data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive integers $n \\leq 2015$ that can be expressed in the form $\\left\\lceil\\frac{x}{2}\\right\\rceil+y+x y$, where $x$ and $y$ are positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "2029906"
} | {
"index": 10900,
"solution": "Lemma: $n$ is expressible as $\\left\\lceil\\frac{x}{2}\\right\\rceil+y+x y$ iff $2 n+1$ is not a Fermat Prime. Proof: Suppose $n$ is expressible. If $x=2 k$, then $2 n+1=(2 k+1)(2 y+1)$, and if $x=2 k-1$, then $n=k(2 y+1)$. Thus, if $2 n+1$ isn't prime, we can factor $2 n+1$ as the p... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 10901,
"solution": "When $n \\geq 12$, we have \\[(n-10)^2 < n^2 -19n + 99 < (n-8)^2.\\]\n So if $n \\geq 12$ and $n^2 -19n + 99$ is a perfect square, then \\[n^2 -19n + 99 = (n-9)^2\\]\n or $n = 18$.\n For $1 \\leq n < 12$, it is easy to check that $n^2 -19n + 99$ is a perfect square when $n = 1, 9$ and... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square.Find the sum of all integers $n$ such that $\\dfrac{12}{n}$ is also an integer."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 10902,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that $1+2+\\cdots+n$ divides $15\\left[(n+1)^{2}+(n+2)^{2}+\\cdots+(2 n)^{2}\\right]$"
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 10903,
"solution": "We can compute that $1+2+\\cdots+n=\\frac{n(n+1)}{2}$ and $(n+1)^{2}+(n+2)^{2}+\\cdots+(2 n)^{2}=\\frac{2 n(2 n+1)(4 n+1)}{6}-\\frac{n(n+1)(2 n+1)}{6}=\\frac{n(2 n+1)(7 n+1)}{6}$, so we need $\\frac{15(2 n+1)(7 n+1)}{3(n+1)}=\\frac{5(2 n+1)(7 n+1)}{n+1}$ to be an integer. The remainde... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that $1.2n-4.4<5.2$."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 10904,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that $1.5n - 6.3 < 7.5$."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 10905,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that $\\sqrt{n^2+85n+2017}$ is an integer."
}
] | math | {
"style": "rule",
"ground_truth": "195"
} | {
"index": 10906,
"solution": "Notice that $(n+42)^2= n^2+84n+1764$. Also note that $(n+45)^2= n^2+90n+2025$. Thus, \\[(n+42)^2< n^2+85n+2017<(n+45)^2\\] where $n^2+85n+2017$ is a perfect square. Hence,\\[n^2+85n+2017= (n+43)^2\\] or \\[n^2+85n+2017= (n+44)^2.\\] Solving the two equations yields the two solutions $... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that there exists an integer $b$ with $|b| \\neq 4$ such that the base -4 representation of $n$ is the same as the base $b$ representation of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "1026"
} | {
"index": 10907,
"solution": "All 1 digit numbers, $0,1,2,3$, are solutions when, say, $b=5$. (Of course, $d \\in \\{0,1,2,3\\}$ works for any base $b$ of absolute value greater than $d$ but not equal to 4 .) Consider now positive integers $n=\\left(a_{d} \\ldots a_{1} a_{0}\\right)_{4}$ with more than one digit, ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that when $1^3+2^3+3^3+\\cdots +n^3$ is divided by $n+5$, the remainder is $17$."
}
] | math | {
"style": "rule",
"ground_truth": "239"
} | {
"index": 10908,
"solution": "The formula for the sum of cubes is as follows: \\[1^3+2^3+3^3+...+n^3=(1+2+3+...+n)^2=\\left(\\frac{n(n+1)}{2}\\right)^2\\] So let's apply this to this problem.\n Let $m=n+5$. Then we have \\[1^3+2^3+3^3+\\cdots+(m-5)^3\\equiv 17 \\mod m\\] \\[\\left(\\frac{(m-5)(m-4)}{2}\\right)^2\\... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $3, n$, and $n+1$ cents, $115$ cents is the greatest postage that cannot be formed."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 10909,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5,n,$ and $n+1$ cents, $91$ cents is the greatest postage that cannot be formed."
