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10^\\circ}.\n\\end{align*}By double angle formula,\n\\begin{align*}\n\\frac{4 \\cos 10^\\circ \\sin 10^\\circ - \\cos 10^\\circ}{\\sin 10^\\circ} &= \\frac{2 \\sin 20^\\circ - \\cos 10^\\circ}{\\sin 10^\\circ} \\\\\n&= \\frac{\\sin 20^\\circ + \\sin 20^\\circ - \\sin 80^\\circ}{\\sin 10^\\circ}.\n\\end{align*}By sum-to-product,\n\\[\\sin 20^\\circ - \\sin 80^\\circ = 2 \\cos 50^\\circ \\sin (-30^\\circ) = -\\cos 50^\\circ,\\]so\n\\begin{align*}\n\\frac{\\sin 20^\\circ + \\sin 20^\\circ - \\sin 80^\\circ}{\\sin 10^\\circ} &= \\frac{\\sin 20^\\circ - \\cos 50^\\circ}{\\sin 10^\\circ} \\\\\n&= \\frac{\\sin 20^\\circ - \\sin 40^\\circ}{\\sin 10^\\circ}.\n\\end{align*}By sum-to-product,\n\\[\\sin 20^\\circ - \\sin 40^\\circ = 2 \\cos 30^\\circ \\sin (-10^\\circ) = -\\sqrt{3} \\sin 10^\\circ,\\]so $\\frac{\\sin 20^\\circ - \\sin 40^\\circ}{\\sin 10^\\circ} = \\boxed{-\\sqrt{3}}.$'] | [] | [[151646, 151646, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 46391, 151644, 50, 70206, 57960, 52591, 220, 16, 15, 15, 24884, 43298, 488, 220, 19, 1124, 15940, 220, 16, 15, 15, 24884, 43298, 2418, 151645]] | {'extractive_match': 1.0} | 0 | 0 | [0] | [] | [(["<think>\nOkay, I need to simplify the expression \\(\\tan 100^\\circ + 4 \\sin 100^\\circ\\). Hmm, that's in degrees, so I don't need to convert anything, which is good. Let me think about how I can manipulate this expression. Maybe I can express both terms in terms of sine and cosine to see if they can be combined or simplified somehow.\n\nFirst, let's write \\(\\tan 100^\\circ\\) as \\(\\frac{\\sin 100^\\circ}{\\cos 100^\\circ}\\). So then, the expression becomes:\n\\[\n\\frac{\\sin 100^\\circ}{\\cos 100^\\circ} + 4 \\sin 100^\\circ\n\\]\nI can factor out \\(\\sin 100^\\circ\\) from both terms:\n\\[\n\\sin 100^\\circ \\left( \\frac{1}{\\cos 100^\\circ} + 4 \\right)\n\\]\nSimplify that insides the parentheses:\n\\[\n\\sin 100^\\circ \\left( \\frac{1 + 4 \\cos 100^\\circ}{\\cos 100^\\circ} \\right)\n\\]\nWhich gives:\n\\[\n\\frac{\\sin 100^\\circ (1 + 4 \\cos 100^\\circ)}{\\cos 100^\\circ}\n\\]\nHmm, not sure if that's helpful. Maybe I can think of another identity or angle addition formula. I remember that sometimes adding angles can lead to simplifications, especially if the resulting terms can be used in known identities.\n\nAlternatively, perhaps instead of factoring, maybe writing everything over a common denominator. Let me try that.\n\nSo, the original expression:\n\\[\n\\tan 100^\\circ + 4 \\sin 100^\\circ = \\frac{\\sin 100^\\circ}{\\cos 100^\\circ} + 4 \\sin 100^\\circ\n\\]\nLet's get a common denominator:\n\\[\n= \\frac{\\sin 100^\\circ + 4 \\sin 100^\\circ \\cos 100^\\circ}{\\cos 100^\\circ}\n\\]\nFactor the numerator:\n\\[\n= \\frac{\\sin 100^\\circ (1 + 4 \\cos 100^\\circ)}{\\cos 100^\\circ}\n\\]\nThis is similar to what I had before. Is \\(\\frac{\\sin 100^\\circ}{\\cos 100^\\circ}\\) which is \\(\\tan 100^\\circ\\), but I don't think that helps.