data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Determine the sum of the real numbers \\( x \\) for which \\(\\frac{2 x}{x^{2}+5 x+3}+\\frac{3 x}{x^{2}+x+3}=1\\)."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 7900,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the tens digit of $17^{1993}$."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 7901,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the time in hours it will take to fill a 32,000 gallon swimming pool using three hoses that deliver 3 gallons of water per minute."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 7902,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the time the copy machine will finish all the paperwork if it starts at 9:00 AM and completes half the paperwork by 12:30 PM."
}
] | math | {
"style": "rule",
"ground_truth": "4:00"
} | {
"index": 7903,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the total degrees that exceed 90 for each interior angle of a regular pentagon."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 7904,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the total number of different selections possible for five donuts when choosing from four types of donuts (glazed, chocolate, powdered, and jelly), with the additional constraint of purchasing at least one jelly donut."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 7905,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the total number of distinct, natural-number factors for the number $4^5 \\cdot 5^2 \\cdot 6^3 \\cdot 7!$."
}
] | math | {
"style": "rule",
"ground_truth": "864"
} | {
"index": 7906,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the total number of real solutions for the equation\n\\[\n\\frac{x}{50} = \\sin x.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 7907,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the total surface area of a cone with a diameter of 8 cm and a height of 12 cm. Express your answer in terms of \\(\\pi\\)."
}
] | math | {
"style": "rule",
"ground_truth": "16\\pi (\\sqrt{10} + 1)"
} | {
"index": 7908,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the total surface area of a cube if the distance between the non-intersecting diagonals of two adjacent faces of this cube is 8. If the answer is not an integer, round it to the nearest whole number."
}
] | math | {
"style": "rule",
"ground_truth": "1152"
} | {
"index": 7909,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of\n\\[\n3003 + \\frac{1}{3} \\left( 3002 + \\frac{1}{3} \\left( 3001 + \\dots + \\frac{1}{3} \\left( 4 + \\frac{1}{3} \\cdot 3 \\right) \\right) \\dotsb \\right).\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "4503.75"
} | {
"index": 7910,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of\n\\[1002 + \\frac{1}{3} \\left( 1001 + \\frac{1}{3} \\left( 1000 + \\dots + \\frac{1}{3} \\left( 3 + \\frac{1}{3} \\cdot 2 \\right) \\right) \\dotsb \\right).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "1502.25"
} | {
"index": 7911,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of\n\\[2002 + \\frac{1}{2} \\left( 2001 + \\frac{1}{2} \\left( 2000 + \\dots + \\frac{1}{2} \\left( 3 + \\frac{1}{2} \\cdot 2 \\right) \\right) \\dotsb \\right).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "4002"
} | {
"index": 7912,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of\n\\[2023 + \\frac{1}{2} \\left( 2022 + \\frac{1}{2} \\left( 2021 + \\dots + \\frac{1}{2} \\left( 4 + \\frac{1}{2} \\cdot (3 + 1) \\right) \\right) \\dotsb \\right).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "4044"
} | {
"index": 7913,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of\n\\[\\frac{\\frac{2016}{1} + \\frac{2015}{2} + \\frac{2014}{3} + \\dots + \\frac{1}{2016}}{\\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\dots + \\frac{1}{2017}}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"index": 7914,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of \n\\[3003 + \\frac{1}{3} \\left( 3002 + \\frac{1}{3} \\left( 3001 + \\dots + \\frac{1}{3} \\left( 4 + \\frac{1}{3} \\cdot 3 \\right) \\right) \\dotsb \\right).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "9006"
} | {
"index": 7915,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $$1 \\cdot 2-2 \\cdot 3+3 \\cdot 4-4 \\cdot 5+\\cdots+2001 \\cdot 2002$$"
}
] | math | {
"style": "rule",
"ground_truth": "2004002"
} | {
"index": 7916,
