Datasets:
data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "\"In a tree with black pearls hidden, this item is only available in May. Travelers who pass by taste one, with a mouthful of sweetness and sourness, never wanting to leave.\" The Dongkui waxberry is a sweet gift in summer. Each batch of Dongkui waxberries must undergo two roun... | math | {
"style": "rule",
"ground_truth": "640"
} | {
"index": 0,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\"Modulo $m$ graph paper\" consists of a grid of $13^2$ points, representing all pairs of integer residues $(x,y)$ where $0\\le x, y <13$. To graph a congruence on modulo $13$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. Consider the graph of $$4x \\e... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 1,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\"Modulo $m$ graph paper\" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0\\le x, y <m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $y\\equiv... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 2,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\"Modulo $m$ graph paper\" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0\\le x<m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $y\\equiv x^2... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 3,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\"My phone number,\" said the trip leader to the kids, \"is a five-digit number. The first digit is a prime number, and the last two digits are obtained from the previous pair (which represents a prime number) by rearrangement, forming a perfect square. The number formed by rev... | math | {
"style": "rule",
"ground_truth": "26116"
} | {
"index": 4,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\"Sun\", \"Xing\", and \"Zhe\" each represent a digit. The product of the two three-digit numbers 'SunXingZhe' and 'ZheXingSun' is 78445. Can you find the sum of these two three-digit numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "686"
} | {
"index": 5,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\"The Nine Chapters on the Mathematical Art\" is an ancient Chinese mathematical text, which records: \"If it can be halved, then halve it; if not, juxtapose the numerator and denominator, subtract the lesser from the greater, continue to subtract in turn, seeking their equalit... | math | {
"style": "rule",
"ground_truth": "273"
} | {
"index": 6,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "\"The Nine Chapters on the Mathematical Art\" is the first mathematical monograph in China, which includes the following problem: \"There is a gold rod, 5 feet in length. Cutting 1 foot from the base, it weighs 4 jin. Cutting 1 foot from the tip, it weighs 2 jin. How much does ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 7,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ (a_n)_{n \\equal{} 1}^\\infty$ is defined on real numbers with $ a_n \\not \\equal{} 0$ , $ a_na_{n \\plus{} 3} \\equal{} a_{n \\plus{} 2}a_{n \\plus{} 5}$ and $ a_1a_2 \\plus{} a_3a_4 \\plus{} a_5a_6 \\equal{} 6$ . So $ a_1a_2 \\plus{} a_3a_4 \\plus{} \\cdots \\plus{}... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 8,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ A$ and $ B$ play the following game with a polynomial of degree at least 4:\r\n\\[ x^{2n} \\plus{} \\_x^{2n \\minus{} 1} \\plus{} \\_x^{2n \\minus{} 2} \\plus{} \\ldots \\plus{} \\_x \\plus{} 1 \\equal{} 0\r\n\\]\r\n$ A$ and $ B$ take turns to fill in one of the blanks with... | math | {
"style": "rule",
"ground_truth": "B"
} | {
"index": 9,
"solution": "\n\nIn this game, Player \\( A \\) and Player \\( B \\) take turns filling in the coefficients of the polynomial \n\\[ P(x) = x^{2n} + a_{2n-1} x^{2n-1} + a_{2n-2} x^{2n-2} + \\ldots + a_1 x + 1. \\]\nPlayer \\( A \\) wins if the resulting polynomial has no real roots, and Player \\( B \\... |
DeepScaleR | [
{
"role": "user",
"content": "$ M$ is a subset of $ \\{1, 2, 3, \\ldots, 15\\}$ such that the product of any three distinct elements of $ M$ is not a square. Determine the maximum number of elements in $ M.$"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 10,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ S$ is a non-empty subset of the set $ \\{ 1, 2, \\cdots, 108 \\}$, satisfying:\r\n\r\n(1) For any two numbers $ a,b \\in S$ ( may not distinct), there exists $ c \\in S$, such that $ \\gcd(a,c)\\equal{}\\gcd(b,c)\\equal{}1$.\r\n\r\n(2) For any two numbers $ a,b \\in S$ ( may ... | math | {
"style": "rule",
"ground_truth": "79"
} | {
"index": 11,
"solution": "\nLet \\( S \\) be a non-empty subset of the set \\( \\{ 1, 2, \\ldots, 108 \\} \\) satisfying the following conditions:\n\n1. For any two numbers \\( a, b \\in S \\) (not necessarily distinct), there exists \\( c \\in S \\) such that \\( \\gcd(a, c) = \\gcd(b, c) = 1 \\).\n2. For any tw... |
DeepScaleR | [
{
"role": "user",
"content": "$ f(x)$ is a given polynomial whose degree at least 2. Define the following polynomial-sequence: $ g_1(x)\\equal{}f(x), g_{n\\plus{}1}(x)\\equal{}f(g_n(x))$ , for all $ n \\in N$ . Let $ r_n$ be the average of $ g_n(x)$ 's roots. If $ r_{19}\\equal{}99$ , find $ r_{99}$... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 12,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ f\\left( x \\right) \\equal{} \\frac {x^5}{5x^4 \\minus{} 10x^3 \\plus{} 10x^2 \\minus{} 5x \\plus{} 1}$ . $ \\sum_{i \\equal{} 1}^{2009} f\\left( \\frac {i}{2009} \\right) \\equal{} ?$"