\n\nWait, so maybe I can write this entire expression as \\(\\sin 100^\\circ / \\cos 100^\\circ + 4 \\sin 100^\\circ\\). Alternatively, perhaps it's useful to write \\(\\tan x\\) as \\(\\sin x / \\cos x\\), but maybe there is a way to write the whole expression using a different trigonometric function.\n\nAlternatively, perhaps I can consider co-function identities or supplementary angles. 100 degrees is in the second quadrant, so I know that sine is positive and tangent is negative there. Wait, is that right? Let me verify: in the second quadrant, angles between 90 and 180 degrees, sine is positive, cosine is negative, so tangent is negative. So yes, \\(\\tan 100^\\circ\\) is negative, and \\(\\sin100^\\circ\\) is positive. So the expression is \\(\\tan 100^\\circ + 4 \\sin 100^\\circ\\). So a negative number plus a positive number. I don't know, but maybe it's a known value or something.\n\nAlternatively, can I express this as a sum or product of angles? Perhaps using angle addition formulas or multiple-angle identities. Maybe I can relate \\(100^\\circ\\) to 60°, 30°, or other standard angles? 100 is 60+40, or 30+70, not sure.\n\nWait, another idea: maybe using some sort of identity for \\(\\tan \\theta + 4 \\sin \\theta\\). Let's set \\(\\theta = 100^\\circ\\), so the expression is \\(\\tan \\theta + 4 \\sin \\theta\\). Maybe there is a way to write this as a single trigonometric function.\n\nFirst, let me compute the numerical value to see if it resembles any familiar value:\n\nCompute each term:\n\nFirst, \\(\\sin 100^\\circ\\) approximately? Let me recall that \\(\\sin 90^\\circ = 1\\), and \\(\\sin 100^\\circ\\) would be a bit less, maybe around 0.9848 or something? Wait, calculator value: sin(100°) ≈ 0.9848077530.\n\nSimilarly, \\(\\tan 100^\\circ\\) is \\(\\tan(180° - 80°)\\) which is \\(-\\tan80°\\). \\(\\tan80° ≈ 5.67128182\\), so \\(\\tan100° ≈ -5.67128182\\). Therefore, \\(\\tan100° ≈ -5.671\\), and \\(4 \\sin100° ≈ 4 * 0.9848 ≈ 3.939\\). So total is approximately \\(-5.671 + 3.939 ≈ -1.732\\). Hmm, that seems familiar.\n\n-1.732 is approximately \\(-\\sqrt{3}\\), since \\(\\sqrt{3} ≈ 1.732\\). So I believe the expression simplifies to \\(-\\sqrt{3}\\). So let me verify that.\n\nIs \\(\\tan 100° + 4 \\sin 100° = -\\sqrt{3}\\)?\n\nWait, if that's the case, we need to verify without calculator, maybe by expressing 100° in terms of known angles.\n\nWait, 100° is 90° + 10°, perhaps you can use co-function identities or some addition formulas. Alternatively, let me consider angles with known tangents.\n\nWait, let me think—could 100° be related to 60° somehow? I know that 60° is a standard angle with \\(\\tan60° = \\sqrt{3}\\), but 100° is double of 50°, but 50° doesn't have standard sine or tangent.\n\nWait, another approach: Maybe express sine and tangent in terms of sine and cosine and try to manipulate. Let's see.