}
] | math | {
"style": "rule",
"ground_truth": "71"
} | {
"index": 10910,
"solution": "Obviously $n\\le 90$. We see that the problem's condition is equivalent to: 96 is the smallest number that can be formed which is 1 mod 5, and 92, 93, 94 can be formed (95 can always be formed). Now divide this up into cases. If $n\\equiv 0\\pmod{5}$, then 91 can be formed by using $n... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $7, n,$ and $n+2$ cents, $120$ cents is the greatest postage that cannot be formed."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 10911,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive integers \\( p \\) such that the expression \\((x-p)(x-13)+4\\) can be expressed in the form \\((x+q)(x+r)\\) for distinct integers \\( q \\) and \\( r \\)."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 10912,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive integers such that their expression in base $5$ digits is the reverse of their expression in base $11$ digits. Express your answer in base $10$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 10913,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive integers such that their expression in base $7$ digits is the reverse of their expression in base $16$ digits. Express your answer in base $10$."
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"index": 10914,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive integral values of $n$ for which $\\frac{n+6}{n}$ is an integer."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 10915,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms."
}
] | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 10916,
"solution": "Note that if $x$ is a solution, then $(300-x)$ is a solution. We know that $\\phi(300) = 80.$ Therefore the answer is $\\frac{80}{2} \\cdot\\frac{300}{30} = \\boxed{400}.$\n The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Com... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive real solutions $x$ to the equation \\[2\\cos2x \\left(\\cos2x - \\cos{\\left( \\frac{2014\\pi^2}{x} \\right) } \\right) = \\cos4x - 1,\\]where $x$ is measured in radians."
}
] | math | {
"style": "rule",
"ground_truth": "1080 \\pi"
} | {
"index": 10917,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all positive two-digit integers that are divisible by each of their digits."
}
] | math | {
"style": "rule",
"ground_truth": "630"
} | {
"index": 10918,
"solution": "Using casework, we can list out all of these numbers: \\[11+12+15+22+24+33+36+44+48+55+66+77+88+99=\\boxed{630}.\\]"
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all possible positive integer values of $b$ such that the quadratic equation $2x^2 + 5x + b = 0$ has rational roots."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 10919,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all possible values of $s$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \\[(\\cos 30^\\circ, \\sin 30^\\circ), (\\cos 45^\\circ, \\sin 45^\\circ), \\text{ and } (\\cos s^\\circ, \\sin s^\\circ)\\] is isosceles and its a... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 10920,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all prime numbers $p$ which satisfy \\[p = a^4 + b^4 + c^4 - 3\\] for some primes (not necessarily distinct) $a$ , $b$ and $c$ ."
}
] | math | {
"style": "rule",
"ground_truth": "719"
} | {
"index": 10921,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all prime numbers between $1$ and $120$ that are simultaneously $1$ greater than a multiple of $3$ and $1$ less than a multiple of $5$."
}
] | math | {
"style": "rule",
"ground_truth": "207"
} | {
"index": 10922,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all prime numbers whose representation in base 14 has the form $101010...101$ (alternating ones and zeros)."
}
] | math | {
"style": "rule",
"ground_truth": "197"
} | {
"index": 10923,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all real numbers $x$ for which $$\\lfloor\\lfloor\\cdots\\lfloor\\lfloor\\lfloor x\\rfloor+x\\rfloor+x\\rfloor \\cdots\\rfloor+x\\rfloor=2017 \\text { and }\\{\\{\\cdots\\{\\{\\{x\\}+x\\}+x\\} \\cdots\\}+x\\}=\\frac{1}{2017}$$ where there are $2017 x$ 's in both... | math | {
"style": "rule",
"ground_truth": "3025 \\frac{1}{2017}"
} | {
"index": 10924,
"solution": "The two equations are equivalent to $2017\\lfloor x\\rfloor=2017$ and $\\{2017 x\\}=\\frac{1}{2017}$, respectively. The first equation reduces to $\\lfloor x\\rfloor=1$, so we must have $x=1+r$ for some real $r$ satisfying $0 \\leq r<1$. From the second equation, we deduce that $\\{20... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all real numbers $x$ such that $5 x^{4}-10 x^{3}+10 x^{2}-5 x-11=0$."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 10925,
"solution": "Rearrange the equation to $x^{5}+(1-x)^{5}-12=0$. It's easy to see this has two real roots, and that $r$ is a root if and only if $1-r$ is a root, so the answer must be 1."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all real numbers $x$ that are not in the domain of the function $$g(x) = \\frac{1}{2 + \\frac{1}{2 + \\frac{1}{x}}}.$$"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{10}"
} | {
"index": 10926,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all real roots \\( x \\) of the equation \\( \\left(2^{x}-4\\right)^{3}+\\left(4^{x}-2\\right)^{3}=\\left(4^{x}+2^{x}-6\\right)^{3} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "3.5"
} | {
"index": 10927,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all real roots of the equation \\(3 \\tan ^{2} x + 8 \\tan x + 3 = 0\\) in the range \\(0 < x < 2\\pi\\)."