}
] | math | {
"style": "rule",
"ground_truth": "1005"
} | {
"index": 13,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ x$ and $ y$ are two distinct positive integers. What is the minimum positive integer value of $ (x + y^2)(x^2 - y)/(xy)$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 14,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$$\n\\text{Consider the system of inequalities:}\n\\begin{cases}\nx + 2y \\leq 6 \\\\\n3x + y \\geq 3 \\\\\nx \\leq 4 \\\\\ny \\geq 0\n\\end{cases}\n$$\nDetermine the number of units in the length of the longest side of the polygonal region formed by this system. Express your a... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"index": 15,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$$\\frac {4}{5} + 9 \\frac {4}{5} + 99 \\frac {4}{5} + 999 \\frac {4}{5} + 9999 \\frac {4}{5} + 1 = \\_\\_\\_\\_\\_\\_.$$"
}
] | math | {
"style": "rule",
"ground_truth": "11111"
} | {
"index": 16,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(-1)^{5^{2}} + 1^{2^{5}} = $"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 17,
"solution": "1. **Evaluate the first term** $(-1)^{5^2}$:\n - Calculate the exponent: $5^2 = 25$.\n - Since $(-1)^{n}$ is $-1$ if $n$ is odd, and $25$ is odd, we have $(-1)^{25} = -1$.\n\n2. **Evaluate the second term** $1^{2^5}$:\n - Calculate the exponent: $2^5 = 32$.\n - Any number raised ... |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ $f(n)$ is a function defined on the set of positive integers, satisfying:<br/>① When $n$ is a positive integer, $f(f(n))=4n+9$;<br/>② When $k$ is a non-negative integer, $f(2^{k})=2^{k+1}+3$. Find the value of $f(1789)$.<br/>$(2)$ The function $f$ is defined on the set of... | math | {
"style": "rule",
"ground_truth": "364"
} | {
"index": 18,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ Calculate: $(\\frac{1}{2})^{-1}+(\\sqrt{2})^{2}-4\\times |-\\frac{1}{2}|$. \n$(2)$ Simplify first, then find the value: $(1+\\frac{4}{a-1})÷\\frac{{a}^{2}+6a+9}{{a}^{2}-a}$, where $a=2$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 19,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ Calculate: $2^{-1}+|\\sqrt{6}-3|+2\\sqrt{3}\\sin 45^{\\circ}-\\left(-2\\right)^{2023}\\cdot (\\frac{1}{2})^{2023}$. \n$(2)$ Simplify and then evaluate: $\\left(\\frac{3}{a+1}-a+1\\right) \\div \\frac{{{a}^{2}}-4}{{{a}^{2}}+2a+1}$, where $a$ takes a suitable value from $-... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 20,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ Calculate: $\\sqrt{12}-(-\\frac{1}{2})^{-1}-|\\sqrt{3}+3|+(2023-\\pi)^0$<br/>$(2)$ Simplify the algebraic expression $\\frac{3x-8}{x-1}-\\frac{x+1}{x}÷\\frac{{x}^{2}-1}{{x}^{2}-3x}$, and select appropriate integers from $0 \\lt x\\leqslant 3$ to substitute and find the va... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 21,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ Calculate: $\\sqrt{3}\\tan 45^{\\circ}-\\left(2023-\\pi \\right)^{0}+|2\\sqrt{3}-2|+(\\frac{1}{4})^{-1}-\\sqrt{27}$; <br/>$(2)$ Simplify first, then evaluate: $\\frac{{{x^2}-x}}{{{x^2}+2x+1}}\\div (\\frac{2}{{x+1}}-\\frac{1}{x})$, simplify it first, then choose an integer... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 22,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ Calculate: $\\sqrt{9}+2\\sin30{}°-(π-3){}°$;<br/>$(2)$ Solve the equation: $\\left(2x-3\\right)^{2}=2\\left(2x-3\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 23,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ Calculate: $tan60°×{({-2})^{-1}}-({\\sqrt{\\frac{3}{4}}-\\sqrt[3]{8}})+|{-\\frac{1}{2}\\sqrt{12}}|$;<br/>$(2)$ Simplify and find the value: $({\\frac{{x+2}}{{{x^2}-2x}}-\\frac{{x-1}}{{{x^2}-4x+4}}})÷\\frac{{x-4}}{x}$, where $x=\\sqrt{2}+2$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 24,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ Calculate: $|2-\\sqrt{3}|+(\\sqrt{2}+1)^0+3\\tan30°+(-1)^{2023}-(\\frac{1}{2})^{-1}$. \n$(2)$ Simplify first and then find the value: $(\\frac{x-1}{x}-\\frac{x-2}{x+1})÷\\frac{2x^2-x}{x^2+2x+1}$, where $x$ satisfies $x^{2}-2x-2=0$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 25,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ Find the value of $x$: $4\\left(x+1\\right)^{2}=49$;<br/>$(2)$ Calculate: $\\sqrt{9}-{({-1})^{2018}}-\\sqrt[3]{{27}}+|{2-\\sqrt{5}}|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5} - 3"
} | {
"index": 26,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ Given $x \\gt 0$, $y \\gt 0$, and $2x+3y=6$, find the maximum value of $xy$;<br/>$(2)$ Given $x＜\\frac{1}{2}$, find the maximum value of $y=2x+\\frac{4}{2x-1}$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 27,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ Given real numbers $x \\gt 0$, $y \\gt 0$, and $\\frac{2}{x}+y=1$, find the minimum value of $2x+\\frac{1}{3y}$;<br/>$(2)$ Given real numbers $x$ and $y$ satisfying $x+y=1$, $x \\gt 0$, $y \\gt 0$, find the minimum value of $\\frac{1}{2x}+\\frac{x}{y+1}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"index": 28,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ Given the function $f(x) = |x+1| + |2x-4|$, find the solution to $f(x) \\geq 6$;<br/>$(2)$ Given positive real numbers $a$, $b$, $c$ satisfying $a+2b+4c=8$, find the minimum value of $\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11+6\\sqrt{2}}{8}"
