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"value": "9.044. $\\frac{\\sqrt{17-15 x-2 x^{2}}}{x+3}>0$.\n\n9.044. $\\frac{\\sqrt{17-15 x-2 x^{2}}}{x+3}>0$."
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"value": "<think>\nOkay, so I need to solve the inequality \\(\\frac{\\sqrt{17 - 15x - 2x^2}}{x + 3} > 0\\). Let me start by understanding e... | 9.044. $\frac{\sqrt{17-15 x-2 x^{2}}}{x+3}>0$.
9.044. $\frac{\sqrt{17-15 x-2 x^{2}}}{x+3}>0$. | null | null | [
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"content": "You like to find multiple answers for a question then deliberate over them saying \"Alternatively,\" between each answer you are deliberating on and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer... | (-3, 1) | alternative_rationalized | [
"To solve the inequality \\(\\frac{\\sqrt{17-15x-2x^2}}{x+3} > 0\\), we need to analyze both the numerator and the denominator separately.\n\n**Step 1: Analyze the Denominator**\n\nThe denominator \\(x + 3\\) must be positive because the fraction needs to be greater than zero:\n\\[ x + 3 > 0 \\]\n\\[ x > -3 \\]\n\n... | [
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"value": "Return your final response within \\boxed{}. In $\\triangle ABC$, $\\angle A = 60^{\\circ}$, the angle bisector of $\\angle A$ intersects side $BC$ at $D$. Line $DE$ is perpendicular to $AB$ at $E$ and line $DF$ is perpendicular to $AC$ at $F$. Let $t = \\frac{S_{\\triangle DEF}}... | In $\triangle ABC$, $\angle A = 60^{\circ}$, the angle bisector of $\angle A$ intersects side $BC$ at $D$. Line $DE$ is perpendicular to $AB$ at $E$ and line $DF$ is perpendicular to $AC$ at $F$. Let $t = \frac{S_{\triangle DEF}}{S_{\triangle ABC}}$. Prove that when $t$ reaches its maximum value, $\triangle ABC$ is an ... | null | null | [
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"content": "You like to find multiple answers for a question then deliberate over them saying \"Alternatively,\" between each answer you are deliberating on and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer... | \text{The maximum value of } t \text{ occurs when } \triangle ABC \text{ is equilateral.} | alternative_rationalized | [
"To solve this problem, we need to analyze the geometric properties of the triangle \\( \\triangle ABC \\) where \\( \\angle A = 60^\\circ \\), and consider how the area ratio \\( t = \\frac{S_{\\triangle DEF}}{S_{\\triangle ABC}} \\) behaves under different configurations of \\( \\triangle ABC \\).\n\n### Step-by-... | [
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"value": "Define $\\lfloor x \\rfloor$ as the largest integer less than or equal to $x$. Define $\\{x \\} = x - \\lfloor x \\rfloor$. For example, $\\{ 3 \\} = 3-3 = 0$, $\\{ \\pi \\} = \\pi - 3$, and $\\{ - \\pi \\} = 4-\\pi$. If $\\{n\\} + \\{ 3n\\} = 1.4$, then find the sum of all possi... | Define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$. Define $\{x \} = x - \lfloor x \rfloor$. For example, $\{ 3 \} = 3-3 = 0$, $\{ \pi \} = \pi - 3$, and $\{ - \pi \} = 4-\pi$. If $\{n\} + \{ 3n\} = 1.4$, then find the sum of all possible values of $100\{n\}$.
[i]Proposed by Isabella Grabski
[... | null | null | [
{
"content": "You like to find multiple answers for a question then deliberate over them saying \"Alternatively,\" between each answer you are deliberating on and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer... | 145 | alternative_rationalized | [
"To solve the problem, we start by expressing \\( n \\) in terms of its integer and fractional parts. Let \\( n = k + f \\), where \\( k = \\lfloor n \\rfloor \\) is the integer part and \\( f = \\{ n \\} \\) is the fractional part such that \\( 0 \\leq f < 1 \\).\n\nGiven:\n\\[ \\{ n \\} + \\{ 3n \\} = 1.4 \\]\n\n... | [
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[
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"value": "Return your final response within \\boxed{}. On the line containing side \\( AB \\), take any point \\( P \\). Draw a perpendicular line from \\( P \\) to \\( AC \\), and let the foot of the perpendicular be \\( Q \\). The intersection of lines \\( BQ \\) and \\( CP \\) is point ... | On the line containing side \( AB \), take any point \( P \). Draw a perpendicular line from \( P \) to \( AC \), and let the foot of the perpendicular be \( Q \). The intersection of lines \( BQ \) and \( CP \) is point \( R \). Find the equation of the locus of point \( R \), and determine what type of curve it is (g... | null | null | [
{
"content": "You like to find multiple answers for a question then deliberate over them saying \"Alternatively,\" between each answer you are deliberating on and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer... | (c^2 + d^2)(bx - cy) = (a c - b c - c^2)xy + d^2 bx | alternative_rationalized | [
"Let's start by setting up our coordinate system where triangle \\(ABC\\) has vertices \\(A(0,0)\\), \\(B(b,0)\\), and \\(C(c,d)\\).\n\n1. **Point \\(P\\)**:\n Take any point \\(P\\) on side \\(AB\\). Let \\(P(x_1, 0)\\) since \\(AB\\) lies along the x-axis.\n\n2. **Foot of Perpendicular \\(Q\\)**:\n Since \\(P... | [
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"value": "6. From a class of 20 people, a team of three students is formed to participate in mathematics, Russian language, and informatics olympiads. In this class, all students excel academically. How many ways are there to form the team of olympiad participants, if each member of the te... | 6. From a class of 20 people, a team of three students is formed to participate in mathematics, Russian language, and informatics olympiads. In this class, all students excel academically. How many ways are there to form the team of olympiad participants, if each member of the team participates in one olympiad? | null | null | [
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"content": "You like to find multiple answers for a question then deliberate over them saying \"Alternatively,\" between each answer you are deliberating on and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer... | 6840 | alternative_rationalized | [
"To solve this problem, we need to determine how many different teams can be formed from a class of 20 people where each team consists of three members who will participate in mathematics, Russian language, and informatics olympiads respectively.\n\nFirstly, let's consider selecting the first student for the team:\... | [
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"value": "Example 2 Let non-negative real numbers $a, b, c$ satisfy $ab + bc + ca = 1$. Find the minimum value of $u = \\frac{1}{a+b} + \\frac{1}{b+c} + \\frac{1}{c+a}$."
