data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum:\n\\[\\sin^2 6^\\circ + \\sin^2 12^\\circ + \\sin^2 18^\\circ + \\dots + \\sin^2 174^\\circ.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 6700,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the sum:\n\\[\\sum_{1 \\le a < b < c} \\frac{1}{3^a 5^b 7^c}.\\]\n(The sum is taken over all triples $(a,b,c)$ of positive integers such that $1 \\le a < b < c.$)"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{21216}"
} | {
"index": 6701,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the surface area of a cube inscribed in a sphere of surface area $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 6702,
"solution": "The sphere's radius $r$ satisfies $4 \\pi r^{2}=\\pi \\Rightarrow r=1 / 2$, so the cube has body diagonal 1 , hence side length $1 / \\sqrt{3}$. So, its surface area is $6(1 / \\sqrt{3})^{2}=2$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the surface integral\n\n$$\n\\iint_{\\Sigma}(-x+3 y+4 z) d \\sigma\n$$\n\nwhere $\\Sigma$ is the part of the plane\n\n$$\nx + 2y + 3z = 1\n$$\n\nlocated in the first octant (i.e., $x \\geq 0, y \\geq 0, z \\geq 0$)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{14}}{18}"
} | {
"index": 6703,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the triple integral \\( I = \\iiint_{G} \\frac{d x d y}{1-x-y} \\), where the region \\( G \\) is bounded by the planes:\n1) \\( x + y + z = 1 \\), \\( x = 0 \\), \\( y = 0 \\), \\( z = 0 \\)\n2) \\( x = 0 \\), \\( x = 1 \\), \\( y = 2 \\), \\( y = 5 \\), \\( z = 2 \\),... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 6704,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the unique ordered pair $(x, y)$ of real numbers satisfying the system of equations $$\\frac{x}{\\sqrt{x^{2}+y^{2}}}-\\frac{1}{x}=7 \\text { and } \\frac{y}{\\sqrt{x^{2}+y^{2}}}+\\frac{1}{y}=4$$"
}
] | math | {
"style": "rule",
"ground_truth": "(-\\frac{13}{96}, \\frac{13}{40})"
} | {
"index": 6705,
"solution": "Solution 1: Consider vectors $$\\binom{x / \\sqrt{x^{2}+y^{2}}}{y / \\sqrt{x^{2}+y^{2}}} \\text { and }\\binom{-1 / x}{1 / y}$$ They are orthogonal and add up to $\\binom{7}{4}$, which have length $\\sqrt{7^{2}+4^{2}}=\\sqrt{65}$. The first vector has length 1, so by Pythagorean's theo... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the unique positive integer $m$ such that\n\\[1 \\cdot 2^1 + 2 \\cdot 2^2 + 3 \\cdot 2^3 + \\dots + m \\cdot 2^m = 2^{m + 8}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "129"
} | {
"index": 6706,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the unique positive integer $n$ such that\n\\[2 \\cdot 2^2 + 3 \\cdot 2^3 + 4 \\cdot 2^4 + \\dots + n \\cdot 2^n = 2^{n + 10}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "513"
} | {
"index": 6707,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the unique positive integer $n$ such that $\\frac{n^{3}-1989}{n}$ is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 6708,
"solution": "We need $n^{2}-\\frac{1989}{n}$ to be a perfect square, so $n \\mid 1989$. Also, this perfect square would be less than $n^{2}$, so it would be at most $(n-1)^{2}=n^{2}-2 n+1$. Thus, $$\\frac{1989}{n} \\geq 2 n-1 \\Longrightarrow 1989 \\geq 2 n^{2}-n$$ so $n \\leq 31$. Moreover, we nee... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the unique positive integer \\( n \\) such that\n\\[\n3 \\cdot 2^3 + 4 \\cdot 2^4 + 5 \\cdot 2^5 + \\dots + n \\cdot 2^n = 2^{n + 11}.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "1025"
} | {
"index": 6709,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of $$64^{-\\frac{1}{3}}+lg0.001$$."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{11}{2}"
} | {
"index": 6710,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of $0.25 \\cdot 0.8 - 0.12$."