"solution": "2004002. Rewrite the expression as $$2+3 \\cdot(4-2)+5 \\cdot(6-4)+\\cdots+2001 \\cdot(2002-2000)$$ $$=2+6+10+\\cdots+4002$$ This is an arithmetic progression with $(4002-2) / 4+1=1001$ terms and average 2002, so its sum is $1001 \\cdot 2002=2004002$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $$2002+\\frac{1}{2}\\left(2001+\\frac{1}{2}\\left(2000+\\cdots+\\frac{1}{2}\\left(3+\\frac{1}{2} \\cdot 2\\right)\\right) \\cdots\\right)$$"
}
] | math | {
"style": "rule",
"ground_truth": "4002"
} | {
"index": 7917,
"solution": "We can show by induction that $n+\\frac{1}{2}\\left([n-1]+\\frac{1}{2}\\left(\\cdots+\\frac{1}{2} \\cdot 2\\right) \\cdots\\right)=2(n-1)$. For $n=3$ we have $3+\\frac{1}{2} \\cdot 2=4$, giving the base case, and if the result holds for $n$, then $(n+1)+\\frac{1}{2} 2(n-1)=2 n=2(n+1)-2... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $-1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 - 9 + \\dots + 10000$, where the signs change after each perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "1000000"
} | {
"index": 7918,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $-1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 - 9 + \\dots + 12100$, where the signs change after each perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "1331000"
} | {
"index": 7919,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $1 - 2 - 3 + 4 + 5 + 6 + 7 + 8 - 9 - 10 - \\dots + 9801$, where the signs change after each perfect square and repeat every two perfect squares."
}
] | math | {
"style": "rule",
"ground_truth": "-9801"
} | {
"index": 7920,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $A + B + C$, where $A$, $B$, and $C$ are the dimensions of a three-dimensional rectangular box with faces having areas $40$, $40$, $90$, $90$, $100$, and $100$ square units."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{83}{3}"
} | {
"index": 7921,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $\\frac{3b^{-1} - \\frac{b^{-1}}{3}}{b^2}$ when $b = \\tfrac{1}{3}$."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 7922,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $\\sin 135^{\\circ}\\cos 15^{\\circ}-\\cos 45^{\\circ}\\sin (-15^{\\circ})$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 7923,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $\\sin {15}^{{}^\\circ }+\\cos {15}^{{}^\\circ }$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"index": 7924,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $a + b$ if the points $(2,a,b),$ $(a,3,b),$ and $(a,b,4)$ are collinear."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 7925,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $a$ such that the equation\n\\[\\frac{x - 3}{ax - 2} = x\\]\nhas exactly one solution."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 7926,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $a^3 + b^3$ given that $a+b=12$ and $ab=20$, and also return the result for $(a+b-c)(a^3+b^3)$, where $c=a-b$."
}
] | math | {
"style": "rule",
"ground_truth": "4032"
} | {
"index": 7927,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $ab$ if $\\log_8a+\\log_4b^2=5$ and $\\log_8b+\\log_4a^2=7$."
}
] | math | {
"style": "rule",
"ground_truth": "512"
} | {
"index": 7928,
"solution": "$\\log_8a+\\log_4b^2=5$ and $\\log_8b+\\log_4a^2=7$ adding both we get \\[\\log_8{ab} +2\\log_4{ab}=12\\] then we see $\\log_4 {ab}$ is 3/2 times $\\log_8 {ab}$ putting $\\log_8 {ab}$ as x we get x+3x=12 so x=3 and $\\log_8 {ab}$ = 3 so ab= $8^3$=\\[\\boxed{512}\\] ~ math_comb01"
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $b$ that satisfies the equation $295_{b} + 467_{b} = 762_{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7929,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $b$, where $b$ is a positive number, such that the terms $10, b, \\frac{10}{9}, \\frac{10}{81}$ are the first four terms, respectively, of a geometric sequence."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7930,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $k$ such that the equation\n\\[\\frac{x + 3}{kx - 2} = x\\] has exactly one solution."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"index": 7931,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $l$ for which\n\\[\\frac{9}{x + y + 1} = \\frac{l}{x + z - 1} = \\frac{13}{z - y + 2}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 7932,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $x$ for which $(2010 + x)^2 = 4x^2$."