}
] | math | {
"style": "rule",
"ground_truth": "5\\pi"
} | {
"index": 10928,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all real solutions for $x$ to the equation $\\left(x^{2}+2 x+3\\right)^{\\left(x^{2}+2 x+3\\right)^{\\left(x^{2}+2 x+3\\right)}}=2012$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 10929,
"solution": "When $y=x^{2}+2 x+3$, note that there is a unique real number $y$ such that $y^{y^{y}}=2012$ because $y^{y^{y}}$ is increasing in $y$. The sum of the real distinct solutions of the equation $x^{2}+2 x+3=y$ is -2 by Vieta's Formulae as long as $2^{2}+4(y-3)>0$, which is equivalent to $... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all real solutions to the equation $(x+1)(2x+1)(3x+1)(4x+1)=17x^{4}$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{25+5\\sqrt{17}}{8}"
} | {
"index": 10930,
"solution": "First, note that $(x+1)(2x+1)(3x+1)(4x+1)=((x+1)(4x+1))((2x+1)(3x+$ $1))=\\left(4x^{2}+5x+1\\right)\\left(6x^{2}+5x+1\\right)=\\left(5x^{2}+5x+1-x^{2}\\right)\\left(5x^{2}+5x+1+x^{2}\\right)=\\left(5x^{2}+5x+1\\right)^{2}-x^{4}$. Therefore, the equation is equivalent to $\\left(5x^{2}... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all real solutions to the equation \\[\\frac{x-2}{x^2+4x+1} = \\frac{x-5}{x^2-10x}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\tfrac{39}{11}"
} | {
"index": 10931,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all real solutions to the equation \\[\\frac{x-3}{x^2+3x+2} = \\frac{x-7}{x^2-7x+12}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{26}{3}"
} | {
"index": 10932,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all real solutions to the equation \\[\\sqrt{x} + \\sqrt{\\frac{4}{x}} + \\sqrt{x + \\frac{4}{x}} = 6.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{64}{9}"
} | {
"index": 10933,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all real values of $x$ that satisfy\n\\[x = 1 - x + x^2 - x^3 + x^4 - x^5 + \\dotsb.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-1 + \\sqrt{5}}{2}"
} | {
"index": 10934,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all roots of the equation:\n\n$$\n\\begin{gathered}\n\\sqrt{2 x^{2}-2024 x+1023131} + \\sqrt{3 x^{2}-2025 x+1023132} + \\sqrt{4 x^{2}-2026 x+1023133} = \\\\\n= \\sqrt{x^{2}-x+1} + \\sqrt{2 x^{2}-2 x+2} + \\sqrt{3 x^{2}-3 x+3}\n\\end{gathered}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "2023"
} | {
"index": 10935,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all solutions to the equation $(x-6)^2=25$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 10936,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all the integer solutions of $x^4 - 25x^2 + 144 = 0$."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 10937,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all the positive integers which have at most three not necessarily distinct prime factors where the primes come from the set $\\{ 2, 3, 5, 7 \\}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "1932"
} | {
"index": 10938,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all the roots of the equation \\( 4x^{2} - 58x + 190 = (29 - 4x - \\log_{2} x) \\cdot \\log_{2} x \\)."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 10939,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all the solutions to $2^{|x|} + 3|x| = 18.$"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 10940,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all three-digit natural numbers that do not contain the digit 0 or the digit 5."
