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33c2c34c4b189843 | Let S = {(x,y) in R^2 : |x| + |y| ≤ 2 and x^2 + y^2 ≥ 2}. Compute the Euclidean area of S. | 8 - 2\u03c0 | algebra | [] | [] | 734a5221a1706d1a | dcc27b5615bc96ed | [0.01190185546875, -0.03460693359375, 0.05499267578125, -0.018096923828125, 0.0010805130004882812, -0.0027179718017578125, 0.0013275146484375, -0.0172576904296875, -0.032928466796875, 0.06134033203125, 0.015594482421875, -0.0640869140625, -0.022064208984375, -0.0047149658203125, 0.0092010498046875, -0.0159912109375, -0... | [1927563484, 699836180, 374307475, 454831603, 538533776, 46124649, 901788442, 234259934, 1649616069, 42110957, 3214735942, 2167087590, 2368046915, 526125254, 825501999, 3712292166, 639960220, 85577997, 662647629, 220174791, 709194029, 2735169933, 2311553915, 218387009, 991927233, 899611864, 1008786526, 273075481, 13638... | {"path": "composer_realizer", "brief": {"primary_concept": "inequality", "secondary_concept": "graph", "answer_form": "FIND_VALUE", "topic": "algebra"}, "brief_id": "734a5221a1706d1a", "scaffold_id": "dcc27b5615bc96ed", "scaffold": {"target_answer": "8 - 2\\u03c0", "intended_solution_sketch": "Interpret the region S = ... | 1 | 1 | 1,777,298,286.846697 |
5825e2794753706f | Let p(x) be a polynomial of degree at most 3 such that p(n)=1/(n+1) for n=0,1,2,3. Find p(4). | 0 | intermediate_algebra | [] | [] | ff79293f0b1f8e8b | fa8fbd3fa1b08307 | [-0.0158233642578125, -0.0191802978515625, 0.034820556640625, -0.007598876953125, 0.045379638671875, -0.01080322265625, 0.0159149169921875, 0.03033447265625, -0.0052642822265625, 0.0187225341796875, 0.04620361328125, 0.0196075439453125, -0.027191162109375, -0.029296875, -0.0092315673828125, -0.005916595458984375, 0.025... | [2158985315, 2150491261, 558582405, 945830331, 1595607114, 2629399279, 101709920, 163231694, 1750678816, 280551609, 1705729439, 499662666, 749830858, 2682457950, 2189982833, 403444620, 2402217078, 627546325, 1413228154, 2097842744, 2249823251, 930300533, 1899506503, 684608703, 2792376493, 1383979244, 532266924, 2691090... | {"path": "composer_realizer", "brief": {"primary_concept": "polynomial", "secondary_concept": "factorization", "answer_form": "FIND_VALUE", "topic": "intermediate_algebra"}, "brief_id": "ff79293f0b1f8e8b", "scaffold_id": "fa8fbd3fa1b08307", "scaffold": {"target_answer": "0", "intended_solution_sketch": "Let q(x) := (x+... | 1 | 1 | 1,777,298,970.477121 |
c128d7766815e486 | Let A=(0,2) and B=(6,4). For a point P on the x-axis between (0,0) and (6,0) define S = PA + PB + 8. Find the minimum possible value of S. | 8 + 6 \sqrt{2} | intermediate_algebra | [] | [] | a86c0d23840ceaec | 7015345a7e187477 | [0.015106201171875, -0.04296875, 0.06591796875, 0.0036449432373046875, -0.0214385986328125, -0.01480865478515625, 0.0016021728515625, -0.003170013427734375, 0.01021575927734375, 0.016998291015625, -0.004947662353515625, -0.046234130859375, -0.0059814453125, -0.0008034706115722656, 0.02459716796875, -0.018646240234375, ... | [174379995, 623452600, 167453604, 2292077561, 1499168544, 1346948128, 2412185933, 893657823, 203698188, 