} | {
"index": 29,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ If $A=\\frac{a-1}{a+2}•\\frac{{a}^{2}-4}{{a}^{2}-2a+1}÷\\frac{1}{a-1}$, simplify $A$;<br/>$(2)$ If $a$ satisfies $a^{2}-a=0$, find the value of $A$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 30,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ Solve the fractional equation: $\\frac{x}{x+1}=\\frac{2x}{3x+3}+1$;<br/>$(2)$ Simplify first ($\\frac{x+2}{{x}^{2}-2x}-\\frac{x-1}{{x}^{2}-4x+4})\\div \\frac{x+2}{{x}^{3}-4x}$, then choose a suitable number from $2$, $0$, $-1$ to substitute into the simplified result for ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"index": 31,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ Solve the inequality: $3A_{x}^{3}≤2A_{x+1}^{2}+6A_{x}^{2}$;<br/>$(2)$ Find the value of $C_{n}^{5-n}+C_{n+1}^{9-n}$;<br/>$(3)$ Given $\\frac{1}{C_{5}^{m}}-\\frac{1}{C_{6}^{m}}=\\frac{7}{10C_{7}^{m}}$, find $C_{8}^{m}$."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 32,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ Solve the system of inequalities: $\\left\\{\\begin{array}{l}{2x-1＜5,①}\\\\{\\frac{4-x}{2}≥1,②}\\end{array}\\right.$<br/>$(2)$ Simplify first, then find the value: $\\frac{1}{a-1}÷(\\frac{{a}^{2}+a}{{a}^{2}-1}+\\frac{1}{a-1})$, where $a=(-2023)^0+(\\frac{1}{2})^{-1}⋅$"
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 33,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ State the operation laws or rules used in each step of the following calculation:<br/>$(-0.4)\\times \\left(-0.8\\right)\\times \\left(-1.25\\right)\\times 2.5$<br/>$=-\\left(0.4\\times 0.8\\times 1.25\\times 2.5\\right) \\text{(Step 1)}$<br/>$=-\\left(0.4\\times 2.5\\tim... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 34,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)$ When throwing two uniformly weighted dice, let $A=$\"the first time an odd number appears,\" $B=$\"the sum of the two dice is a multiple of $3$.\" Determine whether events $A$ and $B$ are independent, and explain the reason;<br/>$(2)$ Athletes A and B are participating in... | math | {
"style": "rule",
"ground_truth": "0.2212"
} | {
"index": 35,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1)\\sqrt{5}-27+|2-\\sqrt{5}|-\\sqrt{9}+(\\frac{1}{2})^{2}$；<br/>$(2)2\\sqrt{40}-5\\sqrt{\\frac{1}{10}}-\\sqrt{10}$；<br/>$(3)(3\\sqrt{12}-2\\sqrt{\\frac{1}{3}}-\\sqrt{48})÷4\\sqrt{3}-{(\\sqrt{2}-1)^0}$；<br/>$(4)(-\\sqrt{5}+\\sqrt{3})(\\sqrt{5}+\\sqrt{3})+(-\\sqrt{3}-1)^{2}$."
... | math | {
"style": "rule",
"ground_truth": "2+2\\sqrt{3}"
} | {
"index": 36,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(1+x^2)(1-x^3)$ equals"
}
] | math | {
"style": "rule",
"ground_truth": "1+x^2-x^3-x^5"
} | {
"index": 37,
"solution": "To solve the problem, we need to expand the expression $(1+x^2)(1-x^3)$ using the distributive property (also known as the FOIL method for binomials).\n\n1. **Distribute each term in the first binomial to each term in the second binomial:**\n \\[\n (1+x^2)(1-x^3) = 1 \\cdot 1 + 1 \\c... |
DeepScaleR | [
{
"role": "user",
"content": "$(100^2-99^2) + (98^2-97^2) + \\ldots + (2^2-1^2) = \\ $"
}
] | math | {
"style": "rule",
"ground_truth": "5050"
} | {
"index": 38,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(2 \\times 3 \\times 4)\\left(\\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4}\\right) = $"
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 39,
"solution": "1. **Calculate the product of the integers:** \n \\[\n 2 \\times 3 \\times 4 = 24\n \\]\n\n2. **Calculate the sum of the fractions:**\n \\[\n \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4}\n \\]\n To add these fractions, find a common denominator. The least common multiple of 2... |
DeepScaleR | [
{
"role": "user",
"content": "$(2x-1)^{10} = a_0 + a_1x + a_2x^2 + \\ldots + a_9x^9 + a_{10}x^{10}$, then $a_2 + a_3 + \\ldots + a_9 + a_{10} =$ \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 40,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(4^{-1} - 3^{-1})^{-1} = $"
}
] | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"index": 41,
"solution": "1. **Rewrite the expression using the properties of exponents and reciprocals:**\n \\[\n (4^{-1} - 3^{-1})^{-1} = \\left(\\frac{1}{4} - \\frac{1}{3}\\right)^{-1}\n \\]\n\n2. **Find a common denominator for the fractions inside the parentheses:**\n \\[\n \\frac{1}{4} - \\frac{1}... |
DeepScaleR | [
{
"role": "user",
"content": "$(81)^{\\frac12}=3^m$. Find $m$."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 42,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(HUN 6)$ Find the positions of three points $A,B,C$ on the boundary of a unit cube such that $min\\{AB,AC,BC\\}$ is the greatest possible."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 43,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any o... | math | {
"style": "rule",
"ground_truth": "1944"
} | {
