5NPzWTrV+tjZu5YDg0rhc4b4hX4lvLKGJwcthgK6Xwzp4t7VT6jNeX6K8viDxHcFiWEcvHtXtGnweVCq46KKSEjRUcU6kXpS5qigoozRmgAqKQEnipc0ZFAMq7T6Ggxn0q3x6U7IpCOf1WyluEIjYqcVn6fotxFFiZy/PeuvIBHQVGQMYwPyosFjzbxP4cllCvACG3Dp9a7HQNMe2sYhK+47a02jVhgqD+FORccAYFFgJgoUdRSE8U3tSnpVgJTqbTqgAooooAKKKKAEpaKKACiiigApr/dp1IRkYoA4Xxnov2yyZkXLE+lc94Qv/AOzL14ZTjJ2816hc2yzRFSOK8s1+xbT9REyDA3ZpbCZ6pA+9FYcgjNWQK5vwxqqX9oMsMqMdfSukHWqRSHYpopx6VGvWkIkooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKSlpDTQCUUUVQBRRRQAU2nU2gApRSUooAUU6minUAIelNxTj0pKCkAooopDA0mKWikJjTRnmg0mKCR4FI/SnCmuM1IFS4HymvJvG3/H3b4/57L/6EK9auOhryXxt/wAfdv8A9dl/9CFJiZ6zpX/IKtf+uYq5VPSv+QVa/wDXMVcqhhRRRigApMUuKMUDExRilxRigCNqY6h0ZT0IIqVl+Wo1+8aBnmHi7QHt7wXVumAMknFavg3WhNFHbSN8/pXV6xZrdWkikZyK8z2vousbugFTsRsetLyM1IDjiqGj3IurON92SavNwwouCHYoxS0YqjQTFGKWigBMUhp1NNACUUUUAFFFFACHpTacTxTaACkxS0UAIaDxQRQTQBEzYNRXFuLiMgipiuTSg4FFiGcNq3haN5PMCc5z0qDT9Sk0Vhbk4Wu8lUSAgisTUdFiuUYhAG9amxNjV07UYtQt12tk1dA8vOK82jubjQ7872byx2rttN1iK/hXBAPei4XNTNOFMGMdaeK1KHU+mCn9qC0FFFFACYpaKKQmJtpacBULtioZDYkkm0ZzXnXxE1fGnrBA37zdgge9dnf3q28LMx6etea2kTa/4smjcb4hgjPTr/8AqpCNnwH4dWxja7KYeYbiSK9ERQAB7VWsbNba3SNRwoxiroGKoocBxRQOlFMoTFGKU0VQxMUYpaO9AmFFFFKxIUhFLRTATFFLRQAUhFLRQA2nU2nVABRRRQAUUUUAFFFFABRRRQAUUUZoAa4G2uT8UaaLm3JC5IFda3SqlxAs0ZU9xQB49oGqzaPqgt5CQrOev1r2Kzu1uohKpBB9K838UeGzFcC6iT7nOcVo+D9eUxpZyOPM7ikhI78twaRetRo+5cjkVMopjHUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFIaWkNNAJRRRVAFFFFABTadTaAClFJSigBRTqaKdQAh6UlKelJQUgooooGFFFHapYDTRQRQOtArDxSNxThTXqRFS4PBryXxt/x92//XZf/QhXrNz0NeTeNv8Aj7t/+uy/+hCkyWes6V/yCrX/AK5irlU9K/5BVr/1zFXKoYZpKMUYoAKKMUYoAKKMUYoAUnjFRYwSaVyaiYntQO4+Qqy49q4TxZpgKSTIOa7I5qpqFr9rtyhHWiwjkfCHiDZcG2fGF4r0LzBIFI7jNeQX1g+i6kZ1G0M4zXo+g6it/bKQwJUAGkl3FE26WgEU8YNMq42m81NgUhAoC5DvNKPm60pAzQMCgLiYpM0E8UygLj80UwGnA0BcUjio81K3Soz1oC4ZozSU3NAXH0VHuo3mgLitUZp2c07bTER00rmQHFTbaMCkBz+t6Qt7Gdo5J/hrAS3m0aRSS20nvXf7BWVrWn/bbYgDJA4osKwmnazBcIB5qk1rxyK3Q5ryKZL/AES73EEIDXYaB4iS8RFZ/mpXC52QYDvS76qJKGAIPBqygyM0XYXY9eafikUYp4IouyrsiLYoV808rmkxtp3FcC2BVW4lCIxJxipJHwPSuZ8Q6ulnbMN3zEVLJucx4x8QnD2kRyzelbXgXSttpHeyL+8cck9a4zTNMk13XEunXKAkV6/ptotnarEo4FCBF0L70/FAxS9qpFIbiilpKZQGig0VQwo70Ud6BMKKKKCQooooAKKKKACiiigBtOptOqACiiigAooooAKKbSjNAC5opaXA9aAG0jdKCwHeo5ZAE680AKCScVKEyKyvtRDdKtR3RI+7QMrataLcW7IRnIryHV4bjQNXe7TcIx2Ar2h2EwIPFYusaDDqFsyEDmh6ktXIvDeuR6jZRDcN+ORmuo4KgivDYpbrwtrbl9xiZ8LnoK9X0fWY9QtUYEE45xQvMa8zboqLfxTt1AD6KbmkyaAH0ZpmTTgKAFooxRigAooxRigAopOaSgB1FNooAdRTaM0AOopu6k3j1oAfRTN49aN49aAH0hpu8etGfemAtGKKSncBaKbRmi4DqbRmlyKLgJSijA9aAB60XAUU6kBFGRRcAPSkpSRikzRcpBRRmjI9aYwo7UZHrRketSAhPFIOtOOKbwKQiQGmv1oBFI5BoEVbgcV5L42/4/Lf/rsv/oQr1qcgjrXk/jb/AI/Lf/rsv/oQpMlnq+lf8gq1/wCuYq5mqulKP7Ktf+uYq3imMSnU08UbqAHUU3dRuoAdRTd1G6gBpAzSqgNMOd1OVyDyKAHG3GKieICpvP4qKRy3SmByXijSxcWxYDkc1zHhHVzY3UltKTlnwM16Tc2/nxsuM8GvK/FGny6TqcV3EpCocnAqWJnrCyErkHipVY1y3hTXhrOmCY4U+ldKjGqRSLAJp2KjD07fSEOxTDS76aTmgBppuDTjSUAIKcBQBT1FACEUw9amI4quTzQApNMNONLtoAiINJtNTbB60eX70ARDil3UrJTdpoAXdS5pAppdpoAXNJnIp2ygIaBmRq+kQXtuysmWPSvOL2wu9Du2kjzsHpXsJjDVm6hpUV1EVZRz61LIaOU0LxQk+I5GIIOOa7W2ullTKtmvM9c8Ly2EhubeR+DuwtWPDPinaDBckIwbA3HmkI9NEnHWlDms+1u1nQMpyD3FXU+Y8Ux3LO/iqVxcbW4NWCcLWPqM4iRnbgCi4yrqWrrawlmYA47mvPLm8m1zUkjQkrvwaXW7+fUbw20QON3UV13hbw0LOMTSfMzDPIpE6m54d0OPT7MAoAx5Fb2MYpsZ2rgDHFPBz1qkVYBTu1N49adkY60xoSkpc0lMoDRQaKoYUd6KO9AmFFFFBIUUUUAFFFFABRRRQA2nU2nVABRRRQAUUUUAGKBSjFKaAENRlgKcelRMOtAEckgFUL7UILSItI+KnunWKJ3bjAzXl/iTWnvtRFrAflbg4obBs359euriUiwCuO2auxXeuCFm8gZxTvCOiGG3jllGT3rrjCgUrjtQhJHllz4x1rT9SSO6jVIieTmvQNJ1FNU05bmNtwPcVyPj/T4Fs5ZQvzKvBo+Gt40miwxE5FLqC3NTxL4ah1eDLD5l5GBXDWGo6j