data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest possible median for the six number set $\\{x, 2x, 3, 2, 5, 4x\\}$ if $x$ can be any positive integer?"
}
] | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"index": 38100,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest possible number of whole 2-by-3 non-overlapping rectangles needed to cover a square region exactly, without extra over-hangs and without gaps?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 38101,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest possible perimeter of a triangle whose side lengths are all squares of distinct positive integers?"
}
] | math | {
"style": "rule",
"ground_truth": "77"
} | {
"index": 38102,
"solution": "There exist a triangle with side lengths $4^{2}, 5^{2}, 6^{2}$, which has perimeter 77. If the sides have lengths $a^{2}, b^{2}, c^{2}$ with $0<a<b<c$, then $a^{2}+b^{2}>c^{2}$ by the triangle inequality. Therefore $(b-1)^{2}+b^{2} \\geq a^{2}+b^{2}>c^{2} \\geq(b+1)^{2}$. Solving this... |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest possible perimeter of a triangle with integer coordinate vertices, area $\\frac12$ , and no side parallel to an axis?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5} + \\sqrt{2} + \\sqrt{13}"
} | {
"index": 38103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest possible perimeter, in units, of a triangle whose side-length measures are consecutive integer values?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 38104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest possible real value of $x^2 + 8x$?"
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"index": 38105,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest possible sum of two consecutive integers whose product is greater than 420?"
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 38106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest possible value of $n$ if a solid cube is made of white plastic and has dimensions $n \\times n \\times n$, the six faces of the cube are completely covered with gold paint, the cube is then cut into $n^{3}$ cubes, each of which has dimensions $1 \\times 1 \... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 38107,
"solution": "We call the $n \\times n \\times n$ cube the \"large cube\", and we call the $1 \\times 1 \\times 1$ cubes \"unit cubes\". The unit cubes that have exactly 0 gold faces are those unit cubes that are on the \"inside\" of the large cube. \n In other words, these are the unit cubes none ... |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest possible value of $x$ such that $2x^2+24x-60=x(x+13)$?"
}
] | math | {
"style": "rule",
"ground_truth": "-15"
} | {
"index": 38108,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest possible value of the sum $\\lvert x + 2\\rvert + \\lvert x + 4\\rvert + \\lvert x + 5\\rvert$?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 38109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest prime divisor of $5^{23} + 7^{17}$?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 38110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest prime factor of 1739?"
}
] | math | {
"style": "rule",
"ground_truth": "1739"
} | {
"index": 38111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest prime factor of 1821?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 38112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest prime number dividing the sum $3^{11}+5^{13}$?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 38113,
"solution": "1. **Identify the Parity of Each Term**: \n - $3^{11}$ is an odd number because any power of an odd number remains odd.\n - $5^{13}$ is also an odd number for the same reason.\n\n2. **Sum of Two Odd Numbers**:\n - The sum of two odd numbers is always even. This can be shown by c... |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest prime whose digits sum to $19$?"
}
] | math | {
"style": "rule",
"ground_truth": "199"
} | {
"index": 38114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest prime whose digits sum to 23?"
}
] | math | {
"style": "rule",
"ground_truth": "1993"
} | {
"index": 38115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest prime whose digits sum to \\(28\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "1999"
} | {
"index": 38116,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest result that can be obtained from the following process?\n\nChoose three different numbers from the set $\\{3,5,7,11,13,17\\}$.\nAdd two of these numbers.\nMultiply their sum by the third number."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 38117,
"solution": "To find the smallest result from the given process, we need to consider the operations involved: addition and multiplication. Both operations are increasing functions when dealing with positive integers. Therefore, to minimize the result, we should ideally start with the smallest numb... |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest solution of the equation $x^4-34x^2+225=0$?"
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 38118,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest square number that, when divided by a cube number, results in a fraction in its simplest form where the numerator is a cube number (other than 1) and the denominator is a square number (other than 1)?"