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"value": "<think>\nOkay, so I have this problem where I need to find the minimum ... | Example 2 Let non-negative real numbers $a, b, c$ satisfy $ab + bc + ca = 1$. Find the minimum value of $u = \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a}$. | null | null | [
{
"content": "You like to find multiple answers for a question then deliberate over them saying \"Alternatively,\" between each answer you are deliberating on and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer... | \dfrac{5}{2} | alternative_rationalized | [
"Firstly, let's consider the simplest case where \\(a = b = c\\). Given \\(ab + bc + ca = 1\\), we have:\n\\[3a^2 = 1 \\implies a^2 = \\frac{1}{3} \\implies a = \\frac{1}{\\sqrt{3}}.\\]\nThus,\n\\[a + b = \\frac{2}{\\sqrt{3}},\\]\nand\n\\[\\frac{1}{a+b} = \\frac{\\sqrt{3}}{2}.\\]\nTherefore,\n\\[u = 3 \\cdot \\frac... | [
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"value": "Among the following functions, the one that is monotonically decreasing on $(0, +\\infty)$ is ( ) \nA: $y=|x-1|$ \nB: $y=\\log_{2}x$ \nC: $y=(x+1)^{2}$ \nD: $y=\\left(\\frac{1}{2}\\right)^{x}$"
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"value": "<think>\nOkay, so I need to figur... | Among the following functions, the one that is monotonically decreasing on $(0, +\infty)$ is ( )
A: $y=|x-1|$
B: $y=\log_{2}x$
C: $y=(x+1)^{2}$
D: $y=\left(\frac{1}{2}\right)^{x}$ | null | null | [
{
"content": "You like to find multiple answers for a question then deliberate over them saying \"Alternatively,\" between each answer you are deliberating on and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer... | D | alternative_rationalized | [
"To determine which function is monotonically decreasing on \\((0, +\\infty)\\), we need to analyze each option step by step.\n\n**Option A: \\( y = |x - 1| \\)**\n\nThe absolute value function \\( y = |x - 1| \\) can be broken down into two parts:\n\\[ \ny = \n\\begin{cases} \nx - 1 & \\text{if } x \\geq 1 \\\\\n-... | [
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[{"from":"user","value":"Return your final response within \\boxed{}. Given positive numbers \\(a, b(...TRUNCATED) | "Given positive numbers \\(a, b, c, x, y, z\\) and \\(k\\) such that\n\\[ a+x = b+y = c+z = k, \\]\n(...TRUNCATED) | null | null | [{"content":"You like to find multiple answers for a question then deliberate over them saying \"Alt(...TRUNCATED) | ay + bz + cx < k^2 | alternative_rationalized | ["First, let's express \\(x\\), \\(y\\), and \\(z\\) in terms of \\(a\\), \\(b\\), \\(c\\), and \\(k(...TRUNCATED) | [
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[{"from":"user","value":"7. Let $x, y$ be real numbers, and $x^{2}-2 x y+y^{2}-\\sqrt{2} x-\\sqrt{2}(...TRUNCATED) | "7. Let $x, y$ be real numbers, and $x^{2}-2 x y+y^{2}-\\sqrt{2} x-\\sqrt{2} y+6=0, u=x+y$ has the m(...TRUNCATED) | null | null | [{"content":"You like to find multiple answers for a question then deliberate over them saying \"Alt(...TRUNCATED) | 3\sqrt{2} | alternative_rationalized | ["Let's start by analyzing the given equation \\(x^2 - 2xy + y^2 - \\sqrt{2}x - \\sqrt{2}y + 6 = 0\\(...TRUNCATED) | [
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