4a1JLSddkbtxknpXsewHrXH+MdAF8vnxJ+8QfKfQ02Nm9Y6xbzW4dn5+lWxqdrn/WV89X+pa3ptyYBMQP901pWCeJtQQNFdDn2pcyFzI92GpWv94/lS/2lbf368YGkeLf+fof98n/GnDR/F/8Az9D/AL5ouFz2b+0bf+8fyqQX0GPvH8q8X/sbxf8A8/I/75/+vS/2Z4wHH2pf++T/AI0XC57R9vg/v0fb4P79eL/2b4x/5+h/3zR/ZvjH/n6H/fNFwue0fb4P79H2+D+/Xi/9m+Mf+fof980f2b4x/wCfof8AfNFwue0C7ifhWz+FL5q+/wCVeLrY+MUbIux/3z/9en+R4y/5/B/3z/8AXouFz2XzU9T+VHnJ6145HB4xDjddjHf5D/jVhx4qik3Nckr6baLhc9b81fWkMi+teTrqOuAYMxz9DS/2lrf/AD2P/fJouFz1XcCaTcK8sGqa4Okx/wC+TS/2prn/AD3P5Gi4XPUtwo3CvLf7U1z/AJ7n8jR/amuf89z+RouFz1LcKVGGec15Z/amuf8APY/kaa2q64Okx/75NO4XPW964703cK8j/trXf+ex/wC+TSf2zrv/AD2P/fJqbhzHrm4U3eK8lOs67/z2P/fJpP7Z1z/nsf8Avk0XDmPW94qJriNTySK8o/trXP8Ansf++TVGbWda8zmYn/gJouHMex/bIf71L9qj9TXkiazqnl/fbP0qe01XU9wMkhI/3TTuFz1X7TH/AHqPtEf96vOG8Qyw/fLflSp4obuH/I0XHdHo4nQnGak3D1rzj/hKivI35oHjB8fx07oakejbh60u4V5x/wAJjJ/t0f8ACYyf7dFxXPR9wo3CvOP+Exk/26X/AITCT/pp+Ro5kO6PRSwA60wuPWvOz4vlP/PT8jTf+Euk/wBui6FdHo4cetIzj1rhY/GCCPkNmkj8YIT8wYijmQXOymcYNeVeNXH2y35/5bL/AOhCunfxdbk/6t/1rz3xVrUd1cxFVPEin9RUsTPc9KuY/wCybX5v+WYq6J4z3rz3T/Fdqmn26lJOEA6GrS+L7Qf8s3/WquO525lQ8A03cPWuO/4TSzTkpJ+v+FQP8SNNRsGGX9f8KLhc7jcPWl3D1rhj8SdNAz5Mv6/4Uz/hZmmf88Zv1/wouFzvNw9aNw9a4P8A4WZpn/PGb9f8KP8AhZmmf88Zv1/wouFzvgQV6ihiD6fnXnjfE2w3cRTY/H/Cn/8ACybHGTHL+R/wouFzu8ilV1HU1wJ+JenY/wBVN+v+FVZviVZ5+WKX8j/hRcLnpazIM+lYnibTbfU9OlTqxXtXByfEWJuESX8jTR42Zv4ZMH2NK6FdFHw7dSeHddj0x/liz3r1q2uEljDKcg14J4g1jfqBvEVxIOhwa6LRvG5SzjWR2LY5FCBHsIcU4yLivMv+E3JHG40n/CaN6Sfkad0O56X5i+tAkU4rzI+NGJ6P+RrRsfGsG0+aGouFzvN4o3iuP/4TSy9G/Oj/AITSzPRWP40XC52KyLnrUiyLnrXGL41shwQ/51IPGtkOzfnRcLnYl1I61ARz/wDXrmV8b2OejfnTz4307+6350XC50ZIHcUpYCuXfxrYHoG/Oo/+ExsZDgA/nRcLnVeYvrR5yetcwPEEDDIJ/OmNr0Wev60XC51Xmqe9HmLXLJr0Xc/rUn9vRf3v1ouFzpg608OtcuNfi/vfrTxr8P8Ae/Wi4XOm3r607zFrmB4hgHVv1pw8QQHo360XC50gcetNdgRWEdftx1P603/hILY9x+dIRoXlulxEysOCDXnut+Dts5ubdW3qSQK7Ua9ZngsPzpzazp7qQSvP+0KQjg9C8US6ZqCWepMI1HXmvStO1C3ul3wvuU15d420+1vEknsyizH+LcDWV4a8Ytpr/ZbmUkpxwcUrge13V1HDA7s2Aoya8x8R+K5rm5a0siHB96j1rxkLuEQ2zHMgxxWRotqsNytxdrkg8k0Bc6/wtoZcrc3SEM2M130RRECDtXJ2Wv2ixKkY6elacWsRORnIz60DOgVhS8HpVSCUSLlWBFWkqkAEGkCnPtUoA9aXHpTGhB0ooFFMoDRQaKoYUd6KO9AmFFFFBIUUUUAFFFFABRRRQA2nU2nVABRRRQAUUUUAITSg03GaUCgBxWmkU7dUZegDm/F1/wDYtOkA6uprzjwbaHU7/wC0SDJDkc/Wuv8AiG5aCJR0xzWX4CRIojj+8f51PUnqemWyLBbhFA4p0kmEY+lQCcdjVDVtQW2tXYsBxVlnCfELUwUeANywxjNW/hnZumlQyHIHFcZqHn65r8aqCyb8HFeweGtNGnaZHCBjFStyUbYWmTxK6lWUEEVKKRgW6VQzgfFfheG7heSNAG7ECuP0i8k0W9MUsRZAcc17NPBvXaw4rk9b8MR3KM6D5uvAqWhNGtpl9ZXsCkKmcc1rRwwf881/KvH47u70G9CnIQtzk9q9E0XXYb+2DGQbqdwOiEMB/wCWa/lSfY4Tz5a/lTI2Dcg8VYD8CgZD9ih/55r+VH2KH/nmv5VPvNG80xkH2KH/AJ5r+VH2KH/nmv5VPvNG80AQfYYf+eS/lS/YIf8Anmn5VNvpfMNIRB9gh/55rTX06B1wY1NWfMNHmGgDOOi25P8Aq1pP7Etv+ea/lWl5lIZKAMo6Lbg/6taQ6Nb/APPNa1c5NLmgDK/sa3x/q1pBo1v/AM81rWzRmgDJ/sW3/wCea0v9i23eNa1c0maAM3+w7T/nktN/sO1/55LWvkU0n3oAyv7EtP8AnktJ/Ydr/wA8krV4oyKAMn+wrbP+qWoH8O2rH/Vp+Vbufem4BNAGGPDtqP8Almv5U8eH7YdEWtqkzQBhv4atpOqLTB4Vth/An5V0Af2pfMNAHP8A/CL2+OVX8qT/AIRi2/uL+VdD5mRSUDSOf/4Ri2/uL+VH/CMW39xfyrfz70uaBnP/APCMW39xfyo/4Rm1/uJ+VdBRmgRzreGLX/nmn5Uz/hGbb/nmtdIeRTCKAsc+PDVv2RfypD4atx/Av5V0IFDCglo5ebw5bgcIn5V5p4u0uOC6gCqOZQP1r2qYcGvJ/G/F5b/9dl/9CFJia0O703w1bvpts5VMmMdqtHwxbf3E/KtLSm/4lVr/ANcxVssadirGC3hi2bqiVXfwbZucmOP8q6bNOosFjlz4NsyMeWn5U3/hCrL/AJ5p+VdVRRYLHK/8IVZf880/Kj/hCrL/AJ5p+VdVRRYLHJnwZZ5/1Uf5Up8G2mB+7T8q6gjmjiiwWOVPguz/AOeSflR/whdn/wA8k/Kuq4o4osFjlR4Kswf9Un5VMPCNmAP3KflXScUcUAcZqXg20ngZBEmfpXk11pkun65JGykRhsD0r6GccniuG8ZaGssSzRL8+7OR7Gk0Jj9D0G1ubRGKoSVra/4RW0/55R/lXPeEtUCEwuRwcV36uGXINCBHPt4Vs/8Ankn5VnX/AIVjKnywB9BXYnmmkAmnYdjzc+Dpc/fNXNN8JmO4zIdwx0Nd3sHoPypQoHYflRYLHn914RZpiVcgZ4xUH/CITf3zXo+0HsPypNg9B+VFgsecf8IhKP42oPhKX/no1ejFBjoPypNg/uj8qLBY84PhGX++1C+EplORIwr0YoPQflSbB6D8qLBY4FfDc6gASNR/wjk//PRq77YP7o/Kk2D+6PyosKxwDeHbgdJWpP8AhH7n/nq1d8Y1P8I/Kl8kelFh2OAHh25/56tTx4ducf6xq74Qr6D8qcsS+g/KiwWPMrrwvev92ZxTrPwvepy0zmvT/LTP3R+VOEaAfdH5UWCx5hfeG711+Sdwazk8Maiq/wDHxJmvWH8vPIX8qqTzwxqSwAx7VBB5NJ4Y1Xef9LlxUMmg6lEuWvpF+rV6DqXiO0tEOGTP0rzvX/GUrhxGFx7UAzndamutPVs3byY9656wt7vUrwmISbmPJFaUcV94huQvl7lb0r1jwT4ISyVZp0IJHekkJIyfCPg6YgS3LM3puq74ut0slaKLCsOwr0sxx2dsxQKMD0ryTxVqAuvEBiJypFNlM2/Bdk0wDSnPHetHxOhsLcyRsQQM8VZ8OSW1pYI28A471keLb8XkXlQkNkYoEX/AWvG9sjHJlm3YBNd6M4rzb4e6HLbW/mupB3Zr0xVzTRSGZNIpO7qam2+1Js5qhijpRQOlFMoDRQaKoYUd6KO9AmFFFFBIUUUUAFFFFABRRRQA2lzSUVAC5ozSUUALmjNJRQA4EUuRURJpQaAHNVZs5NWccVGyjJoA5bxLp5vrUnGdorhdIvjpE5jkJA3d69O1KRIYXZ8YxzmvK9duLe+vvLgZY89xSYmd3b65FNEGRsk+9YerX1xfHykJ54qroOnCNFMl6re26uxt4rGJQzCNj60JgmY/hjwysJ86ZPmySMiu7XC8CsuG8gOEiIyewNaEeSuaY0TZ4qSLB61CAaUEigCdwDVZ0BB+lSqxNO2g0xnI654div0ZtmWHSuEngvtAuMISsY9BXszQgjpWRqekQXaENGpJ9amwjD8OeKYbhFilf5+4Jrso5EdQRg5ryHW/Dd5p0puLaRlGR8q1qeHvFsgxBcqylOMuMZpgem4FLtFZVpqcVwmVYH8a0o5Nw600ND9opPlpxqMg0hDsA00jFIo+ankZoAZkUvFHlmkKEUALS4pOlLuoAOlGaTOTRQAuaM0lFAC5pM0UUAJk0mTT8UYoAjyaCalwKTAoAiyacDT8CmmgAzS02lFADgBTtopgp1ACFQKZUnakxQUhnein4oxQAylp2KMUAJnimk0PTKBEgpG4pRSPQIrTng15N42Oby3/AOuy/wDoQr1e4+6a8m8a/wDH5b/9dl/mKTEz1rSh/wASq1/65ire0VV0o/8AEqtf+uYq3TGHFFFFABRRRQAUUUUANPWkpSKMUAJRS4oxQAlFLijFAEZHNVLy3SeF1YdjitALTHQYpgeN6gs2iauCMhXkr0fQNRF7ZhiwJ4qh4q0aO7h84IMxrmua8Nas1rdC3cEAnvUoSPTqMio7eRZYwwOQalxTGJRS4oxQAlFLijFACGkxS4ooATFIRTs0hoAbikxTs0lAw20UooOKBDd1IXxTSaryyBepxQBZ8w0j3AVGJOMViXerw2yndIuR6muO1TxbJlo4VZs8cA0XC502peIYbbcC3P1rk9Q8Sy3OVhY8+9YX2O91GbzZJmRc9DWlsgsLcAhZXFQQYN+1zId8rcGq1rpB1NtqIT+Fa1to93rN8AqvHHu/CvVfD/hi3021AkRWYjOaB2uY/gzwZBY26PNENw9q7poEhjCxjAApYgkS7VXAp0kgI6UxmPrMhi0udumBXjEVjcavrQlTnOe1eo+MtQWCykizgutc/wCBLRGKyOuTnrSJ6mlaeHr5bVFwOlWLfwrI0wadQcGuyVQBgDinYFWolqBXsbSK0h8tBgVcBwKZRT5R8o8tTd1IaO9FgsPBozSAUYp2KFJopMUtMAo70Ud6BMKKKKCQooooAKKKKACiiigBtLikp1QAmKMUtFACYoxS0UARkc0DrStQvWgB+KQp1qSlzQBi6ppa6hEY2cqCO1cTP8L7Uymb7TLuyf4m/wAa9MZcmmFW64oA8nuPCl9Yv/o3muB/tGr1hpWpzgJPHIor0kL1pwH0/KgLHPaZ4eW2YSGRywPc10ATtTgKcDzQA0KaY4wanqOQHNADENTDpUSjmploAkCDFRSQBqlDjFDOKYFGWxjkQqyqfqK4rXPBkd3IZo8qy9NuRXoBcZpGjJBIA5pAeOG51TQLkKYW8gdXauy0PxXb3qKplXzD2FausaQmoQNFKuQ3XivNdQ8K6hpV00unJgDoaWwtj1mO8V1BBHNWVZXHWvKdL8UXFk4j1FyCK7zTdbtL2ENG2fxp3uO5tlQFyKblqWKdGG0EZqQDNACKaMUBTQcigBdgPek8oUbwOtJ9oTpQA0rijFKTu5FFACYoxS0UAJijFLRQAUUUUAFFFFABTD1p9MPWgBKUUlKKAFFOpop1ABRRRQUgooooAKKKKAI35puKc3WkoEOUUjinLQ9MTKU/3TXk3jb/AI/Lf/rsv8xXrdwPlNeSeNh/plv/ANdl/mKlks9a0r/kFWv/AFzFXKp6V/yCrX/rmK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Multimodal Competition False Equation Open-ended Conic Sections Math Chinese +41 "已知椭圆 $\mathrm{C}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(\mathrm{a}>\mathrm{b}>0)$ 的离心率为 $\frac{\sqrt{3}}{2}$,并且过点 $P(2,-1)$ + + +设点 $\mathrm{Q}$ 在椭圆 $\mathrm{C}$ 上, 且 $\mathrm{PQ}$ 与 $x$ 轴平行, 过 $\mathrm{P}$ 点作两条直线分别交椭圆 $\mathrm{C}$ 于两点 $\mathrm{A}\left(x_{1}, y_{1}\right), \mathrm{B}\left(x_{2}, y_{2}\right)$, 若直线 $\mathrm{PQ}$平分 $\angle \mathrm{APB}$, 其中直线 $\mathrm{AB}$ 的斜率是定值, 并求出这个定值." ['由 $\\frac{c}{a}=\\frac{\\sqrt{3}}{2}$, 得 $\\frac{c^{2}}{a^{2}}=\\frac{a^{2}-b^{2}}{a^{2}}=\\frac{3}{4}$, 即 $a^{2}=4 b^{2}, \\therefore$ 椭圆 $c$ 的方程可化为 $x^{2}+4 y^{2}=4 b^{2}$. 又椭圆 $C$ 过点 $P(2,-1), \\therefore 4+4=4 b^{2}$, 得 $b^{2}=2$, 则 $a^{2}=8 . \\therefore$ 椭圆 $C$ 的方程为 $\\frac{x^{2}}{8}+\\frac{y^{2}}{2}=1$;\n\n由题意, 直线 PA 斜率存在, 设直线 $P A$ 的方程为 $y+1=k(x-2)$,\n\n联立 $\\left\\{\\begin{array}{l}x^{2}+4 y^{2}=8 \\\\ y=k(x-2)-1\\end{array}\\right.