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 38119,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest square number whose first five digits are 4 and the sixth digit is 5?"
}
] | math | {
"style": "rule",
"ground_truth": "666667"
} | {
"index": 38120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest square of an integer that ends with the longest sequence of the same digits?\n\nFor example, if the longest sequence of the same digits were five, then a suitable number would be 24677777 (of course, if it were the smallest square, but it is not). Zero is n... | math | {
"style": "rule",
"ground_truth": "1444"
} | {
"index": 38121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest sum of two $3$-digit numbers that can be obtained by placing each of the six digits $4,5,6,7,8,9$ in one of the six boxes in this addition problem?\n[asy]\nunitsize(12);\ndraw((0,0)--(10,0));\ndraw((-1.5,1.5)--(-1.5,2.5));\ndraw((-1,2)--(-2,2));\ndraw((1,1)... | math | {
"style": "rule",
"ground_truth": "1047"
} | {
"index": 38122,
"solution": "1. **Identify the problem**: We need to find the smallest sum of two 3-digit numbers formed by the digits 4, 5, 6, 7, 8, and 9, each used exactly once.\n\n2. **Understand the sum of two numbers**: If the two numbers are $\\overline{abc}$ and $\\overline{def}$, their sum is given by:\n... |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest sum that nine consecutive natural numbers can have if this sum ends in 2050306?"
}
] | math | {
"style": "rule",
"ground_truth": "22050306"
} | {
"index": 38123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest three-digit multiple of 13?"
}
] | math | {
"style": "rule",
"ground_truth": "104"
} | {
"index": 38124,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest three-digit number in Pascal's triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 38125,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest three-digit positive integer which can be written in the form \\( p q^{2} r \\), where \\( p, q \\), and \\( r \\) are distinct primes?"
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"index": 38126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest two-digit integer $n$ such that switching its digits and then adding 3 results in $2n$?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 38127,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest value of $k$ for which it is possible to mark $k$ cells on a $9 \\times 9$ board such that any placement of a three-cell corner touches at least two marked cells?"
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 38128,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest value of $x$ that satisfies the equation $8x^2 - 38x + 35 = 0$? Express your answer as a decimal."
}
] | math | {
"style": "rule",
"ground_truth": "1.25"
} | {
"index": 38129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest value that the ratio of the areas of two isosceles right triangles can have, given that three vertices of one of the triangles lie on three different sides of the other triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "1/5"
} | {
"index": 38130,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest value that the sum of the digits of the number $3n^2+n+1,$ $n\\in\\mathbb{N}$ can take?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 38131,
"solution": "\nWe need to find the smallest value of the sum of the digits of the expression \\( 3n^2 + n + 1 \\) for \\( n \\in \\mathbb{N} \\).\n\nTo determine this, we will explore values of \\( n \\) and calculate the resulting expression \\( 3n^2 + n + 1 \\), then find the sum of its digits. ... |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest whole number $b$ such that 101 can be expressed in base $b$ using only two digits?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 38132,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest whole number $b$ such that 62 can be expressed in base $b$ using only three digits?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 38133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19$?"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 38134,
"solution": "1. **Understanding the Triangle Inequality Theorem**: The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For a triangle with sides $a$, $b$, and $c$, this can be expressed as:\n - $a + ... |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest whole number larger than the perimeter of any triangle with a side of length $7$ and a side of length $24$?"
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"index": 38135,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the smallest whole number that has a remainder of 1 when divided by 4, a remainder of 1 when divided by 3, and a remainder of 2 when divided by 5?"
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 38136,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum and product of all values of $x$ such that $x^2 = 18x - 16$?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 38137,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum and product of the distinct prime factors of 420?"
}
] | math | {
"style": "rule",
"ground_truth": "210"
} | {
"index": 38138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum and the average of the prime numbers between 20 and 40?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 38139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum and the product of the values of $x$ that satisfy the equation $x^2 - 7x + 12 = 0$?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 38140,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of $1+2+4+8+16+ \\cdots + 1024$?"