}
] | math | {
"style": "rule",
"ground_truth": "-670"
} | {
"index": 7933,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $x$ for which $10^x \\cdot 500^{x} = 1000000^{3}$.\nA) $\\frac{9}{1.699}$\nB) $6$\nC) $\\frac{18}{3.699}$\nD) $5$\nE) $20$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{18}{3.699}"
} | {
"index": 7934,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $x$ for which $9^{x+6} = 5^{x+1}$ can be expressed in the form $x = \\log_b 9^6$. Find the value of $b$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"index": 7935,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $x$ that satisfies $\\sqrt[4]{x\\sqrt{x^4}}=4$."
}
] | math | {
"style": "rule",
"ground_truth": "2^{8/3}"
} | {
"index": 7936,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $x$ that satisfies $\\sqrt[5]{x\\sqrt{x^3}}=3$."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 7937,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of $x^2 + y^2$ if $x - y = 25$ and $xy = 36$. Additionally, find what $x+y$ equals."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{769}"
} | {
"index": 7938,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of \\( n \\) if we know that\n\n$$\n\\binom{n}{5}=\\frac{n(n-1)(n-2)(n-3)(n-4)}{2 \\cdot 3 \\cdot 4 \\cdot 5}\n$$\n\n(which, as we know, is an integer) in the decimal system is of the form \\(\\overline{ababa}\\), where \\( a \\) and \\( b \\) represent digi... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 7939,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of \\( n \\) such that \\( 2^7 \\cdot 3^4 \\cdot n = 10! \\)."
}
] | math | {
"style": "rule",
"ground_truth": "350"
} | {
"index": 7940,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of \\(\\sum_{n=1}^{\\infty} \\ln \\left(1+\\frac{1}{n}\\right) \\cdot \\ln \\left(1+\\frac{1}{2 n}\\right) \\cdot \\ln \\left(1+\\frac{1}{2 n+1}\\right)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3} \\ln ^{3}(2)"
} | {
"index": 7941,
"solution": "Define \\(f(n)=\\ln \\left(\\frac{n+1}{n}\\right)\\) for \\(n \\geq 1\\), and observe that \\(f(2 n)+f(2 n+1)=f(n)\\). The well-known inequality \\(\\ln (1+x) \\leq x\\) implies \\(f(n) \\leq 1 / n\\). Furthermore introduce \\(g(n)=\\sum_{k=n}^{2 n-1} f^{3}(k)<n f^{3}(n) \\leq 1 / n^{2... |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of \\(a\\) if \\(a\\) and \\(b\\) are integers such that \\(x^3 - x - 1\\) is a factor of \\(ax^{19} + bx^{18} + 1\\)."
}
] | math | {
"style": "rule",
"ground_truth": "2584"
} | {
"index": 7942,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of \\(x\\) if \\(x\\) is positive and \\(x \\cdot \\lfloor x \\rfloor = 90\\). Express your answer as a decimal."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7943,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of the expression\n\\[\\log_2 (27 + \\log_2 (27 + \\log_2 (27 + \\cdots))),\\]assuming it is positive."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 7944,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of the expression\n\\[\\log_3 (64 + \\log_3 (64 + \\log_3 (64 + \\cdots))),\\]\nassuming it is positive."
}
] | math | {
"style": "rule",
"ground_truth": "3.8"
} | {
"index": 7945,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of the expression $\\frac{\\log \\sqrt{27}+\\log 8-3 \\log \\sqrt{10}}{\\log 1.2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 7946,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of the expression $\\sin 410^{\\circ}\\sin 550^{\\circ}-\\sin 680^{\\circ}\\cos 370^{\\circ}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 7947,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of the following expression, simplified as a fraction: $$1+\\cfrac{3}{2+\\cfrac{5}{6}}$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{35}{17}"
} | {
"index": 7948,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of the following expression:\n$$\n\\left\\lfloor\\frac{11}{2010}\\right\\rfloor+\\left\\lfloor\\frac{11 \\times 2}{2010}\\right\\rfloor+\\left\\lfloor\\frac{11 \\times 3}{2010}\\right\\rfloor+\\\\left\\lfloor\\frac{11 \\times 4}{2010}\\right\\rfloor+\\cdots+... | math | {
"style": "rule",
"ground_truth": "10045"
} | {
"index": 7949,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of the following expressions:\n$(1)(2 \\frac{7}{9})^{0.5}+0.1^{-2}+(2 \\frac{10}{27})\\,^{- \\frac{2}{3}}-3π^{0}+ \\frac{37}{48}$;\n$(2)(-3 \\frac{3}{8})\\,^{- \\frac{2}{3}}+(0.002)\\,^{- \\frac{1}{2}}-10( \\sqrt{5}-2)^{-1}+( \\sqrt{2}- \\sqrt{3})^{0}$."