$, 得 $\\left(1+4 k^{2}\\right) x^{2}-8\\left(2 k^{2}+k\\right) x+16 k^{2}+16 k-4=0$.\n\n$\\therefore 2 x_{1}=\\frac{16 k^{2}+16 k-4}{1+4 k^{2}}$, 即 $x_{1}=\\frac{8 k^{2}+8 k-2}{1+4 k^{2}} . \\because$ 直线 $P Q$ 平分 $\\angle A P B$, 即直线 $P A$ 与直线 $P B$ 的斜率互为相 反 数, 设 直 线 $P B$ 的方程为 $y+1=-k(x-2)$, 同理求得 $x_{2}=\\frac{8 k^{2}-8 k-2}{1+4 k^{2}} . \\quad x_{1}-x_{2}=\\frac{16 k}{1+4 k^{2}}$\n\n又 $\\left\\{\\begin{array}{l}y_{1}+1=k\\left(x_{1}-2\\right) \\\\ y_{2}+1=-k\\left(x_{2}-2\\right)\\end{array}, \\quad \\therefore y_{1}-y_{2}=k\\left(x_{1}+x_{2}\\right)-4 k\\right.$.\n\n即 $y_{1}-y_{2}=k\\left(x_{1}+x_{2}\\right)-4 k=k \\frac{16 k^{2}-4}{1+4 k^{2}}-4 k=\\frac{8 k}{1+4 k^{2}}$,\n\n$\\therefore$ 直线 $A B$ 的斜率为 $k_{A B}=\\frac{y_{1}-y_{2}}{x_{1}-x_{2}}=\\frac{\\frac{-8 k}{1+4 k^{2}}}{\\frac{16 k}{1+4 k^{2}}}=-\\frac{1}{2}$.'] ['$-\\frac{1}{2}$'] 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Multimodal Competition False Numerical Open-ended Conic Sections Math Chinese +42 "如图, 圆 $C$ 与 $x$ 轴相切于点 $T(2,0)$, 与 $y$ 轴的正半轴相交于 $A 、 B$ 两点 $(A$ 在 $B$ 的上方)，且 $|A B|=3$. + + +求圆 $C$ 的方程;" ['易知 $A C=\\frac{5}{2}$, 故圆方程为 $(x-2)^{2}+\\left(y-\\frac{5}{2}\\right)^{2}=\\frac{25}{4}$;'] ['$(x-2)^{2}+\\left(y-\\frac{5}{2}\\right)^{2}=\\frac{25}{4}$'] 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Multimodal Competition False Equation Open-ended Plane Geometry Math Chinese +43 "已知函数 $f(x)=\frac{x}{\ln x}, g(x)=k(x-1), k \in \mathbf{R}$. + + +若存在 $x \in\left[e, e^{2}\right]$, 使 $f(x) \leq g(x)+\frac{1}{2}$, 求 $k$ 的取值范围." ['$k \\geq \\frac{2 x-\\ln x}{2(x-1) \\ln x} \\triangleq h(x)$, 由题意得: $k \\geq h(x)_{\\text {min }}, x \\in\\left[e, e^{2}\\right]$,\n\n$$\nh^{\\prime}(x)=\\frac{2(x-1) \\ln x \\cdot\\left(2-\\frac{1}{x}\\right)-(2 x-\\ln x)\\left(2 \\ln x+\\frac{2(x-1)}{x}\\right)}{4(x-1)^{2}(\\ln x)^{2}}\n$$\n\n下证 $\\forall x \\in\\left[e, e^{2}\\right]$,\n\n$$\n\\begin{aligned}\n& (x-1) \\ln x \\cdot\\left(2-\\frac{1}{x}\\right)<(2 x-\\ln x)\\left(\\ln x+\\frac{x-1}{x}\\right) \\\\\n& \\Leftrightarrow 2(x-1) \\ln x-\\frac{x-1}{x} \\ln x<2 x \\ln x+2 x(x-1)-(\\ln x)^{2}-\\frac{x-1}{x} \\ln x \\\\\n& \\quad \\Leftrightarrow-2 \\ln x<2(x-1)-(\\ln x)^{2}\n\\end{aligned}\n$$\n\n因为 $\\ln x \\in[1,2]$, 所以 $(\\ln x)^{2}-2 \\ln x \\in[-1,0], 2(x-1) \\in[0,2]$.\n\n综上: $h^{\\prime}(x)<0, \\therefore k \\geq h\\left(e^{2}\\right)=\\frac{1}{2}$.'] ['$[\\frac{1}{2}, +\\infty)$'] 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Multimodal Competition False Interval Open-ended Elementary Functions Math Chinese +44 "如图, 在平面直角坐标系 $x O y$ 中, 已知圆 $O$ 的方程为 $x^{2}+y^{2}=4$, 过点 $P(0,1)$ 的直线 $l$ 与圆 $O$ 交于 $A, B$ 两点, 与 $x$ 轴交于点 $Q$, 设 $\overrightarrow{Q A}=\lambda \overrightarrow{P A}, \overrightarrow{Q B}=\mu \overrightarrow{P B}$, 其中 $\lambda+\mu$ 为定值，求该定值. + +" ['当 $A B$ 与 $x$ 轴垂直时, 此时点 $Q$ 与点 $O$ 重合, 从而\n\n$$\n\\lambda=2, \\mu=\\frac{2}{3}, \\lambda+\\mu=\\frac{8}{3}\n$$\n\n当点 $Q$ 与点 $O$ 不重合时, 直线 $A B$ 的斜率存在.\n\n设 $A B: y=k x+1, A\\left(x_{1}, y_{1}\\right), B\\left(x_{2}, y_{2}\\right)$, 则 $Q\\left(-\\frac{1}{k}, 0\\right)$.