}
] | math | {
"style": "rule",
"ground_truth": "0.08"
} | {
"index": 6711,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of $1^{25}+2^{24}+3^{23}+\\ldots+24^{2}+25^{1}$. If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\\left\\lfloor 25 \\mathrm{~min}\\left(\\left(\\frac{A}{C}\\right)^{2},\\left(\\frac{C}{A}\\right)^{2}\\right)\\right... | math | {
"style": "rule",
"ground_truth": "66071772829247409"
} | {
"index": 6712,
"solution": "The sum is extremely unimodal, so we want to approximate it using its largest term. Taking logs of each term, we see that the max occurs when $(26-n) \\log n$ peaks, and taking derivatives gives $x+x \\log x=26$. From here it's easy to see that the answer is around 10, and slightly les... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of $252^2 - 248^2$."
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"index": 6713,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of $\\left(81\\right)^{0.25} \\cdot \\left(81\\right)^{0.2}$."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\cdot \\sqrt[5]{3^4}"
} | {
"index": 6714,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of $\\sqrt{105^{3}-104^{3}}$, given that it is a positive integer."
}
] | math | {
"style": "rule",
"ground_truth": "181"
} | {
"index": 6715,
"solution": "First compute $105^{3}-104^{3}=105^{2}+105 \\cdot 104+104^{2}=3 \\cdot 105 \\cdot 104+1=32761$. Note that $180^{2}=32400$, so $181^{2}=180^{2}+2 \\cdot 180+1=32761$ as desired."
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of $k$ such that the equation\n\\[\\frac{x + 2}{kx - 1} = x\\]has exactly one solution."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 6716,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of $p$ such that the equation\n\\[\\frac{2x + 3}{px - 2} = x\\]\nhas exactly one solution."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}"
} | {
"index": 6717,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of $x$ such that\n\n$\\left(1+\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}\\cdots\\right)\\left(1-\\frac{1}{2}+\\frac{1}{4}-\\frac{1}{8}+\\cdots\\right)=1+\\frac{1}{x}+\\frac{1}{x^2}+\\frac{1}{x^3}+\\cdots$."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 6718,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of \\(1^{25}+2^{24}+3^{23}+\\ldots+24^{2}+25^{1}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "66071772829247409"
} | {
"index": 6719,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of \\(\\frac{\\cos 30.5^{\\circ}+\\cos 31.5^{\\circ}+\\ldots+\\cos 44.5^{\\circ}}{\\sin 30.5^{\\circ}+\\sin 31.5^{\\circ}+\\ldots+\\sin 44.5^{\\circ}}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "(\\sqrt{2}-1)(\\sqrt{3}+\\sqrt{2})=2-\\sqrt{2}-\\sqrt{3}+\\sqrt{6}"
} | {
"index": 6720,
"solution": "Consider a 360-sided regular polygon with side length 1, rotated so that its sides are at half-degree inclinations (that is, its sides all have inclinations of \\(0.5^{\\circ}, 1.5^{\\circ}, 2.5^{\\circ}\\), and so on. Go to the bottom point on this polygon and then move clockwise, num... |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of \\[M = 50^2 + 48^2 - 46^2 + 44^2 + 42^2 - 40^2 + \\cdots + 4^2 + 2^2 - 0^2,\\] where the additions and subtractions alternate in triplets."
}
] | math | {
"style": "rule",
"ground_truth": "2600"
} | {
"index": 6721,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of \\[N = 100^2 + 99^2 - 98^2 - 97^2 + 96^2 + \\cdots + 4^2 + 3^2 - 2^2 - 1^2,\\]where the additions and subtractions alternate in pairs."