}
] | math | {
"style": "rule",
"ground_truth": "2047"
} | {
"index": 38141,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of $\\left(\\dfrac{1}{3}\\right) + \\left(\\dfrac{1}{3}\\right)^2 + \\left(\\dfrac{1}{3}\\right)^3 + \\left(\\dfrac{1}{3}\\right)^4$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{40}{81}"
} | {
"index": 38142,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of $x + y$ if the sequence $3, ~8, ~13, \\ldots, ~x, ~y, ~33$ forms an arithmetic sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 38143,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all four-digit integers from 1000 to 5000, where each number increases by 4 from the previous one?"
}
] | math | {
"style": "rule",
"ground_truth": "3003000"
} | {
"index": 38144,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all four-digit numbers that are equal to the cube of the sum of their digits (leading zeros are not allowed)?"
}
] | math | {
"style": "rule",
"ground_truth": "10745"
} | {
"index": 38145,
"solution": "We want to find all integers $x$ between 1000 and 9999 that are the cube of the sum of their digits. Of course, our search is only restricted to perfect cubes. The smallest such cube is $10^{3}=1000$ and the largest such cube is $21^{3}=9261$. This means we only have to check 12 diffe... |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all integer solutions to $1<(x-2)^2<25$?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 38146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all integer solutions to $4<(x-3)^2<64$?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 38147,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all integer solutions to $|n| < |n-3| < 9$?"
}
] | math | {
"style": "rule",
"ground_truth": "-14"
} | {
"index": 38148,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all integer solutions to \\( |n| < |n-5| < 10 \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"index": 38149,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all integer solutions to the inequality $|n| < |n-3| < 10$?"
}
] | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"index": 38150,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all integer values $n$ for which $\\binom{20}{n}+\\binom{20}{10}=\\binom{21}{11}$?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 38151,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all integer values $n$ for which $\\binom{26}{13}+\\binom{26}{n}=\\binom{27}{14}$?"
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 38152,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all integer values of $n$ such that $\\frac{20}{2n - 1}$ is an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 38153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all integers from 80 through 90, inclusive?"
}
] | math | {
"style": "rule",
"ground_truth": "935"
} | {
"index": 38154,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all numbers $q$ which can be written in the form $q=\\frac{a}{b}$ where $a$ and $b$ are positive integers with $b \\leq 10$ and for which there are exactly 19 integers $n$ that satisfy $\\sqrt{q}<n<q$?"
}
] | math | {
"style": "rule",
"ground_truth": "777.5"
} | {
"index": 38155,
"solution": "Suppose that a number $q$ has the property that there are exactly 19 integers $n$ with $\\sqrt{q}<n<q$. Suppose that these 19 integers are $m, m+1, m+2, \\ldots, m+17, m+18$. Then $\\sqrt{q}<m<m+1<m+2<\\cdots<m+17<m+18<q$. This tells us that $q-\\sqrt{q}>(m+18)-m=18$ because $q-\\sqrt... |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all numbers in the following square matrix?\n$$\n\\begin{array}{l}\n1, 2, 3, \\ldots, 98, 99, 100 \\\\\n2, 3, 4, \\ldots, 99, 100, 101 \\\\\n3, 4, 5, \\ldots, 100, 101, 102 \\\\\n\\ldots \\\\\n100, 101, 102, \\ldots, 197, 198, 199\n\\end{array}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "1000000"
} | {
"index": 38156,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all odd integers between $400$ and $600$?"
}
] | math | {
"style": "rule",
"ground_truth": "50000"
} | {
"index": 38157,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all of the odd divisors of $180$?"
}
] | math | {
"style": "rule",
"ground_truth": "78"
} | {
"index": 38158,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all of the positive even factors of $504$?"