\n\n由题设得: $x_{1}+\\frac{1}{k}=\\lambda x_{1}, x_{2}+\\frac{1}{k}=\\mu x_{2}$, 即\n\n$$\n\\lambda+\\mu=1+\\frac{1}{k x_{1}}+1+\\frac{1}{k x_{2}}=2+\\frac{x_{1}+x_{2}}{k x_{1} x_{2}}\n$$\n\n将 $y=k x+1$ 代入 $x^{2}+y^{2}=4$, 得 $\\left(1+k^{2}\\right) x^{2}+2 k x-3=0$, 则\n\n$$\n\\Delta>0, x_{1}+x_{2}=\\frac{-2 k}{1+k^{2}}, x_{1} x_{2}=\\frac{-3}{1+k^{2}}\n$$\n\n所以\n\n$$\n\\lambda+\\mu=2+\\frac{-2 k}{-3 k}=\\frac{8}{3}\n$$\n\n综上, $\\lambda+\\mu$ 为定值 $\\frac{8}{3}$.'] ['$\\frac{8}{3}$'] 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Multimodal Competition False Numerical Open-ended Plane Geometry Math Chinese +45 "如图, 矩形 $A B C D$ 沿平行于 $A D$ 的线段 $E F$ 向上翻折 (点 $E$ 在线段 $A B$ 上运动,点 $F$ 在线段 $C D$ 上运动), 得到三棱柱 $A B E-C D F$, 已知 $A B=5, A C=\sqrt{34}$. + + +若 $\triangle A B G$ 是直角三角形, 这里 $G$ 是线段 $E F$ 上的点, 试求线段 $E G$ 的长度 $x$ 的取值范围;" ['由题设条件可知 $\\triangle A E G, \\triangle B E G$ 均为直角三角形, 因此 $A G^{2}=A E^{2}+x^{2}, B G^{2}=B E^{2}+x^{2}$.由余弦定理: $A B^{2}=A E^{2}+B E^{2}-2 A E \\cdot B E \\cos \\angle A E B$.\n\n于是\n\n$$\n\\begin{aligned}\n& 2 x^{2}+A E^{2}+B E^{2}=A B^{2}=A E^{2}+B E^{2}-2 A E \\cdot B E \\cos \\angle A E B \\\\\n& x^{2}=-A E \\cdot B E \\cos \\angle A E B\n\n设 $A=(1,0,0)$ ，则 $B(1,1,0), C(0,2,0), S(0,0,2)$.\n\n$\\overrightarrow{S C}=(0,2,-2), \\overrightarrow{B C}=(-1,1,0)$.\n\n设平面 $S B C$ 的法向量为 $n=(a, b, c)$, 由 $n \\perp \\overrightarrow{S C}, n \\perp \\overrightarrow{B C}$ 得到 $n \\cdot \\overrightarrow{S C}=0, n \\cdot \\overrightarrow{B C}=0$, 故 $b-c=0,-a+b=0$.取 $a=b=c=1$, 则 $n=(1,1,1)$, 又设 $\\overrightarrow{S E}=\\lambda \\overrightarrow{E B}(\\lambda>0)$, 则\n\n$$\nE\\left(\\frac{\\lambda}{1+\\lambda}, \\frac{\\lambda}{1+\\lambda}, \\frac{2}{1+\\lambda}\\right), \\quad \\overrightarrow{D E}=\\left(\\frac{\\lambda}{1+\\lambda}, \\frac{\\lambda}{1+\\lambda}, \\frac{2}{1+\\lambda}\\right), \\quad \\overrightarrow{D C}=(0,2,0)\n$$\n\n设平面 $C D E$ 的法向量为 $m=(x, y, z)$, 由 $m \\perp \\overrightarrow{D E}, m \\perp \\overrightarrow{D C}$, 得到 $m \\cdot \\overrightarrow{D E}=0, m \\cdot \\overrightarrow{D C}=0$, 故\n\n$$\n\\frac{\\lambda x}{1+\\lambda}+\\frac{\\lambda y}{1+\\lambda}+\\frac{2 z}{1+\\lambda}=0, \\quad 2 y=0\n$$\n\n令 $x=2$, 则 $m=(2,0,-\\lambda)$, 由平面 $D E C \\perp$ 平面 $S B C$, 得到 $m \\perp n$, 所以\n\n$$\nm \\cdot n=0, \\quad 2-\\lambda=0, \\quad \\lambda=2\n$$\n\n$\\overrightarrow{D E}=\\left(\\frac{2}{3}, \\frac{2}{3}, \\frac{2}{3}\\right) \\text {, 取 } D E \\text { 的中点 } F \\text {, 则 } F=\\left(\\frac{1}{3}, \\frac{1}{3}, \\frac{1}{3}\\right), \\overrightarrow{F A}=\\left(\\frac{2}{3},-\\frac{1}{3},-\\frac{1}{3}\\right) \\text {, 故 }$\n\n$$\n\\overrightarrow{F A} \\cdot \\overrightarrow{D E}=0 \\Rightarrow \\overrightarrow{F A} \\perp \\overrightarrow{D E}\n$$\n\n又 $\\overrightarrow{E C}=\\left(-\\frac{2}{3}, \\frac{4}{3},-\\frac{2}{3}\\right)$, 故 $\\overrightarrow{E C} \\perp \\overrightarrow{D E}$, 因此向量 $\\overrightarrow{F A}$ 与 $\\overrightarrow{E C}$ 的夹角等于二面角 $A-D E-C$ 的平面角,于是\n\n$$\n\\cos (\\overrightarrow{F A}, \\overrightarrow{E C})=\\frac{\\overrightarrow{F A} \\cdot \\overrightarrow{E C}}{|\\overrightarrow{F A}||\\overrightarrow{E C}|}=-\\frac{1}{2}\n$$\n\n所以二面角 $A-D E-C$ 的大小为 $120^{\\circ}$.'] ['$120$'] 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Multimodal Competition False $^{\circ}$ Numerical Open-ended Solid Geometry Math Chinese +49 "已知四边形 $A B C D$ 满足 $A D \| B C, B A=A D=D C=\frac{1}{2} B C=a, E$ 是 $B C$ 的中点, 将 $\triangle B A E$ 沿着 $A E$ 翻折成 $\triangle B_{1} A E$, 使面 $B_{1} A E \perp$ 面 $A E C D, F$ 为 $B_{1} D$ 的中点. + + +求四棱雉 $B_{1}-A E C D$ 的体积;" ['取 $A E$ 的中点 $M$, 连接 $B_{1} M$, 如下图.