"index": 44,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$(Ⅰ){0.064^{-\\frac{1}{3}}}+\\sqrt{{{(-2)}^4}}-{(π+e)^0}-{9^{\\frac{3}{2}}}×{({\\frac{{\\sqrt{3}}}{3}})^4}$；<br/>$(Ⅱ)\\frac{{lg18+lg5-lg60}}{{{{log}_2}27×lg2-lg8}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 45,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$-14-(-2)^{3}\\times \\dfrac{1}{4}-16\\times \\left(\\dfrac{1}{2}-\\dfrac{1}{4}+\\dfrac{3}{8}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "-22"
} | {
"index": 46,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$-2^{3}+|2-3|-2\\times \\left(-1\\right)^{2023}$."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 47,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$1,000,000,000,000-777,777,777,777=$"
}
] | math | {
"style": "rule",
"ground_truth": "$222,222,222,223$"
} | {
"index": 48,
"solution": "To solve the problem $1,000,000,000,000 - 777,777,777,777$, we can perform the subtraction directly:\n\n1. **Align the numbers for subtraction:**\n \\[\n \\begin{array}{r}\n 1,000,000,000,000 \\\\\n -777,777,777,777 \\\\\n \\end{array}\n \\]\n\n2. **Subtract each correspondin... |
DeepScaleR | [
{
"role": "user",
"content": "$1.$ A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; Thi... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 49,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$10 \\cdot 52 \\quad 1990-1980+1970-1960+\\cdots-20+10$ equals:"
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 50,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$100$ children stand in a line each having $100$ candies. In one move, one of them may take some of their candies and distribute them to a non-empty set of the remaining children. After what least number of moves can it happen that no two children have the same number of candie... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 51,
"solution": "\nTo solve the problem, we consider the process of redistributing candies among $100$ children such that no two children have the same number of candies. Initially, each child has $100$ candies.\n\nThe goal is to reach a state where all $100$ values are distinct. Let's outline the strate... |
DeepScaleR | [
{
"role": "user",
"content": "$100$ numbers $1$, $1/2$, $1/3$, $...$, $1/100$ are written on the blackboard. One may delete two arbitrary numbers $a$ and $b$ among them and replace them by the number $a + b + ab$. After $99$ such operations only one number is left. What is this final number?\n\n(D. Fomin, L... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 52,
"solution": "\nInitially, we have $100$ numbers written on the blackboard: \\(1, \\frac{1}{2}, \\frac{1}{3}, \\ldots, \\frac{1}{100}\\). In each step, two numbers \\(a\\) and \\(b\\) are selected and replaced with the number \\(a + b + ab\\). This process is repeated until only one number remains.\n\... |
DeepScaleR | [
{
"role": "user",
"content": "$1000 \\times 1993 \\times 0.1993 \\times 10 =$"
}
] | math | {
"style": "rule",
"ground_truth": "$(1993)^2$"
} | {
"index": 53,
"solution": "1. **Break down the expression**: We start by simplifying the expression $1000 \\times 1993 \\times 0.1993 \\times 10$. We can rearrange the terms for easier computation:\n \\[\n 1000 \\times 10 \\times 1993 \\times 0.1993\n \\]\n\n2. **Simplify the powers of 10**: Calculate $1000 ... |
DeepScaleR | [
{
"role": "user",
"content": "$100_{10}$ in base $b$ has exactly $5$ digits. What is the value of $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 54,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$101$ people, sitting at a round table in any order, had $1,2,... , 101$ cards, respectively. \nA transfer is someone give one card to one of the two people adjacent to him.\nFind the smallest positive integer $k$ such that there always can through no more than ... | math | {
"style": "rule",
"ground_truth": "42925"
} | {
"index": 55,
"solution": "\nGiven 101 people sitting at a round table, each holding a unique card numbered from 1 to 101, we need to determine the smallest positive integer \\( k \\) such that through no more than \\( k \\) transfers, each person can hold the same number of cards, regardless of the initial sittin... |
DeepScaleR | [
{
"role": "user",
"content": "$12 \\cos ^{4} \\frac{\\pi}{8}+\\cos ^{4} \\frac{3 \\pi}{8}+\\cos ^{4} \\frac{5 \\pi}{8}+\\cos ^{4} \\frac{7 \\pi}{8}=$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 56,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$12$ students need to form five study groups. They will form three study groups with $2$ students each and two study groups with $3$ students each. In how many ways can these groups be formed?"
}
] | math | {
"style": "rule",
"ground_truth": "138600"
} | {
"index": 57,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$13$ fractions are corrected by using each of the numbers $1,2,...,26$ once.**Example:** $\\frac{12}{5},\\frac{18}{26}.... $ What is the maximum number of fractions which are integers?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 58,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$14 N$ is a 5-digit number composed of 5 different non-zero digits, and $N$ is equal to the sum of all three-digit numbers that can be formed using any 3 of these 5 digits. Find all such 5-digit numbers $N$."