}
] | math | {
"style": "rule",
"ground_truth": "10100"
} | {
"index": 6722,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of the expression:\n\\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2)))))))) \\]"
}
] | math | {
"style": "rule",
"ground_truth": "1022"
} | {
"index": 6723,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of the expression:\n\\[ 2(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4))))))))) \\]"
}
] | math | {
"style": "rule",
"ground_truth": "699050"
} | {
"index": 6724,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of the following expression:\n\\[ 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2(1 + 2))))))))). \\]"
}
] | math | {
"style": "rule",
"ground_truth": "2046"
} | {
"index": 6725,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the value of the infinite series \\[\n\\sum_{n=2}^{\\infty} \\frac{n^4+3n^2+10n+10}{2^n \\cdot \\left(n^4+4\\right)}\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{10}"
} | {
"index": 6726,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the values of $\\binom{600}{600}$, $\\binom{600}{0}$, and $\\binom{600}{1}$."
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 6727,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute the volumes of the solids obtained by rotating the regions bounded by the graphs of the functions about the y-axis.\n\n$$\ny=x^{3}, \\quad y=x\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\pi}{15}"
} | {
"index": 6728,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute without using a calculator: $12!/11!$"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 6729,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute without using a calculator: $42!/40!$"
}
] | math | {
"style": "rule",
"ground_truth": "1,\\!722"
} | {
"index": 6730,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute without using a calculator: $8!-7!$"
}
] | math | {
"style": "rule",
"ground_truth": "35,\\!280"
} | {
"index": 6731,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute without using a calculator: $9!/8!$"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 6732,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute without using a calculator: $\\dfrac{9!}{6!3!}$"
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 6733,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute, in terms of $n$, $\\sum_{k=0}^{n}\\binom{n-k}{k} 2^{k}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\cdot 2^{n}+(-1)^{n}}{3}"
} | {
"index": 6734,
"solution": "Let $T_{n}=\\sum_{k=0}^{n}\\binom{n-k}{k} 2^{k}$. From Pascal's recursion for binomial coefficients, we can find $T_{n}=2 T_{n-2}+T_{n-1}$, with $T_{0}=1$ and $T_{1}=1$. The characteristic polynomial of this recursion is $x^{2}-x-2=0$, which has roots 2 and -1. Thus $T_{n}=a \\cdot 2^{... |
DeepScaleR | [
{
"role": "user",
"content": "Compute: $8 + 6(3-8)^2$."
}
] | math | {
"style": "rule",
"ground_truth": "158"
} | {
"index": 6735,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute: $$\\left( \\frac {8}{27}\\right)\\,^{- \\frac {2}{3}}-\\log {\\sqrt {2}}-\\log {\\sqrt {5}}=$$ \\_\\_\\_\\_\\_\\_ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{4}"
} | {
"index": 6736,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute: $$\\left\\lfloor\\frac{2005^{3}}{2003 \\cdot 2004}-\\frac{2003^{3}}{2004 \\cdot 2005}\\right\\rfloor$$"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 6737,
"solution": "Let $x=2004$. Then the expression inside the floor brackets is $$\\frac{(x+1)^{3}}{(x-1) x}-\\frac{(x-1)^{3}}{x(x+1)}=\\frac{(x+1)^{4}-(x-1)^{4}}{(x-1) x(x+1)}=\\frac{8 x^{3}+8 x}{x^{3}-x}=8+\\frac{16 x}{x^{3}-x}$$ Since $x$ is certainly large enough that $0<16 x /(x^{3}-x)<1$, the ans... |
DeepScaleR | [
{
"role": "user",
"content": "Compute: $${0.027}^{− \\frac{1}{3}}−{(− \\frac{1}{7})}^{−2}+{256}^{ \\frac{3}{4}}−{3}^{−1}+{( \\sqrt{2}−1)}^{0} = $$ \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 6738,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute: $(17+10)^2-(17-10)^2$."
}
] | math | {
"style": "rule",
"ground_truth": "680"
} | {
"index": 6739,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute: $104 \\times 96$."