}
] | math | {
"style": "rule",
"ground_truth": "284160"
} | {
"index": 10941,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all values of $a + b$ , where $(a, b)$ is an ordered pair of positive integers and $a^2+\\sqrt{2017-b^2}$ is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 10942,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all values of $x$ such that the set $\\{107, 122,127, 137, 152,x\\}$ has a mean that is equal to its median."
}
] | math | {
"style": "rule",
"ground_truth": "234"
} | {
"index": 10943,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of all values of $x$ such that $2^{x^2-3x-2} = 4^{x - 4}$."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 10944,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of digits of all the numbers in the sequence $1,2,3,4,\\cdots ,10000$."
}
] | math | {
"style": "rule",
"ground_truth": "180001"
} | {
"index": 10945,
"solution": "\nTo find the sum of the digits of all the numbers in the sequence $1, 2, 3, \\ldots, 10000$, we can break down the problem into manageable parts based on the number of digits in the numbers.\n\n#### Step 1: Sum of digits for numbers from 1 to 9\nEach number from 1 to 9 is a single-di... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of distances from a point on the ellipse $7x^{2}+3y^{2}=21$ to its two foci."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{7}"
} | {
"index": 10946,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of distinct residues of the number $2012^n+m^2$ on $\\mod 11$ where $m$ and $n$ are positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 10947,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of every even positive integer less than 233 not divisible by 10."
}
] | math | {
"style": "rule",
"ground_truth": "10812"
} | {
"index": 10948,
"solution": "We find the sum of all positive even integers less than 233 and then subtract all the positive integers less than 233 that are divisible by 10. $2 + 4 + \\ldots + 232 = 2(1 + 2 + \\ldots + 116) = 116 \\cdot 117 = 13572$. The sum of all positive integers less than 233 that are divisibl... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of squares of all distinct complex numbers $x$ satisfying the equation $0=4 x^{10}-7 x^{9}+5 x^{8}-8 x^{7}+12 x^{6}-12 x^{5}+12 x^{4}-8 x^{3}+5 x^{2}-7 x+4$"
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{16}"
} | {
"index": 10949,
"solution": "For convenience denote the polynomial by $P(x)$. Notice $4+8=7+5=12$ and that the consecutive terms $12 x^{6}-12 x^{5}+12 x^{4}$ are the leading terms of $12 \\Phi_{14}(x)$, which is suggestive. Indeed, consider $\\omega$ a primitive 14 -th root of unity; since $\\omega^{7}=-1$, we ha... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the $2007$ roots of $(x-1)^{2007}+2(x-2)^{2006}+3(x-3)^{2005}+\\cdots+2006(x-2006)^2+2007(x-2007)$."
}
] | math | {
"style": "rule",
"ground_truth": "2005"
} | {
"index": 10950,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the $2010$ roots of $(x-1)^{2010} + 2(x-2)^{2009} + 3(x-3)^{2008} + \\cdots + 2009(x-2009)^2 + 2010(x-2010)^1$."
}
] | math | {
"style": "rule",
"ground_truth": "2008"
} | {
"index": 10951,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the $x$-coordinates of the distinct points of intersection of the plane curves given by $x^{2}=x+y+4$ and $y^{2}=y-15 x+36$."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 10952,
"solution": "Substituting $y=x^{2}-x-4$ into the second equation yields $$\\begin{aligned} 0 & =\\left(x^{2}-x-4\\right)^{2}-\\left(x^{2}-x-4\\right)+15 x-36 \\\\ & =x^{4}-2 x^{3}-7 x^{2}+8 x+16-x^{2}+x+4+15 x-36 \\\\ & =x^{4}-2 x^{3}-8 x^{2}+24 x-16 \\\\ & =(x-2)\\left(x^{3}-8 x+8\\right)=(x-2)^{... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-6x+5|$ and $y=\\frac{29}{4}-x$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{2}"
} | {
"index": 10953,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-8x+12|$ and $y=4-x$."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 10954,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-8x+12|$ and $y=\\frac{20}{3}-x$."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 10955,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the \\(1005\\) roots of the polynomial \\((x-1)^{1005} + 2(x-2)^{1004} + 3(x-3)^{1003} + \\cdots + 1004(x-1004)^2 + 1005(x-1005)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "1003"
} | {
"index": 10956,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the absolute values of the roots of $x^4-4x^3-4x^2+16x-8=0$."
