data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The numbers from 1 to 9 are placed at the vertices of a cube such that the sum of the four numbers on each face is the same. Find the common sum."
}
] | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"index": 33500,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers from 1 to 9 are placed in the cells of a $3 \\times 3$ table such that the sum of the numbers on one diagonal is 7, and on the other diagonal is 21. What is the sum of the numbers in the five shaded cells?\n\n![3x3 Grid](https://cdn.mathpix.com/cropped/2024_05_06_a7... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 33501,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers from 1 to 9 are placed in the cells of a 3x3 table so that the sum of the numbers on one diagonal is 7 and the sum on the other diagonal is 21. What is the sum of the numbers in the five shaded cells?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 33502,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers from 1 to 9 are placed in the cells of a \\(3 \\times 3\\) grid such that the sum of the numbers on one diagonal is 7 and on the other diagonal is 21. What is the sum of the numbers in the five shaded cells?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 33503,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers from 1 to 9 are placed in the cells of a \\(3 \\times 3\\) table such that the sum of the numbers on one diagonal equals 7, and the sum on the other diagonal equals 21. What is the sum of the numbers in the five shaded cells?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 33504,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numbers in the sequence $101$, $104$, $109$, $116$,$\\ldots$ are of the form $a_n=100+n^2$, where $n=1,2,3,\\ldots$ For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n+1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers."
... | math | {
"style": "rule",
"ground_truth": "401"
} | {
"index": 33505,
"solution": "We know that $a_n = 100+n^2$ and $a_{n+1} = 100+(n+1)^2 = 100+ n^2+2n+1$. Since we want to find the GCD of $a_n$ and $a_{n+1}$, we can use the Euclidean algorithm:\n$a_{n+1}-a_n = 2n+1$\n Now, the question is to find the GCD of $2n+1$ and $100+n^2$. We subtract $2n+1$ 100 times from $... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is ... | math | {
"style": "rule",
"ground_truth": "220"
} | {
"index": 33506,
"solution": "Let the terms in the sequence be defined as \\[a_1, a_2, ..., a_6.\\]\n Since this is an arithmetic sequence, we have $a_1+a_6=a_2+a_5=a_3+a_4.$ So, \\[\\sum_{i=1}^6 a_i=3(a_1+a_6)=990.\\] Hence, $(a_1+a_6)=330.$ And, since we are given that $a_6=2a_1,$ we get $3a_1=330\\implies a_1=1... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers on the faces of this cube are consecutive whole numbers. The sum of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is"
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 33507,
"solution": "1. **Identify the properties of the cube**: The problem states that the numbers on the faces of the cube are consecutive whole numbers and the sum of the numbers on each pair of opposite faces is equal.\n\n2. **Determine possible sets of numbers**: Since the numbers are consecutive an... |
DeepScaleR | [
{
"role": "user",
"content": "The numbers which contain only even digits in their decimal representations are written in ascending order such that \\[2,4,6,8,20,22,24,26,28,40,42,\\dots\\] What is the $2014^{\\text{th}}$ number in that sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "62048"
} | {
"index": 33508,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numeral $65$ in base $c$ represents the same number as $56$ in base $d$. Assuming that both $c$ and $d$ are positive integers, find the least possible value of $c+d$."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 33509,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The numerator of a fraction is $6x + 1$, then denominator is $7 - 4x$, and $x$ can have any value between $-2$ and $2$, both included. The values of $x$ for which the numerator is greater than the denominator are:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5} < x \\le 2"
} | {
"index": 33510,
"solution": "We are given a fraction with the numerator $6x + 1$ and the denominator $7 - 4x$. We need to find the values of $x$ for which the numerator is greater than the denominator. This can be set up as an inequality:\n\\[ 6x + 1 > 7 - 4x. \\]\n\n1. **Isolate $x$:**\n \\[ 6x + 1 > 7 - 4x \\... |
DeepScaleR | [
{
"role": "user",
"content": "The numerical sequence \\(a_{0}, a_{1}, a_{2}, \\ldots \\) is such that for all non-negative \\(m\\) and \\(n\\) (where \\(m \\geq n\\)), the following relation holds:\n\n\\[a_{m+n} + a_{m-n} = \\frac{1}{2} (a_{2m} + a_{2n})\\]\n\nFind \\(a_{1995}\\) if \\(a_{1} = 1\\)."