}
] | math | {
"style": "rule",
"ground_truth": "9984"
} | {
"index": 6740,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute: $55\\times1212-15\\times1212$ ."
}
] | math | {
"style": "rule",
"ground_truth": "48480"
} | {
"index": 6741,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute: $5^2-3(4)+3^2$."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 6742,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute: $9 \\cdot \\frac{1}{13} \\cdot 26.$"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 6743,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute: $98 \\times 102$."
}
] | math | {
"style": "rule",
"ground_truth": "9996"
} | {
"index": 6744,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute: $\\displaystyle \\frac{66,\\!666^4}{22,\\!222^4}$."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 6745,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute: $\\frac{1}{5} + \\frac{2}{5} + \\frac{3}{5} + \\dots + \\frac{9}{5} + \\frac{10}{5}$."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 6746,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute: $\\frac{\\cos 10^{\\circ} - 2\\sin 20^{\\circ}}{\\sin 10^{\\circ}} = \\_\\_\\_\\_\\_\\_ \\text{.}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 6747,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute: $\\left(\\frac{1}{2} \\right)^{3} \\cdot \\left(\\frac{1}{7} \\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{56}"
} | {
"index": 6748,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute: $\\sin 187^{\\circ}\\cos 52^{\\circ}+\\cos 7^{\\circ}\\sin 52^{\\circ}=\\_\\_\\_\\_\\_\\_ \\cdot$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 6749,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute: \\( 4.165 \\times 4.8 + 4.165 \\times 6.7 - 4.165 \\div \\frac{2}{3} = \\)"
}
] | math | {
"style": "rule",
"ground_truth": "41.65"
} | {
"index": 6750,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute: \\(103 \\times 97\\)."
}
] | math | {
"style": "rule",
"ground_truth": "9991"
} | {
"index": 6751,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Compute: \\(93 \\times 107\\)."
}
] | math | {
"style": "rule",
"ground_truth": "9951"
} | {
"index": 6752,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Concentric circles $\\Omega_1$ and $\\Omega_2$ with radii $1$ and $100$ , respectively, are drawn with center $O$ . Points $A$ and $B$ are chosen independently at random on the circumferences of $\\Omega_1$ and $\\Omega_2$ , respectively. Denote by $\\ell$ the ... | math | {
"style": "rule",
"ground_truth": "10004"
} | {
"index": 6753,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Conduct an experiment by throwing 2 dice, and denote the point P with the coordinates $(x, y)$, where $x$ represents the number shown on the first die, and $y$ represents the number shown on the second die. \n(I) Find the probability that point P lies on the line $y = x$. \n(... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{36}"
} | {
"index": 6754,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Congcong performs a math magic trick by writing the numbers $1, 2, 3, 4, 5, 6, 7$ on the blackboard and lets others select 5 of these numbers. The product of these 5 numbers is then calculated and told to Congcong, who guesses the chosen numbers. If when it is Benben's turn to ... | math | {
"style": "rule",
"ground_truth": "420"
} | {
"index": 6755,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Connecting the centers of adjacent faces of a cube forms a regular octahedron. What is the volume ratio of this octahedron to the cube?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{6}$"
} | {
"index": 6756,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Connecting the right-angled vertex of a right triangle and the two trisection points on the hypotenuse, the lengths of the two resulting line segments are $\\sin \\alpha$ and $\\cos \\alpha$ (where $0 < \\alpha < \\frac{\\pi}{2}$). What is the length of the hypotenuse?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{3}{\\sqrt{5}}$"
} | {
"index": 6757,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Connie has a number of gold bars, all of different weights. She gives the 24 lightest bars, which weigh $45 \\%$ of the total weight, to Brennan. She gives the 13 heaviest bars, which weigh $26 \\%$ of the total weight, to Maya. How many bars did Blair receive?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 6758,
"solution": "Connie gives 24 bars that account for $45 \\%$ of the total weight to Brennan. Thus, each of these 24 bars accounts for an average of $\\frac{45}{24} \\%=\\frac{15}{8} \\%=1.875 \\%$ of the total weight. Connie gives 13 bars that account for $26 \\%$ of the total weight to Maya. Thus, ... |
DeepScaleR | [
{
"role": "user",
"content": "Connie multiplies a number by 2 and gets 60 as her answer. However, she should have divided the number by 2 to get the correct answer. What is the correct answer?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 6759,
"solution": "Let's denote the number Connie should have used as $x$. According to the problem, Connie mistakenly multiplied $x$ by $2$ instead of dividing it by $2$. This gives us two equations based on her actions and what she should have done:\n\n1. **Mistaken Calculation:**\n \\[ 2x = 60 \\]\n... |
DeepScaleR | [
{
"role": "user",
"content": "Connie multiplies a number by 4 and gets 200 as her result. She realizes she should have divided the number by 4 and then added 10 to get the correct answer. Find the correct value of this number."