}
] | math | {
"style": "rule",
"ground_truth": "2+2\\sqrt{2}+2\\sqrt{3}"
} | {
"index": 10957,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the absolute values of the roots of $x^{4}-4 x^{3}-4 x^{2}+16 x-8=0$."
}
] | math | {
"style": "rule",
"ground_truth": "2+2\\sqrt{2}+2\\sqrt{3}"
} | {
"index": 10958,
"solution": "$$\\begin{aligned} x^{4}-4 x^{3}-4 x^{2}+16 x-8 & =\\left(x^{4}-4 x^{3}+4 x^{2}\\right)-\\left(8 x^{2}-16 x+8\\right) \\\\ & =x^{2}(x-2)^{2}-8(x-1)^{2} \\\\ & =\\left(x^{2}-2 x\\right)^{2}-(2 \\sqrt{2} x-2 \\sqrt{2})^{2} \\\\ & =\\left(x^{2}-(2+2 \\sqrt{2}) x+2 \\sqrt{2}\\right)\\left... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the ages of everyone who wrote a problem for this year's HMMT November contest. If your answer is $X$ and the actual value is $Y$, your score will be $\\max (0,20-|X-Y|)$"
}
] | math | {
"style": "rule",
"ground_truth": "258"
} | {
"index": 10959,
"solution": "There was one problem for which I could not determine author information, so I set the author as one of the problem czars at random. Then, I ran the following command on a folder containing TeX solutions files to all four contests: ``` evan@ArchMega ~/Downloads/November $ grep --no-fi... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the areas of all distinct rectangles that can be formed from 9 squares (not necessarily all), if the side of each square is $1 \\text{ cm}$."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 10960,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the arithmetic series\n\\[20+20\\frac{1}{5}+20\\frac{2}{5}+\\cdots+40\\]"
}
] | math | {
"style": "rule",
"ground_truth": "3030"
} | {
"index": 10961,
"solution": "1. **Identify the series type and formula**: The given series is an arithmetic series. The sum of an arithmetic series where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms can be calculated using the formula:\n \\[\n S = \\frac{n(a_1 + a_n)}{2}\n ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the coefficients in the polynomial $-2(x^7 - x^4 + 3x^2 - 5) + 4(x^3 + 2x) - 3(x^5 - 4)$."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 10962,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the coefficients in the polynomial $3(3x^{7} + 8x^4 - 7) + 7(x^5 - 7x^2 + 5)$ when it is fully simplified."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 10963,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the coefficients of the polynomial $P(x)=x^{4}-29 x^{3}+a x^{2}+b x+c$, given that $P(5)=11, P(11)=17$, and $P(17)=23$."
}
] | math | {
"style": "rule",
"ground_truth": "-3193"
} | {
"index": 10964,
"solution": "Define $Q(x)=P(x)-x-6=x^{4}-29 x^{3}+a x^{2}+(b-1)x+(c-6)$ and notice that $Q(5)=Q(11)=Q(17)=0$. $Q(x)$ has degree 4 and by Vieta's Formulas the sum of its roots is 29, so its last root is $29-17-11-5=-4$, giving us $Q(x)=(x-5)(x-11)(x-17)(x+4)$. This means that $P(1)=Q(1)+7=(-4)(-10)... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the digits in the answer to\n$\\underbrace{9999\\cdots 99}_{94\\text{ nines}} \\times \\underbrace{4444\\cdots 44}_{94\\text{ fours}}$\nwhere a string of $94$ nines is multiplied by a string of $94$ fours."
}
] | math | {
"style": "rule",
"ground_truth": "846"
} | {
"index": 10965,
"solution": "1. **Identify the pattern**: We start by observing the multiplication of smaller strings of nines and fours:\n - $9 \\times 4 = 36$, and the sum of the digits is $3 + 6 = 9$.\n - $99 \\times 44 = 4356$, and the sum of the digits is $4 + 3 + 5 + 6 = 18$.\n\n2. **Generalize the patt... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the digits in the number $\\underbrace{44 \\ldots 4}_{2012 \\text{ times}} \\cdot \\underbrace{99 \\ldots 9}_{2012 \\text{ times}}$."