\n\nSo, we have:\n\n\\[\nE = \\tan 100^\\circ + 4 \\sin 100^\\circ\n\\]\n\nExpressed in terms of sine and cosine:\n\n\\[\nE = \\frac{\\sin 100^\\circ}{\\cos 100^\\circ} + 4 \\sin 100^\\circ\n\\]\n\nCombine over a common denominator:\n\n\\[\nE = \\frac{\\sin 100^\\circ + 4 \\sin 100^\\circ \\cos 100^\\circ}{\\cos 100^\\circ}\n\\]\n\nFactor:\n\n\\[\nE = \\frac{\\sin 100^\\circ (1 + 4 \\cos 100^\\circ)}{\\cos 100^\\circ}\n\\]\n\nHmm… Maybe I can write the numerator as something, maybe multiple-angle formula. Let me think about multiple-angle identities for sine.\n\nAlternatively, maybe we can consider Euler's formula or other complex identities, but that might be overcomplicating.\n\nWait, 1 + 4cos100° is in the numerator. Let me compute that:\n\nCompute 1 + 4cos100°. Well cos100° is cos(180° - 80°) = -cos80°, which is approximately, cos80°≈ 0.1736, so 4cos100 ≈4*(-0.1736)= around -0.694. So 1 + 4cos100 ≈ 1 - 0.694≈ 0.306.\n\nThus, the numerator sin100∘ * ≈0.306, and the denominator is cos100°, which is a negative number, approximately -0.1736. So numerator is about 0.9848 * 0.306 ≈ 0.3, so 0.3 / (-0.1736) ≈ -1.732, which is what I had before.\n\nSo, the expression is approximately negative sqrt(3). So, I can tentatively say it's equal to -sqrt(3).\n\nBut I need to verify this via trigonometric identities, because just showing it's a number is not good enough. We need an exact expression.\n\nWait, I know that sqrt(3) is tan60°, so if somehow 100° is involved in an equation or a multiplica...\n\nWait, let me try using degrees with 30°, since tan30° is 1/sqrt(3). Wait, but here we have sqrt(3). Alternatively, maybe I can use some identity to express tan100° + 4 sin100° in terms of an expression that evaluates to sqrt(3). Alternatively, perhaps express 100° as 60° + 40°, but that might not help.\n\nWait, what's tan(60°)? It is sqrt(3). So we have sqrt(3) as part of the problem. Hmm.\n\nAlternatively, perhaps there is an identity for 100°, or 80°, since 80° is 180° - 100°, as I mentioned earlier. Let me think: 80°, 60°, and 20° form a triangle... 80° is from 60°+20°. Alternatively, 100° is 90°+10°, so perhaps that's helpful. Let me see:\n\nWait, to express 100° or 80° in terms that could relate to known angles. 80°=60°+20°, and 20° is a bit of an outlier as a non-special angle. Hmmm.\n\nAlternatively, maybe 80° is 4*20°, but that's not particularly helpful.\n\nWait, perhaps another route: maybe using some trigonometric identities on the expression \\(\\tan\\theta + 4\\sin\\theta\\) to write it as some multiple of another trigonometric function.\n\nLet me just express E as:\n\n\\[\nE = \\tan\\theta + 4 \\sin\\theta\n\\]\nwhere \\(\\theta = 100^\\circ\\)\n\nExpress everything in terms of sine and cosine:\n\n\\[\nE = \\frac{\\sin\\theta}{\\cos\\theta} + 4 \\sin\\theta = \\frac{\\sin\\theta + 4 \\sin\\theta \\cos\\theta}{\\cos\\theta} = \\frac{\\sin\\theta (1 + 4 \\cos\\theta)}{\\cos\\theta}\n\\]\n\nAlternatively, perhaps if I denote phi such that 1 + 4cosθ is related to a sine or cosine of another angle—maybe using the identity for sin(A+B) or something like that. Let me see:\n\nWait, 1 + 4cosθ can be thought of as a multiple-angle identity. Let me see.