}
] | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"index": 6760,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consecutive powers of 3 are added to form this sequence: $3^0,3^0+ 3^1, 3^0+ 3^1+ 3^2$, and so on. What is the simplified value of the fourth term of the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 6761,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider $13$ marbles that are labeled with positive integers such that the product of all $13$ integers is $360$ . Moor randomly picks up $5$ marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 6762,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider $7$ points on a circle. Compute the number of ways there are to draw chords between pairs of points such that two chords never intersect and one point can only belong to one chord. It is acceptable to draw no chords."
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"index": 6763,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider $\\triangle \\natural\\flat\\sharp$ . Let $\\flat\\sharp$ , $\\sharp\\natural$ and $\\natural\\flat$ be the answers to problems $4$ , $5$ , and $6$ , respectively. If the incircle of $\\triangle \\natural\\flat\\sharp$ touches $\\natural\\flat$ at $\\od... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"index": 6764,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider $9$ points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of $\\,n\\,$ such that whenever exactly $\\,n\\,$ edges are co... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 6765,
"solution": "\nConsider a configuration where you have 9 points in space, with each pair of points joined by an edge, for a total of \\(\\binom{9}{2} = 36\\) edges. We want to find the smallest \\( n \\) such that if exactly \\( n \\) edges are colored (either blue or red), there must exist a monoc... |
DeepScaleR | [
{
"role": "user",
"content": "Consider $n$ disks $C_{1}, C_{2}, \\ldots, C_{n}$ in a plane such that for each $1 \\leq i<n$, the center of $C_{i}$ is on the circumference of $C_{i+1}$, and the center of $C_{n}$ is on the circumference of $C_{1}$. Define the score of such an arrangement of $n$ disks to be th... | math | {
"style": "rule",
"ground_truth": "(n-1)(n-2)/2"
} | {
"index": 6766,
"solution": "The answer is $(n-1)(n-2) / 2$. Let's call a set of $n$ disks satisfying the given conditions an $n$-configuration. For an $n$ configuration $\\mathcal{C}=\\left\\{C_{1}, \\ldots, C_{n}\\right\\}$, let $S_{\\mathcal{C}}=\\left\\{(i, j) \\mid C_{i}\\right.$ properly contains $\\left.C_{... |
DeepScaleR | [
{
"role": "user",
"content": "Consider $x^2+px+q=0$, where $p$ and $q$ are positive numbers. If the roots of this equation differ by 1, then $p$ equals"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{4q+1}"
} | {
"index": 6767,
"solution": "1. **Identify the roots of the quadratic equation**: Given the quadratic equation $x^2 + px + q = 0$, the roots can be found using the quadratic formula:\n \\[\n x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}\n \\]\n Here, $a = 1$, $b = p$, and $c = q$. Thus, the roots are:\n \\[\... |
DeepScaleR | [
{
"role": "user",
"content": "Consider \\(A \\in \\mathcal{M}_{2020}(\\mathbb{C})\\) such that\n\\[ A + A^{\\times} = I_{2020} \\]\n\\[ A \\cdot A^{\\times} = I_{2020} \\]\nwhere \\(A^{\\times}\\) is the adjugate matrix of \\(A\\), i.e., the matrix whose elements are \\(a_{ij} = (-1)^{i+j} d_{ji}\\), where ... | math | {
"style": "rule",
"ground_truth": "673"
} | {
"index": 6768,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $2\\times 3$ grid where each entry is either $0$ , $1$ , or $2$ . For how many such grids is the sum of the numbers in every row and in every column a multiple of $3$ ? One valid grid is shown below: $$ \\begin{bmatrix} 1 & 2 & 0 2 & 1 & 0 \\end{bmatrix} $$"
... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 6769,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a \"Modulo $m$ graph paper,\" with $m = 17$, forming a grid of $17^2$ points. Each point $(x, y)$ represents integer pairs where $0 \\leq x, y < 17$. We graph the congruence $$5x \\equiv 3y + 2 \\pmod{17}.$$ The graph has single $x$-intercept $(x_0, 0)$ and single $y$-... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 6770,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $10 \\times 10$ grid of squares. One day, Daniel drops a burrito in the top left square, where a wingless pigeon happens to be looking for food. Every minute, if the pigeon and the burrito are in the same square, the pigeon will eat $10 \\%$ of the burrito's original... | math | {
"style": "rule",
"ground_truth": "71.8"
} | {
"index": 6771,
"solution": "Label the squares using coordinates, letting the top left corner be $(0,0)$. The burrito will end up in 10 (not necessarily different) squares. Call them $p_{1}=\\left(x_{1}, y_{1}\\right)=(0,0), p_{2}=\\left(x_{2}, y_{2}\\right), \\ldots, p_{10}=\\left(x_{10}, y_{10}\\right)$. $p_{2}$... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $10\\times10$ checkerboard with alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard?"
}
] | math | {
"style": "rule",
"ground_truth": "140"
} | {
"index": 6772,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $2 \\times 2$ grid of squares. David writes a positive integer in each of the squares. Next to each row, he writes the product of the numbers in the row, and next to each column, he writes the product of the numbers in each column. If the sum of the eight numbers he ... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 6773,
"solution": "Let the four numbers be $a, b, c, d$, so that the other four numbers are $a b, a d, b c, b d$. The sum of these eight numbers is $a+b+c+d+a b+a d+b c+b d=(a+c)+(b+d)+(a+c)(b+d)=2015$, and so $(a+c+1)(b+d+1)=2016$. Since we seek to minimize $a+b+c+d$, we need to find the two factors of ... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $2 \\times 2$ grid of squares. Each of the squares will be colored with one of 10 colors, and two colorings are considered equivalent if one can be rotated to form the other. How many distinct colorings are there?"
}
] | math | {
"style": "rule",
"ground_truth": "2530"
} | {
"index": 6774,
"solution": "This solution will be presented in the general case with $n$ colors. Our problem asks for $n=10$. We isolate three cases: Case 1: Every unit square has the same color In this case there are clearly $n$ ways to color the square. Case 2: Two non-adjacent squares are the same color, and t... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $2 \\times n$ grid of points and a path consisting of $2 n-1$ straight line segments connecting all these $2 n$ points, starting from the bottom left corner and ending at the upper right corner. Such a path is called efficient if each point is only passed through onc... | math | {
"style": "rule",
"ground_truth": "\\binom{4030}{2015}"
} | {
"index": 6775,
"solution": "The general answer is $\\binom{2(n-1)}{n-1}$ : Simply note that the points in each column must be taken in order, and anything satisfying this avoids intersections, so just choose the steps during which to be in the first column."
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $3 \\times 3$ array where each row and each column is an arithmetic sequence with three terms. The first term of the first row is $3$, and the last term of the first row is $15$. Similarly, the first term of the last row is $9$, and the last term of the last row is $... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 6776,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $3 \\times 3$ block of squares as the center area in an array of unit squares. The first ring around this center block contains unit squares that directly touch the block. If the pattern continues as before, how many unit squares are in the $10^{th}$ ring?"