}... | math | {
"style": "rule",
"ground_truth": "- \\frac{167}{9}"
} | {
"index": 7950,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of the following product with a short calculation:\n\n$$\n\\frac{6 \\cdot 27^{12}+2 \\cdot 81^{9}}{8000000^{2}} \\cdot \\frac{80 \\cdot 32^{3} \\cdot 125^{4}}{9^{19}-729^{6}}\n$$"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7951,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of the following sum:\n\n$$\n\\log _{3}\\left(1-\\frac{1}{15}\\right)+\\log _{3}\\left(1-\\frac{1}{14}\\right)+\\log _{3}\\left(1-\\frac{1}{13}\\right)+\\cdots+\\log _{3}\\left(1-\\frac{1}{8}\\right)+\\log _{3}\\left(1-\\frac{1}{7}\\right)+\\log _{3}\\left(1... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 7952,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of the infinite product $(2^{1/3})(4^{1/9})(8^{1/27})(16^{1/81}) \\dotsm.$ Enter your answer in the form \"\\sqrt[a]{b}\", which stands for $\\sqrt[a]{b}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[4]{8}"
} | {
"index": 7953,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of the infinite product $(3^{1/4})(9^{1/16})(27^{1/64})(81^{1/256}) \\dotsm$ plus 2, the result in the form of \"$\\sqrt[a]{b}$ plus $c$\"."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[9]{81} + 2"
} | {
"index": 7954,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of the sum $$\\frac{3}{1^{2} \\cdot 2^{2}}+\\frac{5}{2^{2} \\cdot 3^{2}}+\\frac{7}{3^{2} \\cdot 4^{2}}+\\cdots+\\frac{29}{14^{2} \\cdot 15^{2}}$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{224}{225}"
} | {
"index": 7955,
"solution": "The sum telescopes as $$\\left(\\frac{1}{1^{2}}-\\frac{1}{2^{2}}\\right)+\\left(\\frac{1}{2^{2}}-\\frac{1}{3^{2}}\\right)+\\cdots+\\left(\\frac{1}{14^{2}}-\\frac{1}{15^{2}}\\right)=\\frac{1}{1^{2}}-\\frac{1}{15^{2}}=\\frac{224}{225}$$"
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of the sum \\[ \\sum_{n=0}^{332} (-1)^{n} {1008 \\choose 3n} \\] and find the remainder when the sum is divided by $500$."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 7956,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the value of:\n\\[3003 + \\frac{1}{3} \\left( 3002 + \\frac{1}{3} \\left( 3001 + \\dots + \\frac{1}{3} \\left( 4 + \\frac{1}{3} \\cdot 3 \\right) \\right) \\dotsb \\right).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "9006.5"
} | {
"index": 7957,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the values of $a$, $b$, and $r$ of the circle given by the equation $x^2 + 14y + 65 = -y^2 - 8x$. Let $a$ and $b$ be the coordinates of the center of the circle, and $r$ be its radius. What is the sum $a + b + r$?"
}
] | math | {
"style": "rule",
"ground_truth": "-11"
} | {
"index": 7958,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the values of $x$ and $y$ given that 15 is the arithmetic mean of the set $\\{8, x, 21, y, 14, 11\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 7959,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the volume in cubic inches of a right, rectangular prism where the areas of the side, front, and bottom faces are 20 square inches, 12 square inches, and 8 square inches, respectively."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{30}"
} | {
"index": 7960,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the volume of the original cube given that one dimension is increased by $3$, another is decreased by $2$, and the third is left unchanged, and the volume of the resulting rectangular solid is $6$ more than that of the original cube."