\n\nAlternatively, maybe write 1 as 2*cos60°, since cos60°=0.5, so 2cos60°=1.\n\nSo, 1 + 4cosθ = 2cos60° +4cosθ.\n\nHmm, not sure if that helps. Alternatively, maybe something else.\n\nWait, If I think of 1 + 4cosθ, perhaps in terms of expressions involving known multiple angles. Alternatively, perhaps using the identity for projection:\n\nWait, like 1 + 4cosθ may be written as a sum that can be simplified.\n\nI think I might be stuck here. Maybe I should use a substitution for theta.\n\nWait, the target value is -sqrt(3). So, 100° is such that tan(theta) + 4 sin(theta) = -sqrt(3). So, let's suppose that:\n\n\\[\n\\tan\\theta + 4\\sin\\theta = -\\sqrt{3}\n\\]\n\nCan I find theta such that this is true?\n\nBut we have theta = 100°, is this an identity that is valid? Not sure.\n\nWait, let me suppose that, perhaps, if I rearrange:\n\n\\[\n\\tan\\theta = -\\sqrt{3} - 4 \\sin\\theta\n\\]\n\nSo,\n\n\\[\n\\frac{\\sin\\theta}{\\cos\\theta} = -\\sqrt{3} -4 \\sin\\theta\n\\]\n\nMultiplying both sides by cosθ:\n\n\\[\n\\sin\\theta = -\\sqrt{3}\\cos\\theta -4 \\sin\\theta \\cos\\theta\n\\]\n\nHmm, move all terms to one side:\n\n\\[\n\\sin\\theta + \\sqrt{3}\\cos\\theta +4 \\sin\\theta \\cos\\theta = 0\n\\]\n\nIs there a way to make this into a recognizable identity? Let me consider, perhaps, this:\n\nLet me try writing 2 terms as one:\n\nFirst term: sinθ + sqrt(3) cosθ, which is a form of a sine addition formula: as A sinθ + B cosθ. So, let me consider that:\n\n\\[\n\\sin\\theta + \\sqrt{3}\\cos\\theta = 2 \\left( \\frac{1}{2}\\sinθ + \\frac{\\sqrt{3}}{2}\\cosθ \\right ) = 2 \\sin\\left(\\theta + 60°\\right )\n\\]\n\nBecause sin(A+B)=sinA cosB + cosA sinB, so 1/2 = cos60°, and sqrt(3)/2=sin60°, thus it's sin(theta + 60°).\n\nSo, that means:\n\nFirst part: sinθ + sqrt(3) cosθ = 2 sin(theta + 60°)\n\nSo equation becomes:\n\n\\[\n2 \\sin(\\theta + 60°) +4 \\sin\\theta \\cos\\theta =0\n\\]\n\nComplete equation:\n\n\\[\n2 \\sin(\\theta + 60°) +4 \\sin\\theta \\cos\\theta =0\n\\]\n\nSimplify 4 sinθ cosθ, which is 2 sin2θ:\n\nSo,\n\n\\[\n2 \\sin(\\theta + 60°) +2 \\sin 2θ =0\n\\]\n\nFactor 2:\n\n\\[\n2 [ \\sin(\\theta +60°) + \\sin2θ ]=0\n\\]\n\nDivide both sides by 2:\n\n\\[\n\\sin(\\theta +60°) + \\sin2θ =0\n\\]\n\nSo,\n\n\\[\n\\sin2θ + \\sin(\\theta + 60°) =0\n\\]\n\nLet me use the identity for sum of sines:\n\n\\[\n\\sin A + \\sin B = 2 \\sin \\left( \\frac{A+B}{2} \\right ) \\cos \\left( \\frac{A - B}{2} \\right )\n\\]\n\nSo set A=2θ, B=θ+60°. Then,\n\nCompute:\n\n\\[\n\\sin(2θ) + \\sin(θ + 60°) = 2 \\sin\\left( \\frac{2θ +θ +60°}{2} \\right ) \\cos\\left( \\frac{2θ - (θ +60°)}{2} \\right )\n\\]\n\nSimplify the arguments:\n\nFirst, for sine term:\n\n\\[\n\\frac{3θ +60°}{2} = \\frac{3θ}{2} + 30°\n\\]\n\nFor cosine term:\n\n\\[\n\\frac{2θ - θ -60°}{2} = \\frac{θ -60°}{2}\n\\]\n\nThus,\n\n\\[\n2 \\sin\\left( \\frac{3θ}{2} +30°\\right ) \\cos \\left( \\frac{θ}{2} -30°\\right )=0\n\\]\n\nSo,\n\nEither:\n\n1. \\(\\sin\\left( \\frac{3θ}{2} +30° \\right )=0\\)\n\nor\n\n2. \\(\\cos\\left( \\frac{θ}{2}-30° \\right )=0\\)\n\nLet me solve each equation.