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 6777,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $3 \\times 3$ grid of squares. A circle is inscribed in the lower left corner, the middle square of the top row, and the rightmost square of the middle row, and a circle $O$ with radius $r$ is drawn such that $O$ is externally tangent to each of the three inscribed c... | math | {
"style": "rule",
"ground_truth": "\\frac{5 \\sqrt{2}-3}{6}"
} | {
"index": 6778,
"solution": "Let $A$ be the center of the square in the lower left corner, let $B$ be the center of the square in the middle of the top row, and let $C$ be the center of the rightmost square in the middle row. It's clear that $O$ is the circumcenter of triangle $A B C$ - hence, the desired radius i... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $4 \\times 4$ grid of squares with 25 grid points. Determine the number of different lines passing through at least 3 of these grid points."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 6779,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $4 \\times 4$ grid of squares, each of which are originally colored red. Every minute, Piet can jump on one of the squares, changing the color of it and any adjacent squares (two squares are adjacent if they share a side) to blue. What is the minimum number of minute... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 6780,
"solution": "Piet can change the colors of at most 5 squares per minute, so as there are 16 squares, it will take him at least four minutes to change the colors of every square. Some experimentation yields that it is indeed possible to make the entire grid blue after 4 minutes; one example is shown... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $4 \\times 4$ grid of squares. Aziraphale and Crowley play a game on this grid, alternating turns, with Aziraphale going first. On Aziraphales turn, he may color any uncolored square red, and on Crowleys turn, he may color any uncolored square blue. The game ends whe... | math | {
"style": "rule",
"ground_truth": "\\[ 6 \\]"
} | {
"index": 6781,
"solution": "We claim that the answer is 6. On Aziraphale's first two turns, it is always possible for him to take 2 adjacent squares from the central four; without loss of generality, suppose they are the squares at $(1,1)$ and $(1,2)$. If allowed, Aziraphale's next turn will be to take one of the... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $4 \\times 4$ matrix filled with each number from $1$ to $16$. Each number must be placed such that the entries in each row and in each column are in increasing order. Find the number of such matrices that exist."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 6782,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $5 \\times 5$ grid of squares, where each square is either colored blue or left blank. The design on the grid is considered symmetric if it remains unchanged under a 90° rotation around the center. How many symmetric designs can be created if there must be at least o... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 6783,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $5 \\times 5$ grid of squares. Vladimir colors some of these squares red, such that the centers of any four red squares do not form an axis-parallel rectangle (i.e. a rectangle whose sides are parallel to those of the squares). What is the maximum number of squares h... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 6784,
"solution": "We claim that the answer is 12. We first show that if 13 squares are colored red, then some four form an axis-parallel rectangle. Note that we can swap both columns and rows without affecting whether four squares form a rectangle, so we may assume without loss of generality that the to... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $6 \\times 6$ grid of squares. Edmond chooses four of these squares uniformly at random. What is the probability that the centers of these four squares form a square?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{561}"
} | {
"index": 6785,
"solution": "Firstly, there are $\\binom{36}{4}$ possible combinations of points. Call a square proper if its sides are parallel to the coordinate axes and improper otherwise. Note that every improper square can be inscribed in a unique proper square. Hence, an $n \\times n$ proper square represent... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $6\\times 6\\times h$ rectangular box containing a larger sphere of radius $3$ and four smaller spheres, each with radius $2$. The smaller spheres are placed at each corner of the bottom square face of the box and are tangent to two adjacent sides of the box. The lar... | math | {
"style": "rule",
"ground_truth": "5 + \\sqrt{23}"
} | {
"index": 6786,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $7 \\times 7$ grid of squares. Let $f:\\{1,2,3,4,5,6,7\\} \\rightarrow\\{1,2,3,4,5,6,7\\}$ be a function; in other words, $f(1), f(2), \\ldots, f(7)$ are each (not necessarily distinct) integers from 1 to 7 . In the top row of the grid, the numbers from 1 to 7 are wr... | math | {