}
] | math | {
"style": "rule",
"ground_truth": "(3 + \\sqrt{15})^3"
} | {
"index": 7961,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the volume of the region enclosed by \\[|x + y + z| + |x + y - z| + |x - y + z| + |-x + y + z| \\le 6.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 7962,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the volume of the region in space defined by\n\\[|x + y + z| + |x + y - z| \\le 12\\]\nand \\(x, y, z \\ge 0.\\)"
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 7963,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the volume of the released gas:\n\n\\[ \\omega\\left(\\mathrm{SO}_{2}\\right) = n\\left(\\mathrm{SO}_{2}\\right) \\cdot V_{m} = 0.1122 \\cdot 22.4 = 2.52 \\text{ liters} \\]"
}
] | math | {
"style": "rule",
"ground_truth": "2.52"
} | {
"index": 7964,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the volume of the right rectangular parallelepiped whose edges are formed by the distances from the orthocenter to the vertices of a triangle, where the radius of the circumcircle $r = 2.35$ and the angles are: $\\alpha = 63^{\\circ} 18^{\\prime} 13^{\\prime \\prime},... | math | {
"style": "rule",
"ground_truth": "12.2"
} | {
"index": 7965,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the volume of the solid formed by the set of vectors $\\mathbf{v}$ such that\n\\[\\mathbf{v} \\cdot \\mathbf{v} = \\mathbf{v} \\cdot \\begin{pmatrix} -6 \\\\ 18 \\\\ 12 \\end{pmatrix}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3} \\pi \\cdot 126 \\sqrt{126}"
} | {
"index": 7966,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine the volume of the solid formed by the set of vectors $\\mathbf{v}$ such that:\n\\[\\mathbf{v} \\cdot \\mathbf{v} = \\mathbf{v} \\cdot \\begin{pmatrix} 12 \\\\ -34 \\\\ 6 \\end{pmatrix}\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3} \\pi (334)^{3/2}"
} | {
"index": 7967,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1, a_2, a_3, \\ldots$ of nonzero integers such that the equality \\[a_k+2a_{2k}+\\ldots+na_{nk}=0\\]holds for every positive integer $k$."
}
] | math | {
"style": "rule",
"ground_truth": "n > 2"
} | {
"index": 7968,
"solution": "Consider the problem to determine which integers \\( n > 1 \\) have the property that there exists an infinite sequence \\( a_1, a_2, a_3, \\ldots \\) of nonzero integers satisfying the equality:\n\n\\[\na_k + 2a_{2k} + \\ldots + na_{nk} = 0\n\\]\n\nfor every positive integer \\( k \\)... |
DeepScaleR | [
{
"role": "user",
"content": "Determine which of the following expressions has the largest value: $4^2$, $4 \\times 2$, $4 - 2$, $\\frac{4}{2}$, or $4 + 2$."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 7969,
"solution": "We evaluate each of the five choices: $4^{2}=16$, $4 \\times 2=8$, $4-2=2$, $\\frac{4}{2}=2$, $4+2=6$. Of these, the largest is $4^{2}=16$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine which of the following numbers is smallest in value: $54 \\sqrt{3}, 144,108 \\sqrt{6}-108 \\sqrt{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "$54 \\sqrt{3}$"
} | {
"index": 7970,
"solution": "We can first compare $54 \\sqrt{3}$ and 144. Note that $\\sqrt{3}<2$ and $\\frac{144}{54}=\\frac{8}{3}>2$. Hence, $54 \\sqrt{3}$ is less. Now, we wish to compare this to $108 \\sqrt{6}-108 \\sqrt{2}$. This is equivalent to comparing $\\sqrt{3}$ to $2(\\sqrt{6}-\\sqrt{2})$. We claim tha... |
DeepScaleR | [
{
"role": "user",
"content": "Determine which positive integers $n$ have the following property: For all integers $m$ that are relatively prime to $n$, there exists a permutation $\\pi\\colon \\{1,2,\\dots,n\\} \\to \\{1,2,\\dots,n\\}$ such that $\\pi(\\pi(k)) \\equiv mk \\pmod{n}$ for all $k \\in \\{1,2,\\... | math | {
"style": "rule",
"ground_truth": "n = 1 \\text{ or } n \\equiv 2 \\pmod{4}"
} | {
"index": 7971,
"solution": "The desired property holds if and only if $n = 1$ or $n \\equiv 2 \\pmod{4}$. Let $\\sigma_{n,m}$ be the permutation of $\\ZZ/n\\ZZ$ induced by multiplication by $m$; the original problem asks for which $n$ does $\\sigma_{n,m}$ always have a square root. For $n=1$, $\\sigma_{n,m}$ is t... |
DeepScaleR | [
{
"role": "user",
"content": "Determine with proof the number of positive integers $n$ such that a convex regular polygon with $n$ sides has interior angles whose measures, in degrees, are integers."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 7972,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine, with proof, the smallest positive integer $c$ such that for any positive integer $n$ , the decimal representation of the number $c^n+2014$ has digits all less than $5$ .\n\n*Proposed by Evan Chen*"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7973,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Determine, with proof, the smallest positive integer \\( n \\) with the following property: For every choice of \\( n \\) integers, there exist at least two whose sum or difference is divisible by 2009."