}
] | math | {
"style": "rule",
"ground_truth": "18108"
} | {
"index": 10966,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the digits of \\(11 \\cdot 101 \\cdot 111 \\cdot 110011\\)."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 10967,
"solution": "There is no regrouping, so the answer is \\(2 \\cdot 2 \\cdot 3 \\cdot 4=48\\). The actual product is 13566666531."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the digits of all counting numbers less than 1000."
}
] | math | {
"style": "rule",
"ground_truth": "13500"
} | {
"index": 10968,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the digits of the greatest prime number that is a divisor of $16,385$."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 10969,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the digits of the number $\\underbrace{44 \\ldots 4}_{2012 \\text { times}} \\cdot \\underbrace{99 \\ldots 9}_{2012 \\text { times}}$."
}
] | math | {
"style": "rule",
"ground_truth": "18108"
} | {
"index": 10970,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the digits of the number \\( A \\), if \\( A=2^{63} \\cdot 4^{25} \\cdot 5^{106}-2^{22} \\cdot 4^{44} \\cdot 5^{105}-1 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "959"
} | {
"index": 10971,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the distances from one vertex of a rectangle with length $3$ and width $4$ to the centers of the opposite sides."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{13} + 2"
} | {
"index": 10972,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the distinct prime factors of $5^5 - 5^3$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 10973,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the distinct prime factors of $7^7 - 7^4$."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 10974,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the even positive divisors of 1000."
}
] | math | {
"style": "rule",
"ground_truth": "2184"
} | {
"index": 10975,
"solution": "Notice that $2 k$ is a divisor of 1000 iff $k$ is a divisor of 500, so we need only find the sum of the divisors of 500 and multiply by 2. This can be done by enumerating the divisors individually, or simply by using the formula: $\\sigma\\left(2^{2} \\cdot 5^{3}\\right)=\\left(1+2+2^... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the first 10 elements that appear both among the terms of the arithmetic progression $\\{4,7,10,13, \\ldots\\}$ and the geometric progression $\\{20,40,80,160, \\ldots\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "13981000"
} | {
"index": 10976,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the first eight prime numbers that have a units digit of 3."
}
] | math | {
"style": "rule",
"ground_truth": "404"
} | {
"index": 10977,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the first five terms in the geometric sequence $\\frac13,\\frac19,\\frac1{27},\\dots$. Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{121}{243}"
} | {
"index": 10978,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the first seven prime numbers that have a units digit of 7."
}
] | math | {
"style": "rule",
"ground_truth": "379"
} | {
"index": 10979,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the first six terms in the geometric sequence $\\frac12,\\frac14,\\frac18,\\dots$. Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{63}{64}"
} | {
"index": 10980,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the infinite series $$1+2\\left(\\frac{1}{1998}\\right)+3\\left(\\frac{1}{1998}\\right)^{2}+4\\left(\\frac{1}{1998}\\right)^{3}+\\ldots$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\frac{1998}{1997}\\right)^{2} \\text{ or } \\frac{3992004}{3988009}"
} | {
"index": 10981,
"solution": "We can rewrite the sum as \\(\\left(1+\\frac{1}{1998}+\\left(\\frac{1}{1998}\\right)^{2}+\\ldots\\right)+\\left(\\frac{1}{1998}+\\left(\\frac{1}{1998}\\right)^{2}+\\left(\\frac{1}{1998}\\right)^{3}+\\ldots\\right)+\\left(\\left(\\frac{1}{1998}\\right)^{2}+\\left(\\frac{1}{1998}\\right... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the infinite series $1+2\\left(\\dfrac{1}{1998}\\right)+3\\left(\\dfrac{1}{1998}\\right)^2+4\\left(\\dfrac{1}{1998}\\right)^3+\\cdots$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3992004}{3988009}"
} | {
"index": 10982,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the infinite series $\\frac{1}{3^{2}-1^{2}}\\left(\\frac{1}{1^{2}}-\\frac{1}{3^{2}}\\right)+\\frac{1}{5^{2}-3^{2}}\\left(\\frac{1}{3^{2}}-\\frac{1}{5^{2}}\\right)+\\frac{1}{7^{2}-5^{2}}\\left(\\frac{1}{5^{2}}-\\frac{1}{7^{2}}\\right)+$"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 10983,
"solution": "$1-\\frac{1}{3}+\\frac{1}{3}-\\frac{1}{5}+\\cdots=1$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the positive divisors of 18."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 10984,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the rational roots of $g(x)=x^3-9x^2+16x-4$."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 10985,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the real roots of $x^4 - 4x - 1 = 0.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 10986,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the reciprocals of all the (positive) divisors of 144."