\n\n\n\n因为 $B A=A D=D C=\\frac{1}{2} B C=a, \\triangle A B E$ 为等边三角形,则 $B_{1} M=\\frac{\\sqrt{3}}{2} a$, 又因为面 $B_{1} A E \\perp$ 面 $A E C D$, 所以 $B_{1} M \\perp$ 面 $A E C D$, 所以\n\n$$\nV=\\frac{1}{3} \\times \\frac{\\sqrt{3}}{2} a \\times a \\times a \\times \\sin \\frac{\\pi}{3}=\\frac{a^{3}}{4}\n$$'] ['$\\frac{a^{3}}{4}$'] 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Multimodal Competition False Expression Open-ended Solid Geometry Math Chinese +50 "已知四边形 $A B C D$ 满足 $A D \| B C, B A=A D=D C=\frac{1}{2} B C=a, E$ 是 $B C$ 的中点, 将 $\triangle B A E$ 沿着 $A E$ 翻折成 $\triangle B_{1} A E$, 使面 $B_{1} A E \perp$ 面 $A E C D, F$ 为 $B_{1} D$ 的中点. + + +求平面 $A D B_{1}$ 与平面 $E C B_{1}$ 所成角的正弦值." ['连接 $M D$, 分别以 $M E, M D, M B_{1}$ 为 $x, y, z$ 轴, 如下图.\n\n\n\n则有\n$$\n\\begin{gathered}\nE\\left(\\frac{a}{2}, 0,0\\right), C\\left(a, \\frac{\\sqrt{3}}{2} a, 0\\right), A\\left(-\\frac{a}{2}, 0,0\\right), D\\left(0, \\frac{\\sqrt{3}}{2} a, 0\\right), B_{1}\\left(0,0, \\frac{\\sqrt{3}}{2} a\\right) \\\\\n\\overrightarrow{E C}=\\left(\\frac{a}{2}, \\frac{\\sqrt{3} a}{2}, 0\\right), \\overrightarrow{E B_{1}}=\\left(-\\frac{a}{2}, 0, \\frac{\\sqrt{3} a}{2}\\right), \\overrightarrow{A D}=\\left(\\frac{a}{2}, \\frac{\\sqrt{3} a}{2}, 0\\right), \\overrightarrow{A B_{1}}=\\left(\\frac{a}{2}, 0, \\frac{\\sqrt{3} a}{2}\\right)\n\\end{gathered}\n$$\n\n设面 $E C B_{1}$ 的法向量 $\\vec{v}=\\left(x^{\\prime}, y^{\\prime}, z^{\\prime}\\right),\\left\\{\\begin{array}{l}\\frac{a}{2} x^{\\prime}+\\frac{\\sqrt{3}}{2} a y^{\\prime}=0 \\\\ -\\frac{a}{2} x^{\\prime}+\\frac{\\sqrt{3}}{2} a z^{\\prime}=0\\end{array}\\right.$, 令 $x^{\\prime}=1$, 则 $\\vec{v}=\\left(1,-\\frac{\\sqrt{3}}{3}, \\frac{\\sqrt{3}}{3}\\right)$;\n\n设面 $A D B_{1}$ 的法向量为 $\\vec{u}=(x, y, z),\\left\\{\\begin{array}{l}\\frac{a}{2} x+\\frac{\\sqrt{3}}{2} a y=0 \\\\ \\frac{a}{2} x+\\frac{\\sqrt{3}}{2} a z=0\\end{array}\\right.$, 令 $x=1$, 则 $\\vec{u}=\\left(1,-\\frac{\\sqrt{3}}{3},-\\frac{\\sqrt{3}}{3}\\right)$, 则\n\n$$\n\\cos <\\vec{u}, \\vec{v}>=\\frac{1+\\frac{1}{3}-\\frac{1}{3}}{\\sqrt{1+\\frac{1}{3}+\\frac{1}{3}} \\times \\sqrt{1+\\frac{1}{3}+\\frac{1}{3}}}=\\frac{3}{5}\n$$\n\n所以二面角的正弦值为 $\\frac{4}{5}$.'] ['$\\frac{4}{5}$'] 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Multimodal Competition False Numerical Open-ended Solid Geometry Math Chinese +51 "如图所示, 点 $A$ 和 $A^{\prime}$ 在 $x$ 轴上, 且关于 $y$ 轴对称, 过点 $A^{\prime}$ 垂直于 $x$ 轴的直线于抛物线 $y^{2}=2 x$交于两点 $B$ 和 $C$, 点 $D$ 为线段 $A B$ 上的动点, 点 $E$ 在线段 $A C$ 上, 满足 $\frac{|C E|}{|C A|}=\frac{|A D|}{|A B|}$. + + +直线 $D E$ 与此抛物线有且只有一个公共点，求该点坐标;" ['设 $A\\left(-2 a^{2}, 0\\right), A^{\\prime}\\left(2 a^{2}, 0\\right)$,所以可得 $B\\left(2 a^{2}, 2 a\\right), C\\left(2 a^{2},-2 a\\right)$,设 $D\\left(x_{1}, y_{1}\\right), \\overrightarrow{A D}=\\lambda \\overrightarrow{A B}$, 则 $\\overrightarrow{C E}=\\lambda \\overrightarrow{C A}$.于是 $\\left(x_{1}+2 a^{2}, y_{1}\\right)=\\lambda\\left(4 a^{2}, 2 a\\right)$, 故 $D$ 点的坐标为: $\\left((4 \\lambda-2) a^{2}, 2 \\lambda a\\right)$.\n\n设 $E\\left(x_{2}, y_{2}\\right)$, 同理可得, $E$ 坐标 $\\left((2-4 \\lambda) a^{2},(2 \\lambda-2) a\\right)$.\n\n因此直线 $D E$ 的斜率为: $k_{D E}=\\frac{2 a}{(8 \\lambda-4) a^{2}}=\\frac{1}{(4 \\lambda-2) a}$,\n\n所以直线 $D E$ 的方程为: $y-2 \\lambda a=\\frac{1}{(4 \\lambda-2) a}\\left(x-(4 \\lambda-2) a^{2}\\right)$, 化简得:\n\n$(4 \\lambda-2) a y-2 \\lambda a(4 \\lambda-2) a=x-(4 \\lambda-2) a^{2}$, 即 $x=2 a(2 \\lambda-1) y-2 a^{2}(2 \\lambda-1)^{2}$. (1)\n\n与抛物线方程联立, 得 $y^{2}=2\\left[2 a(2 \\lambda-1) y-2 a^{2}(2 \\lambda-1)^{2}\\right]$,\n\n即 $y^{2}-4 a(2 \\lambda-1) y+4 a^{2}(2 \\lambda-1)^{2}=0$, (2)\n\n此时方程有两个相等的根: $y=2 a(2 \\lambda-1)$, 代入 (1)式得: $x=2 a^{2}(2 \\lambda-1)^{2}$.