}
] | math | {
"style": "rule",
"ground_truth": "1006"
} | {
"index": 7974,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dewa writes down a list of four integers. He calculates the average of each group of three of the four integers. These averages are $32,39,40,44$. What is the largest of the four integers?"
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 7975,
"solution": "Suppose that Dewa's four numbers are $w, x, y, z$. The averages of the four possible groups of three of these are $\\frac{w+x+y}{3}, \\frac{w+x+z}{3}, \\frac{w+y+z}{3}, \\frac{x+y+z}{3}$. These averages are equal to $32,39,40,44$, in some order. The sums of the groups of three are equa... |
DeepScaleR | [
{
"role": "user",
"content": "Define the determinant $D_1$ = $|1|$ , the determinant $D_2$ =\f \f \f \f \f \f $|1 1|$ $|1 3|$ \f \f \f \f \f \f\n, and the determinant $D_3=$ \f \f \f \f \f \f \f \f \f\n|1 1 1|\n|1 3 3|\n|1 3 5| \f \f \f \f \f \f \f \f \f\n.\nIn general, for positive integer n, let t... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 7976,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the sequence of positive integers $a_n$ recursively by $a_1=7$ and $a_n=7^{a_{n-1}}$ for all $n\\geq 2$ . Determine the last two digits of $a_{2007}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 7977,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Diagonal $DB$ of rectangle $ABCD$ is divided into three segments of length $1$ by parallel lines $L$ and $L'$ that pass through $A$ and $C$ and are perpendicular to $DB$. The area of $ABCD$, rounded to the one decimal place, is"
}
] | math | {
"style": "rule",
"ground_truth": "4.2"
} | {
"index": 7978,
"solution": "1. **Identify the Geometry and Key Points**: \n Let $ABCD$ be a rectangle with diagonal $DB$. Lines $L$ and $L'$ are parallel to each other, pass through points $A$ and $C$ respectively, and are perpendicular to diagonal $DB$. Let $E$ and $F$ be the points where lines $L$ and $L'$ i... |
DeepScaleR | [
{
"role": "user",
"content": "Diagonals $A C$ and $C E$ of a regular hexagon $A B C D E F$ are divided by points $M$ and $N$ such that $A M : A C = C N : C E = \\lambda$. Find $\\lambda$ if it is known that points $B, M$, and $N$ are collinear."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{3}}"
} | {
"index": 7979,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Diagonals \\( AC \\) and \\( BD \\) of the cyclic quadrilateral \\( ABCD \\) are perpendicular and intersect at point \\( M \\). It is known that \\( AM = 3 \\), \\( BM = 4 \\), and \\( CM = 6 \\). Find \\( CD \\)."