\n\nCase 1: \\(\\sin\\left( \\frac{3θ}{2} +30° \\right )=0\\)\n\nSo,\n\n\\[\n\\frac{3θ}{2} +30°=n*180° \\quad \\text{for some integer }n\n\\]\n\nThus,\n\n\\[\n\\frac{3θ}{2}=n*180° -30°\n\\]\n\nMultiply both sides by 2:\n\n\\[\n3θ=360°n -60°\n\\]\n\nThus,\n\n\\[\nθ=120°n -20°\n\\]\n\nFor integer n.\n\nCase 2: \\(\\cos\\left( \\frac{θ}{2} -30° \\right )=0\\)\n\nSo,\n\n\\[\n\\frac{θ}{2} -30°=90° +180°k \\quad \\text{for some integer }k\n\\]\n\nThen,\n\n\\[\n\\frac{θ}{2}=90° +180°k +30°=120° +180°k\n\\]\n\nHence,\n\n\\[\nθ=240° +360°k\n\\]\n\nSo solutions are θ=240° +360°k, which we can write as θ=240°, but since θ is given as 100°, around 10-20%, which is 100°, perhaps n=1 in case 1:\n\nFor Case1: θ=120°*1 -20°=100°, which matches the given theta=100°. So, the solution θ=100° comes from Case1.\n\nSo that means that when theta=100°, sin(theta +60°) + sin2theta =0, so overall equation is 0, so that goes back to the starting point that E= -sqrt(3). Therefore, the initial equation is valid when theta=100°, which is the case we are given.\n\nThus, connecting back, since we have that \\(\\tan 100^\\circ +4 \\sin100^\\circ =-\\sqrt{3}\\). So, it's valid because following through the identities, when theta=100°, this identity holds.\n\nHence, the expression simplifies to \\(-\\sqrt{3}\\).\n\n**Final Answer**\nThe simplified expression is \\(\\boxed{-\\sqrt{3}}\\).\n</think>\n\nTo simplify the expression \\(\\tan 100^\\circ + 4 \\sin 100^\\circ\\), we start by expressing \\(\\tan 100^\\circ\\) in terms of sine and cosine:\n\n\\[\n\\tan 100^\\circ = \\frac{\\sin 100^\\circ}{\\cos 100^\\circ}\n\\]\n\nThus, the expression becomes:\n\n\\[\n\\frac{\\sin 100^\\circ}{\\cos 100^\\circ} + 4 \\sin 100^\\circ\n\\]\n\nWe factor out \\(\\sin 100^\\circ\\):\n\n\\[\n\\sin 100^\\circ \\left( \\frac{1}{\\cos 100^\\circ} + 4 \\right)\n\\]\n\nCombining the terms over a common denominator:\n\n\\[\n\\frac{\\sin 100^\\circ (1 + 4 \\cos 100^\\circ)}{\\cos 100^\\circ}\n\\]\n\nWe recognize that the expression might be simplified using trigonometric identities. 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-5.125986263010418e-06, -1.2993727978027891e-05, -8.05822346592322e-05, -5.245195097813848e-06, -0.2860549986362457, 0.0, -3.40932747349143e-05, -0.008652695454657078, -0.00015066919149830937, -1.1920922133867862e-06, -0.003262912854552269, -7.033323527139146e-06, -0.0001722425949992612, -1.4305104514278355e-06, -1.6093124941107817e-05, -4.172316494077677e-06, -4.255681051290594e-05, -1.0728830375228426e-06, -6.6756979322235566e-06, -1.9073468138230965e-06, -0.0012673450401052833, -2.8729025871143676e-05, -2.372236667724792e-05, -4.768370445162873e-07])] | {'extracted_predictions': ['-sqrt(3)', '-\\sqrt{3}'], 'extracted_golds': ['-sqrt(3)', '-\\sqrt{3}']} | [0] |