}
] | math | {
"style": "rule",
"ground_truth": "403/144"
} | {
"index": 10987,
"solution": "As $d$ ranges over the divisors of 144 , so does $144 / d$, so the sum of $1 / d$ is $1 / 144$ times the sum of the divisors of 144 . Using the formula for the sum of the divisors of a number (or just counting them out by hand), we get that this sum is 403, so the answer is 403/144."
... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the reciprocals of the roots of $x^2-13x+4=0$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{4}"
} | {
"index": 10988,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the roots of $\\tan^2 x - 8\\tan x + \\sqrt{2} = 0$ that are between $x=0$ and $x=2\\pi$ radians."
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"index": 10989,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the roots of $\\tan^2x-9\\tan x+1=0$ that are between $x=0$ and $x=2\\pi$ radians."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\pi"
} | {
"index": 10990,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the roots of the equation $\\tan^2x - 8\\tan x + 2 = 0$ that are between $x = 0$ and $x = 2\\pi$ radians."
}
] | math | {
"style": "rule",
"ground_truth": "3\\pi"
} | {
"index": 10991,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the roots of the equation \\[(2x^3 + x^2 - 8x + 20)(5x^3 - 25x^2 + 19) = 0.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\tfrac{9}{2}"
} | {
"index": 10992,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the roots, real and non-real, of the equation $x^{2001}+\\left(\\frac 12-x\\right)^{2001}=0$, given that there are no multiple roots."
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"index": 10993,
"solution": "We find that the given equation has a $2000^{\\text{th}}$ degree polynomial. Note that there are no multiple roots. Thus, if $\\frac{1}{2} - x$ is a root, $x$ is also a root. Thus, we pair up $1000$ pairs of roots that sum to $\\frac{1}{2}$ to get a sum of $\\boxed{500}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the series $\\sum_{n=1}^{\\infty} \\frac{1}{n^{2}+2n}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 10994,
"solution": "We know that $\\frac{1}{n^{2}+2n}=\\frac{1}{n(n+2)}=\\frac{\\frac{1}{n}-\\frac{1}{n+2}}{2}$. So, if we sum this from 1 to $\\infty$, all terms except for $\\frac{1}{2}+\\frac{\\frac{1}{2}}{2}$ will cancel out (a 'telescoping' series). Therefore, the sum will be $\\frac{3}{4}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the solutions of the equation $-32x^2 + 84x + 135=0$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{21}{8}"
} | {
"index": 10995,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the solutions to\n\\[\\frac{1}{\\sin x} + \\frac{1}{\\cos x} = 2 \\sqrt{2}\\]in the interval $0 \\le x \\le 2 \\pi.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11 \\pi}{4}"
} | {
"index": 10996,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the squares of all real numbers satisfying the equation $x^{256} - 256^{32} = 0$."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 10997,
"solution": "1. **Identify the form of the equation**: The given equation is $x^{256} - 256^{32} = 0$. We can rewrite this as $x^{256} = 256^{32}$.\n\n2. **Simplify the equation**: Notice that $256 = 2^8$, so $256^{32} = (2^8)^{32} = 2^{256}$. Thus, the equation becomes $x^{256} = 2^{256}$.\n\n3. ... |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the squares of the solutions of $x^2-13x+4=0$."
}
] | math | {
"style": "rule",
"ground_truth": "161"
} | {
"index": 10998,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Find the sum of the squares of the solutions to\n\\[\\left| x^2 - x + \\frac{1}{2008} \\right| = \\frac{1}{2008}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1003}{502}"
} | {
"index": 10999,
"solution": ""
} |
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