}
] | math | {
"style": "rule",
"ground_truth": "10.5"
} | {
"index": 7980,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 7981,
"solution": "Let $AB=10x+y$ and $CD=10y+x$. It follows that $CO=\\frac{AB}{2}=\\frac{10x+y}{2}$ and $CH=\\frac{CD}{2}=\\frac{10y+x}{2}$. Scale up this triangle by 2 to ease the arithmetic. Applying the Pythagorean Theorem on $2CO$, $2OH$ and $2CH$, we deduce \\[(2OH)^2=\\left(10x+y\\right)^2-\\left... |
DeepScaleR | [
{
"role": "user",
"content": "Diana and Apollo each roll a standard die obtaining a number at random from $1$ to $6$. What is the probability that Diana's number is larger than Apollo's number?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"index": 7982,
"solution": "To find the probability that Diana's number is larger than Apollo's number, we consider all possible outcomes when both Diana and Apollo roll a standard six-sided die.\n\n1. **List the possible outcomes for each roll of Apollo:**\n - If Apollo rolls a 1, Diana can roll a 2, 3, 4, 5, ... |
DeepScaleR | [
{
"role": "user",
"content": "Diana drew a rectangular grid of 2 by 1009 squares. Some of the squares were then painted black. In each white square, she wrote the number of black squares that shared an edge with it (a whole edge, not just a vertex). What is the maximum value that she could obtain as the res... | math | {
"style": "rule",
"ground_truth": "3025"
} | {
"index": 7983,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Diana is playing a card game against a computer. She starts with a deck consisting of a single card labeled 0.9. Each turn, Diana draws a random card from her deck, while the computer generates a card with a random real number drawn uniformly from the interval $[0,1]$. If the n... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 7984,
"solution": "By linearity of expectation, we can treat the number of turns each card contributes to the total independently. Let $f(x)$ be the expected number of turns a card of value $x$ contributes (we want $f(0.9)$). If we have a card of value $x$, we lose it after 1 turn with probability $1-x$.... |
DeepScaleR | [
{
"role": "user",
"content": "Diana is three years younger than Eduardo. Eduardo is four years older than Chad. Faye is three years older than Chad. Diana is 14 years old. How many years old is Faye?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 7985,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Diane has one 1-cent stamp, two identical 2-cent stamps, and so on, up to nine identical 9-cent stamps. In how many different arrangements can Diane paste exactly 10 cents worth of postage in a row across the top of an envelope? (Note, however, that simply rotating or inverting... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 7986,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Digital clocks display hours and minutes (for example, 16:15). While practicing arithmetic, Buratino finds the sum of the digits on the clock $(1+6+1+5=13)$. Write down such a time of day when the sum of the digits on the clock will be the greatest."
}
] | math | {
"style": "rule",
"ground_truth": "19:59"
} | {
"index": 7987,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Digits are placed in the two boxes of $2 \\square \\square$, with one digit in each box, to create a three-digit positive integer. In how many ways can this be done so that the three-digit positive integer is larger than 217?"
}
] | math | {
"style": "rule",
"ground_truth": "82"
} | {
"index": 7988,
"solution": "The question is equivalent to asking how many three-digit positive integers beginning with 2 are larger than 217. These integers are 218 through 299 inclusive. There are $299 - 217 = 82$ such integers."
} |
DeepScaleR | [
{
"role": "user",
"content": "Dima and Sergey were picking berries from a raspberry bush that had 900 berries. Dima alternated his actions: he put one berry in the basket and ate the next one. Sergey also alternated his actions: he put two berries in the basket and ate the next one. It is known that Dima pi... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 7989,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dima and Sergey were picking raspberries from a bush that had 900 berries. Dima alternated his actions while picking: he put one berry in the basket, and then he ate the next one. Sergey also alternated: he put two berries in the basket, and then he ate the next one. It is know... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 7990,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dima calculated the factorials of all natural numbers from 80 to 99, found the numbers that are reciprocals to them, and printed the resulting decimal fractions on 20 infinite ribbons (for example, the last ribbon had printed the number $\\frac{1}{99!}=0, \\underbrace{00 \\ldot... | math | {