|
[] | "[[(151648, 198, 32313, 11, 773, 358, 1184, 311, 11625, 279, 31205, 17767, 856, 61, 17, 481, 220, 20(...TRUNCATED) | "For what values of $x$ is it true that $x^2 - 5x - 4 \\le 10$? Express your answer in interval nota(...TRUNCATED) | "<｜begin▁of▁sentence｜>Please reason step by step, and put your final answer within \\boxed{}(...TRUNCATED) | "['Re-arranging, $x^2 - 5x - 14 \\\\le 0$. The left-hand quadratic factors as $x^2 - 5x - 14 = (x - (...TRUNCATED) | [] | "[[151646, 151646, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 4(...TRUNCATED) | {'extractive_match': 0.0} | 0 | 0 | [0] | [] | "[([\"<think>\\nOkay, so I need to solve the inequality \\\\( x^2 - 5x - 4 \\\\le 10 \\\\). Hmm, let(...TRUNCATED) | "{'extracted_predictions': ['{Interval(-2, 7)}', '[-2, 7],[-2, 7]'], 'extracted_golds': ['Eq(x, Inte(...TRUNCATED) | [0] |
|
[] | "[[(151648, 198, 32313, 11, 773, 358, 1184, 311, 11625, 419, 3491, 25, 758, 1290, 21495, 19360, 448,(...TRUNCATED) | "In right triangle $ABC$ with $\\angle B = 90^\\circ$, we have $\\sin A = 2\\cos A$. What is $\\tan(...TRUNCATED) | "<｜begin▁of▁sentence｜>Please reason step by step, and put your final answer within \\boxed{}(...TRUNCATED) | "['The triangle is shown below:\\n\\n[asy]\\npair A,B,C;\\nA = (0,0);\\nB = (5,0);\\nC = (5,10);\\nd(...TRUNCATED) | [] | "[[151646, 151646, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 4(...TRUNCATED) | {'extractive_match': 1.0} | 0 | 0 | [0] | [] | "[([\"<think>\\nOkay, so I need to solve this problem: In right triangle ABC with angle B being 90 d(...TRUNCATED) | {'extracted_predictions': ['{2}', '2,2'], 'extracted_golds': ['2', '2']} | [0] |
|
[] | "[[(151648, 198, 32313, 11, 773, 358, 3003, 2684, 419, 3491, 1588, 448, 1378, 38280, 11, 362, 323, 4(...TRUNCATED) | "Two runners, $A$ and $B,$ start at a point $O$ on a linear track, and start running in the same dir(...TRUNCATED) | "<｜begin▁of▁sentence｜>Please reason step by step, and put your final answer within \\boxed{}(...TRUNCATED) | "['Without loss of generality, assume that $OP = 1.$ Let $OA = x$ and $OB = 3x.$ Let $\\\\alpha = (...TRUNCATED) | [] | "[[151646, 151646, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 4(...TRUNCATED) | {'extractive_match': 1.0} | 0 | 0 | [0] | [] | "[([\"<think>\\nOkay, so I've got this problem here with two runners, A and B, starting at the same (...TRUNCATED) | {'extracted_predictions': ['{30}', '30,30'], 'extracted_golds': ['30', '30^\\circ']} | [0] |
|