}
] | math | {
"style": "rule",
"ground_truth": "1456"
} | {
"index": 38159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all of the possibilities for Sam's number if Sam thinks of a 5-digit number, Sam's friend Sally tries to guess his number, Sam writes the number of matching digits beside each of Sally's guesses, and a digit is considered \"matching\" when it is the correct d... | math | {
"style": "rule",
"ground_truth": "526758"
} | {
"index": 38160,
"solution": "We label the digits of the unknown number as vwxyz. \n Since vwxyz and 71794 have 0 matching digits, then $v \\neq 7$ and $w \\neq 1$ and $x \\neq 7$ and $y \\neq 9$ and $z \\neq 4$. \n Since vwxyz and 71744 have 1 matching digit, then the preceding information tells us that $y=4$. \n... |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all of the solutions of the equation $\\frac{4x}{20}=\\frac{5}{x}$?"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 38161,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all of the two-digit primes that are greater than 12 but less than 99 and are still prime when their two digits are interchanged?"
}
] | math | {
"style": "rule",
"ground_truth": "418"
} | {
"index": 38162,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all positive integer divisors of 77?"
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 38163,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all positive integer solutions less than or equal to $20$ to the congruence $13(3x-2)\\equiv 26\\pmod 8$?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 38164,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all positive integer solutions less than or equal to $30$ to the congruence $15(3x-4) \\equiv 30 \\pmod{10}$?"
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 38165,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all positive integer values of $n$ such that $\\frac{n+18}{n}$ is an integer?"
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 38166,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all positive integers $n$ such that $\\text{lcm}(2n, n^2) = 14n - 24$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 38167,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all positive integers $\\nu$ for which $\\mathop{\\text{lcm}}[\\nu, 18] = 72$?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 38168,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all positive integers $\\nu$ for which $\\mathop{\\text{lcm}}[\\nu, 45]=180$?"
}
] | math | {
"style": "rule",
"ground_truth": "292"
} | {
"index": 38169,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all positive integers $\\nu$ for which $\\mathop{\\text{lcm}}[\\nu,20]=60$?"
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"index": 38170,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all positive integers $n$ that satisfy $$\\mathop{\\text{lcm}}[n,100] = \\gcd(n,100)+450~?$$"
}
] | math | {
"style": "rule",
"ground_truth": "250"
} | {
"index": 38171,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all positive integers $n$ that satisfy $$\\mathop{\\text{lcm}}[n,120] = \\gcd(n,120) + 600~?$$"
}
] | math | {
"style": "rule",
"ground_truth": "2520"
} | {
"index": 38172,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all positive integers less than 100 that are squares of perfect squares?"
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 38173,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all positive integers less than 500 that are fourth powers of even perfect squares?"
}
] | math | {
"style": "rule",
"ground_truth": "272"
} | {
"index": 38174,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all possible values of $k$ for which the polynomials $x^2 - 3x + 2$ and $x^2 - 5x + k$ have a root in common?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 38175,
"solution": "1. **Factor the first polynomial**: \n The polynomial $x^2 - 3x + 2$ can be factored as follows:\n \\[\n x^2 - 3x + 2 = (x - 1)(x - 2)\n \\]\n This implies that the roots of $x^2 - 3x + 2$ are $x = 1$ and $x = 2$.\n\n2. **Determine the value of $k$ for each common root**:\n ... |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are $(\\cos 40^\\circ,\\sin 40^\\circ)$, $(\\cos 60^\\circ,\\sin 60^\\circ)$, and $(\\cos t^\\circ,\\sin t^\\circ)$ is isosceles?"