"style": "rule",
"ground_truth": "155"
} | {
"index": 7991,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dima calculated the factorials of all natural numbers from 80 to 99, found the reciprocals of them, and printed the resulting decimal fractions on 20 endless ribbons (for example, the last ribbon had the number \\(\\frac{1}{99!}=0. \\underbrace{00\\ldots00}_{155 \\text{ zeros}}... | math | {
"style": "rule",
"ground_truth": "155"
} | {
"index": 7992,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dima took the fractional-linear function \\(\\frac{a x + 2b}{c x + 2d}\\), where \\(a, b, c, d\\) are positive numbers, and summed it with the remaining 23 functions obtained from it by permuting the numbers \\(a, b, c, d\\). Find the root of the sum of all these functions, ind... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 7993,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Distinct lines $\\ell$ and $m$ lie in the $xy$-plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $\\ell$ to point $P'$, and then $P'$ is reflected about line $m$ to point $P''$. The equation of line $\\ell$ is $5x - y = 0$, and the coordinates of $P'... | math | {
"style": "rule",
"ground_truth": "2x-3y=0"
} | {
"index": 7994,
"solution": "1. **Identify the given information:**\n - Line $\\ell$ has the equation $5x - y = 0$.\n - Point $P$ has coordinates $(-1, 4)$.\n - Point $P''$ has coordinates $(4, 1)$.\n - Lines $\\ell$ and $m$ intersect at the origin $O$.\n\n2. **Understand the geometric transformations:**\n... |
DeepScaleR | [
{
"role": "user",
"content": "Distinct planes $p_1, p_2, \\dots, p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $P = \\bigcup_{j=1}^{k} p_j$. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belong... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 7995,
"solution": "To solve this problem, we need to analyze the different types of planes that can intersect the cube $Q$ such that their intersection with the surface $S$ of the cube consists of segments joining the midpoints of edges on the same face of $Q$. We categorize these planes based on the typ... |
DeepScaleR | [
{
"role": "user",
"content": "Distinct points $A$, $B$, $C$, and $D$ lie on a line, with $AB=BC=CD=1$. Points $E$ and $F$ lie on a second line, parallel to the first, with $EF=1$. A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of th... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 7996,
"solution": "1. **Identify Possible Bases and Heights**: \n - The points $A$, $B$, $C$, and $D$ are collinear with equal distances between consecutive points, i.e., $AB = BC = CD = 1$.\n - Points $E$ and $F$ are also collinear on a different line parallel to the line containing $A$, $B$, $C$, a... |
DeepScaleR | [
{
"role": "user",
"content": "Distinct points $A, B, C, D$ are given such that triangles $A B C$ and $A B D$ are equilateral and both are of side length 10 . Point $E$ lies inside triangle $A B C$ such that $E A=8$ and $E B=3$, and point $F$ lies inside triangle $A B D$ such that $F D=8$ and $F B=3$. What i... | math | {
"style": "rule",
"ground_truth": "\\frac{91 \\sqrt{3}}{4}"
} | {
"index": 7997,
"solution": "$\\angle F B D+\\angle A B F=\\angle A B D=60^{\\circ}$. Since $E B=B F=3$, this means that $E B F$ is an equilateral triangle of side length 3. Now we have $[A E F D]=[A E B D]-[E B F]-[F B D]=[A E B]+[A B D]-[E B F]-$ $[F B D]=[A B D]-[E B F]=\\frac{\\sqrt{3}}{4}\\left(10^{2}-3^{2}\\... |
DeepScaleR | [
{
"role": "user",
"content": "Distinct points $P$, $Q$, $R$, $S$ lie on the circle $x^{2}+y^{2}=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\\frac{PQ}{RS}$?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 7998,
"solution": "1. **Identify Possible Points**: The circle given by the equation $x^2 + y^2 = 25$ has a radius of 5. We need to find points $(x, y)$ with integer coordinates that lie on this circle. Solving $x^2 + y^2 = 25$ for integer solutions, we find the points $(\\pm 3, \\pm 4), (\\pm 4, \\pm 3)... |
DeepScaleR | [
{
"role": "user",
"content": "Distinct prime numbers $p, q, r$ satisfy the equation $2 p q r+50 p q=7 p q r+55 p r=8 p q r+12 q r=A$ for some positive integer $A$. What is $A$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "1980"
} | {
"index": 7999,
"solution": "Note that $A$ is a multiple of $p, q$, and $r$, so $K=\\frac{A}{p q r}$ is an integer. Dividing through, we have that $$K=8+\\frac{12}{p}=7+\\frac{55}{q}=2+\\frac{50}{r}$$ Then $p \\in\\{2,3\\}, q \\in\\{5,11\\}$, and $r \\in\\{2,5\\}$. These values give $K \\in\\{14,12\\}, K \\in\\{18... |
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