}
] | math | {
"style": "rule",
"ground_truth": "1995^2"
} | {
"index": 33511,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The octagon $P_1P_2P_3P_4P_5P_6P_7P_8$ is inscribed in a circle, with the vertices around the circumference in the given order. Given that the polygon $P_1P_3P_5P_7$ is a square of area 5, and the polygon $P_2P_4P_6P_8$ is a rectangle of area 4, find the maximum possible area ... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{5}"
} | {
"index": 33512,
"solution": "The maximum area is $3 \\sqrt{5}$.\n\nWe deduce from the area of $P_1P_3P_5P_7$ that the radius of the circle is $\\sqrt{5/2}$. An easy calculation using the Pythagorean Theorem then shows that the rectangle $P_2P_4P_6P_8$ has sides $\\sqrt{2}$ and $2\\sqrt{2}$.\nFor notational ease,... |
DeepScaleR | [
{
"role": "user",
"content": "The odd function $y=f(x)$ has a domain of $\\mathbb{R}$, and when $x \\geq 0$, $f(x) = 2x - x^2$. If the range of the function $y=f(x)$, where $x \\in [a, b]$, is $[\\frac{1}{b}, \\frac{1}{a}]$, then the minimum value of $b$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 33513,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The odd numbers from 5 to 21 are used to build a 3 by 3 magic square. If 5, 9 and 17 are placed as shown, what is the value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 33514,
"solution": "The sum of the odd numbers from 5 to 21 is $5+7+9+11+13+15+17+19+21=117$. Therefore, the sum of the numbers in any row is one-third of this total, or 39. This means as well that the sum of the numbers in any column or diagonal is also 39. Since the numbers in the middle row add to 39,... |
DeepScaleR | [
{
"role": "user",
"content": "The older brother and the younger brother each bought several apples. The older brother said to the younger brother, \"If I give you one apple, we will have the same number of apples.\" The younger brother thought for a moment and said to the older brother, \"If I give you one ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33515,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The on-time arrival rate of bus No. 101 in a certain city is 90%. Calculate the probability that the bus arrives on time exactly 4 times out of 5 rides for a person."
}
] | math | {
"style": "rule",
"ground_truth": "0.32805"
} | {
"index": 33516,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The only prime factors of an integer $n$ are 2 and 3. If the sum of the divisors of $n$ (including itself) is $1815$ , find $n$ ."
}
] | math | {
"style": "rule",
"ground_truth": "648"
} | {
"index": 33517,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The operation $ \\diamond $ is defined for positive integers $a$ and $b$ such that $a \\diamond b = a^2 - b$. Determine how many positive integers $x$ exist such that $20 \\diamond x$ is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 33518,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The operation $*$ is defined by\n\\[a * b = \\frac{a - b}{1 - ab}.\\]Compute\n\\[1 * (2 * (3 * (\\dotsb (999 * 1000) \\dotsb))).\\]"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 33519,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The operation $*$ is defined for non-zero integers as follows: $a * b = \\frac{1}{a} + \\frac{1}{b}$. If $a+b = 9$ and $ a \\times b = 20$, what is the value of $a*b$? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{20}"
} | {
"index": 33520,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The operation $\\#$ is defined as $a \\# b = a + \\frac{a}{b}$. What is the value of $6 \\# 2$?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 33521,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The operation $\\&$ is defined as $a \\& b = (a+b)(a-b)$. What is the value of $6 \\& 3$?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 33522,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The operation $\\Diamond$ is defined by $a\\Diamond b=ab^2-b+1$. What is the value of $(-1)\\Diamond 6$?"