[] | "[[(151648, 198, 32313, 11, 773, 358, 614, 311, 12564, 279, 66750, 482, 315, 481, 16, 14, 17, 323, 3(...TRUNCATED) | Compute $\arcsin \left( -\frac{1}{2} \right).$ Express your answer in radians. | "<｜begin▁of▁sentence｜>Please reason step by step, and put your final answer within \\boxed{}(...TRUNCATED) | "['Since $\\\\sin \\\\left( -\\\\frac{\\\\pi}{6} \\\\right) = -\\\\frac{1}{2},$ $\\\\arcsin \\\\left(...TRUNCATED) | [] | "[[151646, 151646, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 4(...TRUNCATED) | {'extractive_match': 1.0} | 0 | 0 | [0] | [] | "[([\"<think>\\nOkay, so I have to compute the arcsine of -1/2 and express it in radians. Hmm, arcsi(...TRUNCATED) | "{'extracted_predictions': ['-1*pi/6', '-\\\\frac{\\\\pi}{6}'], 'extracted_golds': ['-1*pi/6', '-\\\(...TRUNCATED) | [0] |
|
[] | "[[(151648, 198, 71486, 11, 773, 358, 3003, 2684, 419, 6888, 3491, 1588, 429, 358, 1184, 311, 11625,(...TRUNCATED) | "What is the length, in units, of the radius of a sphere whose volume and surface area, in cubic uni(...TRUNCATED) | "<｜begin▁of▁sentence｜>Please reason step by step, and put your final answer within \\boxed{}(...TRUNCATED) | "['The volume of a sphere is $\\\\frac{4}{3}\\\\pi r^3$ and the surface area is $4\\\\pi r^2$, so\\n(...TRUNCATED) | [] | "[[151646, 151646, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 4(...TRUNCATED) | {'extractive_match': 1.0} | 0 | 0 | [0] | [] | "[(['<think>\\nAlright, so I\\'ve got this math problem here that I need to solve. It says: \"What i(...TRUNCATED) | {'extracted_predictions': ['{3}', '3,3'], 'extracted_golds': ['3', '3']} | [0] |
|
[] | "[[(151648, 198, 71486, 11, 773, 358, 614, 419, 3491, 1380, 358, 1184, 311, 1477, 6785, 25780, 264, (...TRUNCATED) | "Positive integers $a$, $b$, and $2009$, with $a<b<2009$, form a geometric sequence with an integer (...TRUNCATED) | "<｜begin▁of▁sentence｜>Please reason step by step, and put your final answer within \\boxed{}(...TRUNCATED) | "['The prime factorization of $2009$ is $2009 = 7\\\\cdot 7\\\\cdot 41$. As $a<b<2009$, the ratio mu(...TRUNCATED) | [] | "[[151646, 151646, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 4(...TRUNCATED) | {'extractive_match': 1.0} | 0 | 0 | [0] | [] | "[([\"<think>\\nAlright, so I have this problem where I need to find positive integers a, b, and 200(...TRUNCATED) | {'extracted_predictions': ['{41}', '41,41'], 'extracted_golds': ['41', '41']} | [0] |
|
[] | "[[(151648, 198, 32313, 11, 1077, 594, 1490, 13, 358, 614, 419, 3491, 1380, 264, 1378, 48342, 7546, (...TRUNCATED) | When the two-digit integer $``\text{AB}"$ is cubed, the value is $912,\!673$. What is $A + B$? | "<｜begin▁of▁sentence｜>Please reason step by step, and put your final answer within \\boxed{}(...TRUNCATED) | "['Since $90^3=729,\\\\!000$, $\\\\text{AB}$ is greater than 90. Therefore, $\\\\text{A}=9$. Since(...TRUNCATED) | [] | "[[151646, 151646, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 4(...TRUNCATED) | {'extractive_match': 1.0} | 0 | 0 | [0] | [] | "[(['<think>\\nOkay, let\\'s see. I have this problem where a two-digit integer \"AB\" is cubed and (...TRUNCATED) | {'extracted_predictions': ['{16}', '16,16'], 'extracted_golds': ['16', '16']} | [0] |
|
[] | "[[(151648, 198, 32313, 11, 773, 358, 614, 419, 7493, 311, 39721, 25, 1124, 11520, 37018, 35702, 502(...TRUNCATED) | Simplify