}
] | math | {
"style": "rule",
"ground_truth": "35964"
} | {
"index": 59,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$15\\times\u0002 36$ -checkerboard is covered with square tiles. There are two kinds of tiles, with side $7$ or $5.$ \nTiles are supposed to cover whole squares of the board and be non-overlapping. \nWhat is the maximum number of squares to be covered?"
}
] | math | {
"style": "rule",
"ground_truth": "540"
} | {
"index": 60,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$174 S-A B C D-S^{\\prime}$ is a regular octahedron with edge length 1. Sphere $O$ is outside this octahedron, tangent to face $S A B$, and tangent to the extended planes of faces $S B C$, $S A D$, and $S^{\\prime} A B$. Find the radius $r$ of sphere $O$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4} \\sqrt{\\frac{2}{3}}"
} | {
"index": 61,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$18 \\cdot 92$ A square $ABCD$ has side lengths of 1. Points $E$ and $F$ lie on sides $AB$ and $AD$ respectively, such that $AE = AF$. If the quadrilateral $CDFE$ has the maximum area, what is the maximum area?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{5}{8}$"
} | {
"index": 62,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$2(81+83+85+87+89+91+93+95+97+99)= $"
}
] | math | {
"style": "rule",
"ground_truth": "1800"
} | {
"index": 63,
"solution": "1. **Identify the sequence and its properties**: The sequence given in the problem is $81, 83, 85, 87, 89, 91, 93, 95, 97, 99$. This is an arithmetic sequence where each term increases by $2$.\n\n2. **Sum the sequence**: The sum of an arithmetic sequence can be calculated using the formu... |
DeepScaleR | [
{
"role": "user",
"content": "$2.46 \\times 8.163 \\times (5.17 + 4.829)$ is closest to"
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 64,
"solution": "To solve the problem, we will approximate the values and calculate the product.\n\n1. **Approximate the values:**\n - $2.46 \\approx 2.5$\n - $8.163 \\approx 8$\n - $5.17 + 4.829 = 10$ (This is exact since $5.17 + 4.829$ is designed to sum to $10$).\n\n2. **Calculate the product us... |
DeepScaleR | [
{
"role": "user",
"content": "$2.46\\times 8.163\\times (5.17+4.829)$ is approximately equal to what value?"
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 65,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$2014$ points are placed on a circumference. On each of the segments with end points on two of the $2014$ points is written a non-negative real number. For any convex polygon with vertices on some of the $2014$ points, the sum of the numbers written on their sides is less or eq... | math | {
"style": "rule",
"ground_truth": "507024.5"
} | {
"index": 66,
"solution": "\nGiven the problem, we are tasked with finding the maximum possible sum of numbers written on segments between 2014 points uniformly placed on a circumference, under the condition that for any convex polygon formed using these points as vertices, the sum of the numbers on its sides must... |
DeepScaleR | [
{
"role": "user",
"content": "$2016$ bugs are sitting in different places of $1$ -meter stick. Each bug runs in one or another direction with constant and equal speed. If two bugs face each other, then both of them change direction but not speed. If bug reaches one of the ends of the stick, then it flies ... | math | {
"style": "rule",
"ground_truth": "1008^2"
} | {
"index": 67,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$2018$ people (call them $A, B, C, \\ldots$ ) stand in a line with each permutation equally likely. Given that $A$ stands before $B$ , what is the probability that $C$ stands after $B$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 68,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$2020$ positive integers are written in one line. Each of them starting with the third is divisible by previous and by the sum of two previous numbers. What is the smallest value the last number can take?\n\nA. Gribalko"
}
] | math | {
"style": "rule",
"ground_truth": "2019!"
} | {
"index": 69,
"solution": "\nGiven the problem, we have a sequence of \\(2020\\) positive integers, say \\(a_1, a_2, \\ldots, a_{2020}\\). Each term in the sequence starting with the third term (\\(a_i\\) for \\(i \\geq 3\\)) is divisible by its preceding term and the sum of its two immediate predecessors. Formall... |
DeepScaleR | [
{
"role": "user",
"content": "$21$ Savage has a $12$ car garage, with a row of spaces numbered $1,2,3,\\ldots,12$ . How many ways can he choose $6$ of them to park his $6$ identical cars in, if no $3$ spaces with consecutive numbers may be all occupied?\n\n*2018 CCA Math Bonanza Team Round #9*"
... | math | {
"style": "rule",
"ground_truth": "357"
} | {
"index": 70,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$24$ lemons are required to make $32$ gallons of lemonade. How many lemons are needed to make $4$ gallons of lemonade?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 71,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$25$ checkers are placed on $25$ leftmost squares of $1 \\times N$ board. Checker can either move to the empty adjacent square to its right or jump over adjacent right checker to the next square if it is empty. Moves to the left are not allowed. Find minimal $N$ such tha... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 72,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$2\\left(1-\\frac{1}{2}\\right) + 3\\left(1-\\frac{1}{3}\\right) + 4\\left(1-\\frac{1}{4}\\right) + \\cdots + 10\\left(1-\\frac{1}{10}\\right)=$"
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 73,
"solution": "1. **Identify the General Term**: \n Each term in the sequence is of the form $n\\left(1-\\frac{1}{n}\\right)$ for $n$ ranging from $2$ to $10$. Simplifying the expression inside the parentheses:\n \\[\n 1 - \\frac{1}{n} = \\frac{n-1}{n}\n \\]\n Multiplying this by $n$ gives:\... |
DeepScaleR | [
{
"role": "user",
"content": "$3-i(i)$ is a root of the equation $x^{2}+px+10=0(p∈R)$ with respect to $x$. Find the value of $p$."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 74,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$30$ same balls are put into four boxes $A$, $B$, $C$, $D$ in such a way that sum of number of balls in $A$ and $B$ is greater than sum of in $C$ and $D$. How many possible ways are there?"