3332502008, 2591317742, 926743950, 359405592, 219585319, 17449682, 997690324, 1432567722, 858039962, 1132540922, 2708713387, 247238816, 2469092771, 1658769873, 1081204520, 1769785641, 1199194744, 458718062, 200657208... | {"path": "composer_realizer", "brief": {"primary_concept": "inequality", "secondary_concept": "point", "answer_form": "FIND_VALUE", "topic": "intermediate_algebra"}, "brief_id": "a86c0d23840ceaec", "scaffold_id": "7015345a7e187477", "scaffold": {"target_answer": "8 + 6 \\sqrt{2}", "intended_solution_sketch": "Let P = (... | 1 | 1 | 1,777,299,159.514955 |
590bd2f0f8e7cc5a | Let R be a rational function of the form R(x)=\frac{ax+b}{cx+d} with real numbers a,b,c,d, and assume cx+d\neq0 for x=1,2,3. If R(1)=1, R(2)=2, and R(3)=3, find R(0). | 0 | algebra | [] | [] | 643d5f6a56e1600e | 79370bd1ecf43fdc | [0.03228759765625, -0.04248046875, 0.04779052734375, 0.00574493408203125, 0.0194244384765625, -0.043304443359375, -0.0172119140625, 0.021942138671875, 0.04010009765625, 0.053680419921875, 0.00713348388671875, -0.027252197265625, -0.033477783203125, 0.0039043426513671875, 0.009735107421875, -0.0306854248046875, 0.035675... | [228628556, 2354785121, 366994307, 2505429503, 2348340910, 1229229632, 1160413227, 812697415, 145664914, 1164717408, 2570478093, 1487322942, 638061139, 1263859066, 113710239, 1459900584, 2045919297, 104395597, 2121732587, 215419353, 553848473, 5914877, 1086321177, 542647667, 2230655643, 2323999429, 1250834955, 12520192... | {"path": "composer_realizer", "brief": {"primary_concept": "rational_function", "secondary_concept": "polynomial", "answer_form": "FIND_VALUE", "topic": "algebra"}, "brief_id": "643d5f6a56e1600e", "scaffold_id": "79370bd1ecf43fdc", "scaffold": {"target_answer": "0", "intended_solution_sketch": "Write the rational funct... | 1 | 1 | 1,777,302,483.45645 |
7d43ad19a3e327cf | Let z be a complex number satisfying z^4+z^3+z^2+z+1=0 and lying in the first quadrant (Re z>0 and Im z>0). Find the value of z+z^4. | (sqrt(5)-1)/2 | precalculus | [] | [] | 049b49aac85526c7 | a38d28340241b599 | [0.0235137939453125, 0.0028514862060546875, -0.0201263427734375, 0.06109619140625, 0.01155853271484375, -0.0095367431640625, 0.00423431396484375, 0.019195556640625, -0.03759765625, 0.049713134765625, 0.02899169921875, 0.0299224853515625, -0.045501708984375, 0.03765869140625, 0.0228271484375, -0.0004425048828125, -0.023... | [213495000, 338684828, 73732478, 451989508, 6144197, 668502924, 797855093, 376156192, 855081117, 77426467, 788776230, 387527962, 1352807777, 2795994681, 30312106, 57140769, 779181815, 88930243, 86142697, 1068145220, 2043520564, 652867864, 1961980512, 723872290, 557560579, 2867098293, 280262982, 674184903, 381382712, 17... | {"path": "composer_realizer", "brief": {"primary_concept": "complex_number", "secondary_concept": "polynomial", "answer_form": "FIND_VALUE", "topic": "precalculus"}, "brief_id": "049b49aac85526c7", "scaffold_id": "a38d28340241b599", "scaffold": {"target_answer": "(sqrt(5)-1)/2", "intended_solution_sketch": "Multiply z^... | 1 | 1 | 1,777,302,950.515046 |
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