}
] | math | {
"style": "rule",
"ground_truth": "-41"
} | {
"index": 33523,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The operation $\\dagger$ is defined as $\\frac{m}{n}\\dagger\\frac{p}{q} = (m)(p)(\\frac{q}{n}).$ What is the simplified value of $\\frac{7}{12}\\dagger\\frac{8}{3}$?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 33524,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The operation $\\nabla$ is defined by $a \\nabla b=4 a+b$. What is the value of $(5 \\nabla 2) \\nabla 2$?"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 33525,
"solution": "Using the definition, $(5 \\nabla 2) \\nabla 2=(4 \\times 5+2) \\nabla 2=22 \\nabla 2=4 \\times 22+2=90$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The operation $\\nabla$ is defined by $g \\nabla h=g^{2}-h^{2}$. If $g>0$ and $g \\nabla 6=45$, what is the value of $g$?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 33526,
"solution": "Using the definition of the operation, $g \\nabla 6=45$ gives $g^{2}-6^{2}=45$. Thus, $g^{2}=45+36=81$. Since $g>0$, then $g=\\sqrt{81}=9$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The operation $\\odot$ is defined as $a \\odot b = a + \\frac{3a}{2b}$. What is the value of $8 \\odot 6$?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 33527,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The operation $\\otimes$ is defined for all nonzero numbers by $a \\otimes b = \\frac{a^{2}}{b}$. Determine $[(1 \\otimes 2) \\otimes 3] - [1 \\otimes (2 \\otimes 3)]$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{3}"
} | {
"index": 33528,
"solution": "1. **Apply the operation $\\otimes$ to the innermost parentheses first:**\n \\[\n (1 \\otimes 2) \\otimes 3 = \\left(\\frac{1^2}{2}\\right) \\otimes 3 = \\frac{1}{2} \\otimes 3\n \\]\n \\[\n 1 \\otimes (2 \\otimes 3) = 1 \\otimes \\left(\\frac{2^2}{3}\\right) = 1 \\otimes \\... |
DeepScaleR | [
{
"role": "user",
"content": "The operation $\\star$ is defined as $a \\star b = a + \\frac{a}{b}$. What is the value of $12 \\star 3$?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 33529,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The operation $\\star$ is defined as $a \\star b = a^2 \\div b$. For how many integer values of $x$ will the value of $10 \\star x$ be a positive integer?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 33530,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The operation $\\star$ is defined as $a \\star b = a^2 \\div b$. For how many negative integer values of $x$ will the value of $12 \\star x$ be a positive integer?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 33531,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The operation $a \\nabla b$ is defined by $a \\nabla b=\\frac{a+b}{a-b}$ for all integers $a$ and $b$ with $a \\neq b$. If $3 \\nabla b=-4$, what is the value of $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 33532,
"solution": "Using the definition, $3 \\nabla b=\\frac{3+b}{3-b}$. Assuming $b \\neq 3$, the following equations are equivalent: $3 \\nabla b =-4$, $\\frac{3+b}{3-b} =-4$, $3+b =-4(3-b)$, $3+b =-12+4 b$, $15 =3 b$ and so $b=5$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The operation \\( \\otimes \\) is defined by \\( a \\otimes b = \\frac{a}{b} + \\frac{b}{a} \\). What is the value of \\( 4 \\otimes 8 \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 33533,
"solution": "From the given definition, \\( 4 \\otimes 8 = \\frac{4}{8} + \\frac{8}{4} = \\frac{1}{2} + 2 = \\frac{5}{2} \\)."
} |
DeepScaleR | [
{
"role": "user",
"content": "The opposite number of $-1 \\frac{1}{2}$ is ______, its reciprocal is ______, and its absolute value is ______."
}
] | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"index": 33534,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The opposite number of $2- \\sqrt{3}$ is ______, and its absolute value is ______."
}
] | math | {
"style": "rule",
"ground_truth": "2- \\sqrt{3}"
} | {
"index": 33535,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The opposite of $-23$ is ______; the reciprocal is ______; the absolute value is ______."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 33536,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The opposite of the arithmetic square root of $\\sqrt{81}$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"index": 33537,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The opposite of the real number $2023$ is"
}
] | math | {
"style": "rule",
"ground_truth": "-2023"
} | {
"index": 33538,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The oranges are placed in a pyramid-like stack with the base being a rectangle that is 5 oranges wide and 7 oranges long. Each orange above the first layer rests in a pocket formed by four oranges in the layer below. Determine the total number of oranges in the stack."