}
] | math | {
"style": "rule",
"ground_truth": "2600"
} | {
"index": 75,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$361+2(19)(6)+36=x$. Solve for $x$."
}
] | math | {
"style": "rule",
"ground_truth": "625"
} | {
"index": 76,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$3^3 + 3^3 + 3^3 =$"
}
] | math | {
"style": "rule",
"ground_truth": "$3^4$"
} | {
"index": 77,
"solution": "1. **Identify the expression and simplify**: The given expression is $3^3 + 3^3 + 3^3$.\n\n2. **Factor out the common term**: Notice that each term in the sum is $3^3$. We can factor out $3^3$ as follows:\n \\[\n 3^3 + 3^3 + 3^3 = 3 \\times 3^3\n \\]\n\n3. **Apply the exponent rule... |
DeepScaleR | [
{
"role": "user",
"content": "$3^n = 3 \\cdot 9^3 \\cdot 81^2$. What is the value of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 78,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$44 \\times 22$ is equal to"
}
] | math | {
"style": "rule",
"ground_truth": "$88 \\times 11$"
} | {
"index": 79,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$48n$ is the smallest positive integer that satisfies the following conditions:\n1. $n$ is a multiple of 75;\n2. $n$ has exactly 75 positive divisors (including 1 and itself).\n\nFind $\\frac{n}{75}$."
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"index": 80,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$5y$ varies inversely as the square of $x$. When $y=16$, $x=1$. When $x=8$, $y$ equals:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 81,
"solution": "1. **Understanding the relationship**: Given that $5y$ varies inversely as the square of $x$, we can express this relationship mathematically as:\n \\[\n 5y = \\frac{k}{x^2}\n \\]\n where $k$ is a constant.\n\n2. **Finding the constant $k$**: We know that when $y = 16$ and $x = 1... |
DeepScaleR | [
{
"role": "user",
"content": "$6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6 = $"
}
] | math | {
"style": "rule",
"ground_truth": "$6^7$"
} | {
"index": 82,
"solution": "1. **Identify the Expression**: The problem gives the expression $6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6$.\n\n2. **Simplify the Expression**: We observe that the expression consists of six terms, each equal to $6^6$. This can be rewritten using the distributive property of multiplication over... |
DeepScaleR | [
{
"role": "user",
"content": "$8[x]$ represents the greatest integer not exceeding the real number $x$. Then, $\\left[\\log _{2} 1\\right]+\\left[\\log _{2} 2\\right]+\\left[\\log _{2} 3\\right]+\\cdots+\\left[\\log _{2} 2012\\right]=$ $\\qquad$ ."
}
] | math | {
"style": "rule",
"ground_truth": "18084"
} | {
"index": 83,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$908 \\times 501 - [731 \\times 1389 - (547 \\times 236 + 842 \\times 731 - 495 \\times 361)] =$"
}
] | math | {
"style": "rule",
"ground_truth": "5448"
} | {
"index": 84,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$A B C D E$ is a cyclic convex pentagon, and $A C=B D=C E . A C$ and $B D$ intersect at $X$, and $B D$ and $C E$ intersect at $Y$. If $A X=6, X Y=4$, and $Y E=7$, then the area of pentagon $A B C D E$ can be written as $\\frac{a \\sqrt{b}}{c}$, where $a, b, c$ are integers, $c$... | math | {
"style": "rule",
"ground_truth": "2852"
} | {
"index": 85,
"solution": "Since $A C=B D, A B C D$ is an isosceles trapezoid. Similarly, $B C D E$ is also an isosceles trapezoid. Using this, we can now calculate that $C Y=D Y=D X-X Y=A X-X Y=2$, and similarly $B X=C X=3$. By applying Heron's formula we find that the area of triangle $C X Y$ is $\\frac{3}{4} \\... |
DeepScaleR | [
{
"role": "user",
"content": "$A B C D$ is a cyclic quadrilateral in which $A B=3, B C=5, C D=6$, and $A D=10 . M, I$, and $T$ are the feet of the perpendiculars from $D$ to lines $A B, A C$, and $B C$ respectively. Determine the value of $M I / I T$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{9}"
} | {
"index": 86,
"solution": "Quadrilaterals $A M I D$ and $D I C T$ are cyclic, having right angles $\\angle A M D, \\angle A I D$, and $\\angle C I D, \\angle C T D$ respectively. We see that $M, I$, and $T$ are collinear. For, $m \\angle M I D=\\pi-m \\angle D A M=$ $\\pi-m \\angle D A B=m \\angle B C D=\\pi-m \\a... |
DeepScaleR | [
{
"role": "user",
"content": "$A B C D$ is a cyclic quadrilateral in which $A B=4, B C=3, C D=2$, and $A D=5$. Diagonals $A C$ and $B D$ intersect at $X$. A circle $\\omega$ passes through $A$ and is tangent to $B D$ at $X . \\omega$ intersects $A B$ and $A D$ at $Y$ and $Z$ respectively. Compute $Y Z / B D... | math | {
"style": "rule",
"ground_truth": "\\frac{115}{143}"
} | {
"index": 87,