"style": "rule",
"ground_truth": "1470"
} | {
"index": 6787,
"solution": "Consider the directed graph with $1,2,3,4,5,6,7$ as vertices, and there is an edge from $i$ to $j$ if and only if $f(i)=j$. Since the bottom row is equivalent to the top one, we have $f^{6}(x)=x$. Therefore, the graph must decompose into cycles of length $6,3,2$, or 1 . Furthermore, si... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a $9 \\times 9$ grid of squares. Haruki fills each square in this grid with an integer between 1 and 9 , inclusive. The grid is called a super-sudoku if each of the following three conditions hold: - Each column in the grid contains each of the numbers $1,2,3,4,5,6,7,8... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 6788,
"solution": "Without loss of generality, suppose that the top left corner contains a 1 , and examine the top left $3 \\times 4$ : \\begin{tabular}{|c|c|c|c|} \\hline 1 & x & x & x \\\\ \\hline x & x & x & $*$ \\\\ \\hline x & x & x & $*$ \\\\ \\hline \\end{tabular} There cannot be another 1 in any ... |
DeepScaleR | [
{
"role": "user",
"content": "Consider a 4-by-4 grid where each of the unit squares can be colored either purple or green. Each color choice is equally likely independent of the others. Compute the probability that the grid does not contain a 3-by-3 grid of squares all colored purple. Express your result in... | math | {
"style": "rule",
"ground_truth": "255"
} | {
"index": 6789,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a 4x4 grid of points (equally spaced). How many rectangles, of any size, can be formed where each of its four vertices are points on this grid?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 6790,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a 4x4 grid with points that are equally spaced horizontally and vertically, where the distance between two neighboring points is 1 unit. Two triangles are formed: Triangle A connects points at (0,0), (3,2), and (2,3), while Triangle B connects points at (0,3), (3,3), a... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 6791,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a 5-minute interval. In this period, an average of 5 bites occur on the first fishing rod, and 1 bite on the second fishing rod. Therefore, the total average number of bites on both rods during these 5 minutes is 6. Determine the average waiting time for the first bite... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 6792,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a 5x5 grid of squares. How many different squares can be traced using the lines in this grid?"
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 6793,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a 6x3 grid where you can move only to the right or down. How many valid paths are there from top-left corner $A$ to bottom-right corner $B$, if paths passing through segment from $(4,3)$ to $(4,2)$ and from $(2,1)$ to $(2,0)$ are forbidden? [The coordinates are given i... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 6794,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a 9x9 chessboard where the squares are labelled from a starting square at the bottom left (1,1) increasing incrementally across each row to the top right (9,9). Each square at position $(i,j)$ is labelled with $\\frac{1}{i+j-1}$. Nine squares are chosen such that there... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{362880}"
} | {
"index": 6795,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a beam of light starting from point $A$. It propagates within a plane and reflects between the lines $AD$ and $CD$. After several reflections, it perpendicularly hits point $B$ (which may lie on either $AD$ or $CD$), then reflects back to point $A$ along the original p... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 6796,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a board on $2013 \\times 2013$ squares, what is the maximum number of chess knights that can be placed so that no $2$ attack each other?"
}
] | math | {
"style": "rule",
"ground_truth": "2026085"
} | {
"index": 6797,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a bug starting at vertex $A$ of a cube, where each edge of the cube is 1 meter long. At each vertex, the bug can move along any of the three edges emanating from that vertex, with each edge equally likely to be chosen. Let $p = \\frac{n}{6561}$ represent the probabilit... | math | {
"style": "rule",
"ground_truth": "1641"
} | {
"index": 6798,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Consider a central regular hexagon surrounded by six regular hexagons, each of side length $\\sqrt{2}$. Three of these surrounding hexagons are selected at random, and their centers are connected to form a triangle. Calculate the area of this triangle."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 6799,
"solution": ""
} |
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