\n\n所以直线 $D E$ 与抛物线有且只有一个公共点 $F\\left(2 a^{2}(2 \\lambda-1)^{2}, 2 a(2 \\lambda-1)\\right)$.'] ['$\\left(2 a^{2}(2 \\lambda-1)^{2}, 2 a(2 \\lambda-1)\\right)$'] 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Multimodal Competition False Tuple Open-ended Geometry Math Chinese +52 "如图所示, 点 $A$ 和 $A^{\prime}$ 在 $x$ 轴上, 且关于 $y$ 轴对称, 过点 $A^{\prime}$ 垂直于 $x$ 轴的直线于抛物线 $y^{2}=2 x$交于两点 $B$ 和 $C$, 点 $D$ 为线段 $A B$ 上的动点, 点 $E$ 在线段 $A C$ 上, 满足 $\frac{|C E|}{|C A|}=\frac{|A D|}{|A B|}$. + + +设直线 $D E$ 与此抛物线的公共点为 $F$, 记 $\triangle B C F$ 与 $\triangle A D E$ 的面积分别为 $S_{1}, S_{2}$, 求 $\frac{S_{1}}{S_{2}}$ 的值." ['设 $A\\left(-2 a^{2}, 0\\right), A^{\\prime}\\left(2 a^{2}, 0\\right)$,所以可得 $B\\left(2 a^{2}, 2 a\\right), C\\left(2 a^{2},-2 a\\right)$,设 $D\\left(x_{1}, y_{1}\\right), \\overrightarrow{A D}=\\lambda \\overrightarrow{A B}$, 则 $\\overrightarrow{C E}=\\lambda \\overrightarrow{C A}$.于是 $\\left(x_{1}+2 a^{2}, y_{1}\\right)=\\lambda\\left(4 a^{2}, 2 a\\right)$, 故 $D$ 点的坐标为: $\\left((4 \\lambda-2) a^{2}, 2 \\lambda a\\right)$.\n\n设 $E\\left(x_{2}, y_{2}\\right)$, 同理可得, $E$ 坐标 $\\left((2-4 \\lambda) a^{2},(2 \\lambda-2) a\\right)$.\n\n因此直线 $D E$ 的斜率为: $k_{D E}=\\frac{2 a}{(8 \\lambda-4) a^{2}}=\\frac{1}{(4 \\lambda-2) a}$,\n\n所以直线 $D E$ 的方程为: $y-2 \\lambda a=\\frac{1}{(4 \\lambda-2) a}\\left(x-(4 \\lambda-2) a^{2}\\right)$, 化简得:\n\n$(4 \\lambda-2) a y-2 \\lambda a(4 \\lambda-2) a=x-(4 \\lambda-2) a^{2}$, 即 $x=2 a(2 \\lambda-1) y-2 a^{2}(2 \\lambda-1)^{2}$. (1)\n\n$S_{1}=S_{\\triangle B C F}=\\frac{1}{2} B C \\cdot h=\\frac{1}{2} \\cdot 4 a \\cdot\\left(2 a^{2}-x_{F}\\right)=4 a^{3}\\left(4 \\lambda-4 \\lambda^{2}\\right)$.\n\n设直线 $D E$ 与 $x$ 轴交于点 $G$, 令 $y=0$, 代入方程(1), $x=2 a(2 \\lambda-1) y-2 a^{2}(2 \\lambda-1)^{2}$, 解得:\n\n$x=-2 a^{2}(2 \\lambda-1)^{2}$, 故 $|A G|=2 a^{2}-2 a^{2}(2 \\lambda-1)^{2}=2 a^{2}\\left(4 \\lambda-4 \\lambda^{2}\\right)$, 于是\n\n$S_{2}=S_{\\triangle A D E}=S_{\\triangle A D G}+S_{\\triangle A E G}=\\frac{1}{2} \\cdot|A G| \\cdot\\left|y_{D}-y_{E}\\right|=a^{2}\\left(4 \\lambda-4 \\lambda^{2}\\right) \\cdot|2 \\lambda a-(2 \\lambda-2) a|=2 a^{3}\\left(4 \\lambda-4 \\lambda^{2}\\right)$\n\n所以 $\\frac{S_{1}}{S_{2}}=2$.'] ['$2$'] 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Multimodal Competition False Numerical Open-ended Geometry Math Chinese +53 "设 $k>0$ 且 $k \neq 1$, 直线 $l: y=k x+1$ 与 $l_{1}: y=k_{1} x+1$ 关于直线 $y=x+1$ 对称,直线 $l$ 与 $l_{1}$ 分别交椭圆 $E: \frac{x^{2}}{4}+y^{2}=1$ 于点 $A 、 M$ 和 $A 、 N$. + + +求 $k \cdot k_{1}$ 的值." ['直线 $l$ 与 $l_{1}$ 的交点为 $A(0,1)$.\n\n设点 $P(x, y)$ 是直线 $l$ 上异于点 $A(0,1)$ 的任意一点, 点 $P_{0}\\left(x_{0}, y_{0}\\right)$ 是点 $P$ 关于直线 $y=x+1$ 的对称点.由 $\\frac{y+y_{0}}{2}=\\frac{x+x_{0}}{2}+1$ 得 $y-x=x_{0}-y_{0}+2$.\n\n由 $\\frac{y-y_{0}}{x-x_{0}}=-1$ 得 $y+x=y_{0}+x_{0}$.\n\n联合解得 $\\left\\{\\begin{array}{l}x=y_{0}-1 \\\\ y=x_{0}+1\\end{array}\\right.$.\n\n代入直线 $l: y=k x+1$ 可得 $x_{0}=k\\left(y_{0}-1\\right)$.\n\n又由点 $P_{0}\\left(x_{0}, y_{0}\\right)$ 在直线 $l_{1}: y=k_{1} x+1$ 上, 有 $y_{0}=k_{1} x_{0}+1$, 则 $y_{0}-1=k_{1} x_{0}$.\n\n所以有 $x_{0}=k k_{1} x_{0}$, 从而由 $x_{0} \\neq 0$ 可得 $k k_{1}=1$.'] ['$1$'] 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Multimodal Competition False Numerical Open-ended Plane Geometry Math Chinese +54 "设 $k>0$ 且 $k \neq 1$, 直线 $l: y=k x+1$ 与 $l_{1}: y=k_{1} x+1$ 关于直线 $y=x+1$ 对称,直线 $l$ 与 $l_{1}$ 分别交椭圆 $E: \frac{x^{2}}{4}+y^{2}=1$ 于点 $A 、 M$ 和 $A 、 N$. + + +对任意的 $k$ ，直线 $M N$ 恒过定点，求该定点." ['直线 $l$ 与 $l_{1}$ 的交点为 $A(0,1)$.\n\n设点 $P(x, y)$ 是直线 $l$ 上异于点 $A(0,1)$ 的任意一点, 点 $P_{0}\\left(x_{0}, y_{0}\\right)$ 是点 $P$ 关于直线 $y=x+1$ 的对称点.由 $\\frac{y+y_{0}}{2}=\\frac{x+x_{0}}{2}+1$ 得 $y-x=x_{0}-y_{0}+2$.\n\n由 $\\frac{y-y_{0}}{x-x_{0}}=-1$ 得 $y+x=y_{0}+x_{0}$.\n\n联合解得 $\\left\\{\\begin{array}{l}x=y_{0}-1 \\\\ y=x_{0}+1\\end{array}\\right.