"solution": "Denote the lengths $A B, B C, C D$, and $D A$ by $a, b, c$, and $d$ respectively. Because $A B C D$ is cyclic, $\\triangle A B X \\sim \\triangle D C X$ and $\\triangle A D X \\sim \\triangle B C X$. It follows that $\\frac{A X}{D X}=\\frac{B X}{C X}=\\frac{a}{c}$ and $\\frac{A X}{B X}=\... |
DeepScaleR | [
{
"role": "user",
"content": "$A B C D$ is a cyclic quadrilateral with sides $A B=10, B C=8, C D=25$, and $D A=12$. A circle $\\omega$ is tangent to segments $D A, A B$, and $B C$. Find the radius of $\\omega$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{1209}{7}} \\text{ OR } \\frac{\\sqrt{8463}}{7}"
} | {
"index": 88,
"solution": "Denote $E$ an intersection point of $A D$ and $B C$. Let $x=E A$ and $y=E B$. Because $A B C D$ is a cyclic quadrilateral, $\\triangle E A B$ is similar to $\\triangle E C D$. Therefore, $\\frac{y+8}{x}=\\frac{25}{10}$ and $\\frac{x+12}{y}=\\frac{25}{10}$. We get $x=\\frac{128}{21}$ and ... |
DeepScaleR | [
{
"role": "user",
"content": "$A B C D$ is a parallelogram satisfying $A B=7, B C=2$, and $\\angle D A B=120^{\\circ}$. Parallelogram $E C F A$ is contained in $A B C D$ and is similar to it. Find the ratio of the area of $E C F A$ to the area of $A B C D$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{39}{67}"
} | {
"index": 89,
"solution": "First, note that $B D$ is the long diagonal of $A B C D$, and $A C$ is the long diagonal of $E C F A$. Because the ratio of the areas of similar figures is equal to the square of the ratio of their side lengths, we know that the ratio of the area of $E C F A$ to the area of $A B C D$ is ... |
DeepScaleR | [
{
"role": "user",
"content": "$A B C D$ is a rectangle with $A B=20$ and $B C=3$. A circle with radius 5, centered at the midpoint of $D C$, meets the rectangle at four points: $W, X, Y$, and $Z$. Find the area of quadrilateral $W X Y Z$."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 90,
"solution": "Suppose that $X$ and $Y$ are located on $A B$ with $X$ closer to $A$ than $B$. Let $O$ be the center of the circle, and let $P$ be the midpoint of $A B$. We have $O P \\perp A B$ so $O P X$ and $O P Y$ are right triangles with right angles at $P$. Because $O X=O Y=5$ and $O P=3$, we have... |
DeepScaleR | [
{
"role": "user",
"content": "$A B C D$ is a regular tetrahedron of volume 1. Maria glues regular tetrahedra $A^{\\prime} B C D, A B^{\\prime} C D$, $A B C^{\\prime} D$, and $A B C D^{\\prime}$ to the faces of $A B C D$. What is the volume of the tetrahedron $A^{\\prime} B^{\\prime} C^{\\prime} D^{\\prime}$... | math | {
"style": "rule",
"ground_truth": "\\frac{125}{27}"
} | {
"index": 91,
"solution": "Consider the tetrahedron with vertices at $W=(1,0,0), X=(0,1,0), Y=(0,0,1)$, and $Z=(1,1,1)$. This tetrahedron is similar to $A B C D$. It has center $O=\\left(\\frac{1}{2}, \\frac{1}{2}, \\frac{1}{2}\\right)$. We can construct a tetrahedron $W^{\\prime} X^{\\prime} Y^{\\prime} Z^{\\prim... |
DeepScaleR | [
{
"role": "user",
"content": "$A B C$ is a right triangle with $\\angle A=30^{\\circ}$ and circumcircle $O$. Circles $\\omega_{1}, \\omega_{2}$, and $\\omega_{3}$ lie outside $A B C$ and are tangent to $O$ at $T_{1}, T_{2}$, and $T_{3}$ respectively and to $A B, B C$, and $C A$ at $S_{1}, S_{2}$, and $S_{3}... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}+1}{2}"
} | {
"index": 92,
"solution": "Let $[P Q R]$ denote the area of $\\triangle P Q R$. The key to this problem is following fact: $[P Q R]=\\frac{1}{2} P Q \\cdot P R \\sin \\angle Q P R$. Assume that the radius of $O$ is 1. Since $\\angle A=30^{\\circ}$, we have $B C=1$ and $A B=\\sqrt{3}$. So $[A B C]=\\frac{\\sqrt{3}}... |
DeepScaleR | [
{
"role": "user",
"content": "$A B C$ is a triangle with $A B=15, B C=14$, and $C A=13$. The altitude from $A$ to $B C$ is extended to meet the circumcircle of $A B C$ at $D$. Find $A D$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{63}{4}"
} | {
"index": 93,
"solution": "Let the altitude from $A$ to $B C$ meet $B C$ at $E$. The altitude $A E$ has length 12 ; one way to see this is that it splits the triangle $A B C$ into a $9-12-15$ right triangle and a $5-12-13$ right triangle; from this, we also know that $B E=9$ and $C E=5$. Now, by Power of a Point, ... |
DeepScaleR | [
{
"role": "user",
"content": "$A B C$ is a triangle with points $E, F$ on sides $A C, A B$, respectively. Suppose that $B E, C F$ intersect at $X$. It is given that $A F / F B=(A E / E C)^{2}$ and that $X$ is the midpoint of $B E$. Find the ratio $C X / X F$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 94,
"solution": "Let $x=A E / E C$. By Menelaus's theorem applied to triangle $A B E$ and line $C X F$, $$1=\\frac{A F}{F B} \\cdot \\frac{B X}{X E} \\cdot \\frac{E C}{C A}=\\frac{x^{2}}{x+1}$$ Thus, $x^{2}=x+1$, and $x$ must be positive, so $x=(1+\\sqrt{5}) / 2$. Now apply Menelaus to triangle $A C F$ a... |
DeepScaleR | [
{
"role": "user",