}
] | math | {
"style": "rule",
"ground_truth": "380"
} | {
"index": 38176,
"solution": "Let $A = (\\cos 40^\\circ, \\sin 40^\\circ)$, $B = (\\cos 60^\\circ, \\sin 60^\\circ)$, and $C = (\\cos t^\\circ, \\sin t^\\circ)$. We need to find the values of $t$ such that $\\triangle ABC$ is isosceles. We consider three cases based on which sides of the triangle are equal.\n\n**C... |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 38177,
"solution": "1. **Convert the given equation to a simpler form by completing the square:**\n \\[\n |x^2 - 12x + 34| = 2\n \\]\n Completing the square for the quadratic expression:\n \\[\n x^2 - 12x + 34 = (x^2 - 12x + 36) - 2 = (x-6)^2 - 2\n \\]\n Thus, the equation becomes:\n \\... |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 38178,
"solution": "1. **Calculate the Mean:**\n The mean of the numbers $4, 6, 8, 17,$ and $x$ is given by:\n \\[\n \\text{Mean} = \\frac{4+6+8+17+x}{5} = \\frac{35+x}{5}\n \\]\n\n2. **Identify Possible Medians:**\n The median of these numbers depends on the value of $x$. The possible medians ... |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all real numbers \\(x\\) for which \\(|x^2 - 14x + 45| = 3?\\)\nA) 12\nB) 14\nC) 16\nD) 18"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 38179,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the even integers between $200$ and $400$?"
}
] | math | {
"style": "rule",
"ground_truth": "30100"
} | {
"index": 38180,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the four-digit positive integers that end in 0?"
}
] | math | {
"style": "rule",
"ground_truth": "4945500"
} | {
"index": 38181,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the four-digit positive integers?"
}
] | math | {
"style": "rule",
"ground_truth": "49495500"
} | {
"index": 38182,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the integers between -12.1 and 3.3?"
}
] | math | {
"style": "rule",
"ground_truth": "-72"
} | {
"index": 38183,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the integers between -20.5 and -1.5?"
}
] | math | {
"style": "rule",
"ground_truth": "-210"
} | {
"index": 38184,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the integers between -25.4 and 15.8, excluding the integer zero?"
}
] | math | {
"style": "rule",
"ground_truth": "-200"
} | {
"index": 38185,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the integers from 100 to 2000?"
}
] | math | {
"style": "rule",
"ground_truth": "1996050"
} | {
"index": 38186,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the odd divisors of $360$?"
}
] | math | {
"style": "rule",
"ground_truth": "78"
} | {
"index": 38187,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the odd integers between $200$ and $600$?"
}
] | math | {
"style": "rule",
"ground_truth": "80000"
} | {
"index": 38188,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the odd integers between $300$ and $500$?"
}
] | math | {
"style": "rule",
"ground_truth": "40000"
} | {
"index": 38189,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the positive divisors of 91?"
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 38190,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the positive two-digit integers divisible by both the sum and product of their digits?"
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 38191,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the prime numbers between 1 and 10?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 38192,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the solutions of $x = \\left|2x-|60-2x|\\right|$?"
}
] | math | {
"style": "rule",
"ground_truth": "92"
} | {
"index": 38193,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the solutions of \\( x = |2x - |50-2x|| \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{170}{3}"
} | {
"index": 38194,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the three-digit positive integers?"
}
] | math | {
"style": "rule",
"ground_truth": "494550"
} | {
"index": 38195,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all the two-digit primes that are greater than 20 but less than 80 and are still prime when their two digits are interchanged?"
}
] | math | {
"style": "rule",
"ground_truth": "291"
} | {
"index": 38196,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all three-digit and four-digit positive integers up to 2000?"
}
] | math | {
"style": "rule",
"ground_truth": "1996050"
} | {
"index": 38197,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all three-digit numbers \\( n \\) for which \\(\\frac{3n+2}{5n+1}\\) is not in its simplest form?"
}
] | math | {
"style": "rule",
"ground_truth": "70950"
} | {
"index": 38198,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "What is the sum of all two-digit positive integers whose squares end with the digits 01?"
}
] | math | {
"style": "rule",
"ground_truth": "199"
} | {
"index": 38199,
"solution": ""
} |
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