$.\n\n代入直线 $l: y=k x+1$ 可得 $x_{0}=k\\left(y_{0}-1\\right)$.\n\n又由点 $P_{0}\\left(x_{0}, y_{0}\\right)$ 在直线 $l_{1}: y=k_{1} x+1$ 上, 有 $y_{0}=k_{1} x_{0}+1$, 则 $y_{0}-1=k_{1} x_{0}$.\n\n所以有 $x_{0}=k k_{1} x_{0}$, 从而由 $x_{0} \\neq 0$ 可得 $k k_{1}=1$.\n\n设点 $M 、 N$ 的坐标分别为 $\\left(x_{1}, y_{1}\\right)$ 与 $\\left(x_{2}, y_{2}\\right)$.\n\n由 $\\left\\{\\begin{array}{l}y_{1}=k x_{1}+1 \\\\ \\frac{x_{1}^{2}}{4}+y_{1}^{2}=1\\end{array}\\right.$ 可得 $\\left(4 k^{2}+1\\right) x_{1}^{2}+8 k x_{1}=0$. 所以有 $x_{1}=\\frac{-8 k}{4 k^{2}+1}, y_{1}=\\frac{1-4 k^{2}}{4 k^{2}+1}$.\n\n同理求得 $x_{2}=\\frac{-8 k_{1}}{4 k_{1}^{2}+1}, y_{2}=\\frac{1-4 k_{1}^{2}}{4 k_{1}^{2}+1}$.\n\n由 $k k_{1}=1$ 可得 $x_{2}=\\frac{-8 k}{4+k^{2}}, y_{2}=\\frac{k^{2}-4}{4+k^{2}}$.\n\n则直线 $M N$ 的斜率为 $k_{M N}=\\frac{y_{1}-y_{2}}{x_{1}-x_{2}}=\\frac{\\frac{1-4 k^{2}}{4 k^{2}+1}-\\frac{k^{2}-4}{4+k^{2}}}{\\frac{-8 k}{4 k^{2}+1}-\\frac{-8 k}{4+k^{2}}}=\\frac{8-8 k^{4}}{8 k\\left(3 k^{2}-3\\right)}=-\\frac{k^{2}+1}{3 k}$.\n\n所以直线 $M N$ 的方程为 $y-\\frac{1-4 k^{2}}{4 k^{2}+1}=-\\frac{k^{2}+1}{3 k}\\left(x-\\frac{-8 k}{4 k^{2}+1}\\right)$, 化简得 $y=-\\frac{k^{2}+1}{3 k} x-\\frac{5}{3}$.\n\n因此, 对任意的 $k$, 直线 $M N$ 恒过定点 $\\left(0,-\\frac{5}{3}\\right)$.'] ['$\\left(0,-\\frac{5}{3}\\right)$'] 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Multimodal Competition False Tuple Open-ended Plane Geometry Math Chinese +55 "已知 $F$ 为椭圆 $C: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ 的右焦点, 点 $P$ 为直线 $x=4$ 上动点, 过点 $P$ 作椭圆 $C$的切线 $P A 、 P B, A 、 B$ 为切点. + + +求 $\triangle P A B$ 面积的最小值." ['$F(1,0)$, 设 $P(4, t), A\\left(x_{1}, y_{1}\\right), B\\left(x_{2}, y_{2}\\right)$ .\n\n则切线 $P A, P B$ 的方程分别为 $\\frac{x_{1} x}{4}+\\frac{y_{1} y}{3}=1, \\frac{x_{2} x}{4}+\\frac{y_{2} y}{3}=1$ .\n\n由切线 $P A, P B$ 过点 $P(4, t)$, 得 $x_{1}+\\frac{y_{1} t}{3}=1, x_{2}+\\frac{y_{2} t}{3}=1$, 即 $x_{1}+\\frac{t}{3} y_{1}=1, x_{2}+\\frac{t}{3} y_{2}=1$ .\n\n由此可得直线 $A B$ 方程为 $x+\\frac{t}{3} y=1$ .\n\n由 $\\left\\{\\begin{array}{l}x+\\frac{t}{3} y=1 \\\\ \\frac{x^{2}}{4}+\\frac{y^{2}}{3}=1\\end{array}\\right.$, 得 $\\left(t^{2}+12\\right) y^{2}-6 t y-27=0$ .\n\n$\\triangle=36 t^{2}+4 \\times 27 \\times\\left(t^{2}+12\\right)>0, \\quad y_{1}+y_{2}=\\frac{6 t}{t^{2}+12}, \\quad y_{1} y_{2}=\\frac{-27}{t^{2}+12} .$\n\n于是, $|A B|=\\sqrt{1+\\left(-\\frac{t}{3}\\right)^{2}} \\cdot\\left|y_{1}-y_{2}\\right|=\\sqrt{1+\\left(-\\frac{t}{3}\\right)^{2}} \\cdot \\sqrt{\\left(\\frac{6 t}{t^{2}+12}\\right)^{2}-4 \\times \\frac{-27}{t^{2}+12}}=\\frac{4\\left(t^{2}+9\\right)}{t^{2}+12}$ .\n\n又点 $P(4, t)$ 到直线 $A B$ 方程的距离 $d=\\sqrt{9+t^{2}}$ .\n\n$\\therefore \\quad \\mathrm{S}_{\\triangle P A B}=\\frac{1}{2}|A B| \\cdot d=\\frac{1}{2} \\times \\frac{4\\left(t^{2}+9\\right)}{t^{2}+12} \\times \\sqrt{9+t^{2}}=\\frac{2\\left(t^{2}+9\\right) \\sqrt{9+t^{2}}}{t^{2}+12}$ .\n\n设 $\\sqrt{t^{2}+9}=\\lambda$, 由 $t \\in R$ 知, $\\lambda \\geq 3$, 且 $\\mathrm{S}_{\\triangle P A B}=f(\\lambda)=\\frac{2 \\lambda^{3}}{\\lambda^{2}+3}$ .\n\n$\\because \\quad f^{\\prime}(\\lambda)=\\frac{6 \\lambda^{2}\\left(\\lambda^{2}+3\\right)-2 \\lambda^{3}(2 \\lambda)}{\\left(\\lambda^{2}+3\\right)^{2}}=\\frac{2 \\lambda^{4}+18 \\lambda^{2}}{\\left(\\lambda^{2}+3\\right)^{2}}>0$,\n\n$\\therefore f(\\lambda)$ 在区间 $[3,+\\infty)$ 上为增函数, $f(\\lambda)$ 的最小值为 $f(3)=\\frac{9}{2}$, 此时 $t=0$ .\n\n$\\therefore \\triangle P A B$ 面积的最小值为 $\\frac{9}{2}$ .'] ['$\\frac{9}{2}$'] 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Multimodal Competition False Numerical Open-ended Conic Sections Math Chinese