"content": "$A B C$ is an acute triangle with incircle $\\omega$. $\\omega$ is tangent to sides $\\overline{B C}, \\overline{C A}$, and $\\overline{A B}$ at $D, E$, and $F$ respectively. $P$ is a point on the altitude from $A$ such that $\\Gamma$, the circle with diameter $\\overline{A P}$... | math | {
"style": "rule",
"ground_truth": "\\frac{675}{4}"
} | {
"index": 95,
"solution": "By the Law of Sines we have $\\sin \\angle A=\\frac{X Y}{A P}=\\frac{4}{5}$. Let $I, T$, and $Q$ denote the center of $\\omega$, the point of tangency between $\\omega$ and $\\Gamma$, and the center of $\\Gamma$ respectively. Since we are told $A B C$ is acute, we can compute $\\tan \\an... |
DeepScaleR | [
{
"role": "user",
"content": "$A B$ is a diameter of circle $O . X$ is a point on $A B$ such that $A X=3 B X$. Distinct circles $\\omega_{1}$ and $\\omega_{2}$ are tangent to $O$ at $T_{1}$ and $T_{2}$ and to $A B$ at $X$. The lines $T_{1} X$ and $T_{2} X$ intersect $O$ again at $S_{1}$ and $S_{2}$. What is... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 96,
"solution": "Since the problem only deals with ratios, we can assume that the radius of $O$ is 1. As we have proven in Problem 5, points $S_{1}$ and $S_{2}$ are midpoints of arc $A B$. Since $A B$ is a diameter, $S_{1} S_{2}$ is also a diameter, and thus $S_{1} S_{2}=2$. Let $O_{1}, O_{2}$, and $P$ d... |
DeepScaleR | [
{
"role": "user",
"content": "$A$ and $B$ are on a circle of radius $20$ centered at $C$ , and $\\angle ACB = 60^\\circ$ . $D$ is chosen so that $D$ is also on the circle, $\\angle ACD = 160^\\circ$ , and $\\angle DCB = 100^\\circ$ . Let $E$ be the intersection of lines $AC$ and $BD$ . Wha... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 97,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$A$ and $B$ plays a game, with $A$ choosing a positive integer $n \\in \\{1, 2, \\dots, 1001\\} = S$ . $B$ must guess the value of $n$ by choosing several subsets of $S$ , then $A$ will tell $B$ how many subsets $n$ is in. $B$ will do this three times selecti... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 98,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$A$ and $B$ are $46$ kilometers apart. Person A rides a bicycle from point $A$ to point $B$ at a speed of $15$ kilometers per hour. One hour later, person B rides a motorcycle along the same route from point $A$ to point $B$ at a speed of $40$ kilometers per hour. \n$(1)$ Afte... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 99,
"solution": ""
} |
End of preview. Expand in Data Studio
DeepScaleR-Verl-Clean
A single cleaned, deduplicated, verl-ready parquet built from four partial dumps of the
DeepScaleR math dataset (agentica-org/DeepScaleR-Preview-Dataset, MIT licensed). It is
meant for rule-based-reward GRPO/RL training with the
verl framework.
Files
| File | Rows |
|---|---|
data/verl_deepscaler_clean_39179.parquet |
39179 |
data/cleaning_report.json |
full paper trail of every dropped row |
How it was built
Sources were read in priority order (deepscaler_part1.jsonl, deepscaler_part2.jsonl,
deepscaler_export.json, deepscaler_legacy.jsonl; within a file, earlier rows win):
- Validity - a row is garbage unless its statement and its answer are both present
and non-blank after trimming. Invalid rows are removed before deduplication, so a
broken copy can never steal the slot of a good copy. Rows whose statement is unusable
are recorded as
missing_problem, rows with a usable statement but no answer asmissing_answer. A missingsolutionis not garbage: it is stored as an empty string. - Identity / deduplication - identity is the trimmed statement text. Only the first
occurrence (by source priority, then row order) is kept; every later copy is dropped as
duplicate, including legacy copies that carry stale answers. - Ordering / indexing - survivors are sorted by statement text ascending (plain
code-point ordering) and numbered 0..N-1;
extra_info.indexequals the row's position and the parquet is physically in that order. Prompts carry the trimmed statement andreward_model.ground_truththe trimmed answer.
Totals: 43608 input rows -> 39179 kept, 4390 dropped as duplicates, 39 dropped as invalid.
See data/cleaning_report.json for the per-row detail of everything that was thrown away.
Schema (verl)
Five columns, in this order:
| Column | Content |
|---|---|
data_source |
"DeepScaleR" |
prompt |
[{"role": "user", "content": "<trimmed problem statement>"}] |
ability |
"math" |
reward_model |
{"style": "rule", "ground_truth": "<trimmed answer>"} |
extra_info |
{"index": <row position>, "solution": "<solution, or empty string>"} |
Usage
from datasets import load_dataset
ds = load_dataset("dusersad12/DeepScaleR-Verl-Clean", split="train")
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