\[\frac{\sec x}{\sin x} - \frac{\sin x}{\cos x}.\] | "<｜begin▁of▁sentence｜>Please reason step by step, and put your final answer within \\boxed{}(...TRUNCATED) | "['We can write\\n\\\\begin{align*}\\n\\\\frac{\\\\sec x}{\\\\sin x} - \\\\frac{\\\\sin x}{\\\\cos x(...TRUNCATED) | [] | "[[151646, 151646, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 4(...TRUNCATED) | {'extractive_match': 1.0} | 0 | 0 | [0] | [] | "[([\"<think>\\nOkay, so I have this expression to simplify: \\\\(\\\\frac{\\\\sec x}{\\\\sin x} - \(...TRUNCATED) | "{'extracted_predictions': ['{cot(x)}', '\\\\cot x,\\\\cot x'], 'extracted_golds': ['cot(x)', '\\\\c(...TRUNCATED) | [0] |
|
[] | "[[(151648, 198, 71486, 11, 773, 358, 614, 419, 3491, 1588, 1380, 358, 1184, 311, 1477, 279, 897, 31(...TRUNCATED) | "If\n\\[\\sqrt[3]{2} = a + \\cfrac{1}{b + \\cfrac{1}{c + \\cfrac{1}{d + \\dotsb}}},\\]where $a,$ $b,(...TRUNCATED) | "<｜begin▁of▁sentence｜>Please reason step by step, and put your final answer within \\boxed{}(...TRUNCATED) | "['We know that\\n\\\\[\\\\sqrt[3]{2} = a + \\\\cfrac{1}{b + \\\\cfrac{1}{c + \\\\cfrac{1}{d + \\\\d(...TRUNCATED) | [] | "[[151646, 151646, 5501, 2874, 3019, 553, 3019, 11, 323, 2182, 697, 1590, 4226, 2878, 1124, 79075, 4(...TRUNCATED) | {'extractive_match': 1.0} | 0 | 0 | [0] | [] | "[([\"<think>\\nAlright, so I have this problem here where I need to find the value of \\\\( b \\\\)(...TRUNCATED) | {'extracted_predictions': ['{3}', '3,3'], 'extracted_golds': ['3', '3']} | [0] |
End of preview. Expand
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Dataset Card for Evaluation run of deepseek-ai/DeepSeek-R1-Distill-Qwen-32B
Dataset automatically created during the evaluation run of model deepseek-ai/DeepSeek-R1-Distill-Qwen-32B.
The dataset is composed of 1 configuration, each one coresponding to one of the evaluated task.
The dataset has been created from 3 run(s). Each run can be found as a specific split in each configuration, the split being named using the timestamp of the run.The "train" split is always pointing to the latest results.
An additional configuration "results" store all the aggregated results of the run.
To load the details from a run, you can for instance do the following:
from datasets import load_dataset
data = load_dataset("debug-r1/details_deepseek-ai__DeepSeek-R1-Distill-Qwen-32B_private",
"results",
split="train")
Latest results
These are the latest results from run 2025-02-05T22:09:49.535552(note that their might be results for other tasks in the repos if successive evals didn't cover the same tasks. You find each in the results and the "latest" split for each eval):
{
"all": {
"extractive_match": 0.852,
"extractive_match_stderr": 0.015896458561251246
},
"custom|math_500|0": {
"extractive_match": 0.852,
"extractive_match_stderr": 0.015896458561251246
}
}
Dataset Details
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