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"index": 33539,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The orchestra has more than 150 members but fewer than 300 members. When they line up in rows of 6 there are two extra people; when they line up in rows of 8 there are four extra people; and when they line up in rows of 9 there are six extra people. How many members are in the ... | math | {
"style": "rule",
"ground_truth": "212"
} | {
"index": 33540,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The organizers of a ping-pong tournament have only one table. They call two participants to play, who have not yet played against each other. If after the game the losing participant suffers their second defeat, they are eliminated from the tournament (since there are no ties i... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 33541,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The organizing committee of the sports meeting needs to select four volunteers from Xiao Zhang, Xiao Zhao, Xiao Li, Xiao Luo, and Xiao Wang to take on four different tasks: translation, tour guide, etiquette, and driver. If Xiao Zhang and Xiao Zhao can only take on the first tw... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 33542,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The orthocenter of triangle $ABC$ divides altitude $\\overline{CF}$ into segments with lengths $HF = 6$ and $HC = 15.$ Calculate $\\tan A \\tan B.$\n\n[asy]\nunitsize (1 cm);\n\npair A, B, C, D, E, F, H;\n\nA = (0,0);\nB = (5,0);\nC = (4,4);\nD = (A + reflect(B,C)*(A))/2;\nE =... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"index": 33543,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The orthocenter of triangle $DEF$ divides altitude $\\overline{DM}$ into segments with lengths $HM = 10$ and $HD = 24.$ Calculate $\\tan E \\tan F.$"
}
] | math | {
"style": "rule",
"ground_truth": "3.4"
} | {
"index": 33544,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The orthogonal projections of the triangle \\(ABC\\) onto two mutually perpendicular planes are equilateral triangles with sides of length 1. Find the perimeter of triangle \\(ABC\\), given that \\(AB = \\frac{\\sqrt{5}}{2}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + \\sqrt{5}"
} | {
"index": 33545,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The output of a factory last year is denoted as $1$. If it is planned that the output of each of the next five years will increase by $10\\%$ compared to the previous year, then the total output of this factory for the five years starting from this year will be approximately \\... | math | {
"style": "rule",
"ground_truth": "6.6"
} | {
"index": 33546,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The overall idea is a common method in mathematical problem-solving. Below is the train of thought for factoring the polynomial $(a^{2}+2a)(a^{2}+2a+2)+1$: Consider \"$a^{2}+2a$\" as a whole, let $a^{2}+2a=x$, then the expression $=x(x+2)+1=x^{2}+2x+1=(x+1)^{2}$, then restore \... | math | {
"style": "rule",
"ground_truth": "2024"
} | {
"index": 33547,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The owner of an individual clothing store purchased 30 dresses for $32 each. The selling price of the 30 dresses varies for different customers. Using $47 as the standard price, any excess amount is recorded as positive and any shortfall is recorded as negative. The results are... | math | {
"style": "rule",
"ground_truth": "472"
} | {
"index": 33548,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The pages of a book are numbered $1_{}^{}$ through $n_{}^{}$. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of $1986_{}^{}$. What was the number of the page that was added twice?"
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 33549,
"solution": "Denote the page number as $x$, with $x < n$. The sum formula shows that $\\frac{n(n + 1)}{2} + x = 1986$. Since $x$ cannot be very large, disregard it for now and solve $\\frac{n(n+1)}{2} = 1986$. The positive root for $n \\approx \\sqrt{3972} \\approx 63$. Quickly testing, we find th... |
DeepScaleR | [
{
"role": "user",
"content": "The pairwise greatest common divisors of five positive integers are $2,3,4,5,6,7,8, p, q, r$ in some order, for some positive integers $p, q, r$. Compute the minimum possible value of $p+q+r$."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 33550,
"solution": "To see that 9 can be achieved, take the set $\\{6,12,40,56,105\\}$, which gives $$\\{p, q, r\\}=\\{2,3,4\\}$$ Now we show it's impossible to get lower. Notice that if $m$ of the five numbers are even, then exactly $\\binom{m}{2}$ of the gcd's will be even. Since we're shown four even ... |
DeepScaleR | [
{
"role": "user",
"content": "The pairwise products $a b, b c, c d$, and $d a$ of positive integers $a, b, c$, and $d$ are $64,88,120$, and 165 in some order. Find $a+b+c+d$."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 33551,
"solution": "The sum $a b+b c+c d+d a=(a+c)(b+d)=437=19 \\cdot 23$, so $\\{a+c, b+d\\}=\\{19,23\\}$ as having either pair sum to 1 is impossible. Then the sum of all 4 is $19+23=42$. (In fact, it is not difficult to see that the only possible solutions are $(a, b, c, d)=(8,8,11,15)$ or its cyclic ... |
DeepScaleR | [
{
"role": "user",
"content": "The pan containing 24-inch by 15-inch brownies is cut into pieces that measure 3 inches by 2 inches. Calculate the total number of pieces of brownies the pan contains."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 33552,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\\in P$ with integer coordinates is it true that $|4x+3y| \\leq 1000$?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 33553,
"solution": "1. **Identify the axis of symmetry**: Given the focus of the parabola $P$ at $(0,0)$ and points $(4,3)$ and $(-4,-3)$ on $P$, we observe that the line connecting these points has a slope of $\\frac{3 - (-3)}{4 - (-4)} = \\frac{6}{8} = \\frac{3}{4}$. This suggests that the axis of symm... |
DeepScaleR | [
{
"role": "user",
"content": "The parabola $y = x^2+2$ and the hyperbola $y^2 - mx^2 = 1$ are tangent. Find $m.$"
}
] | math | {
"style": "rule",
"ground_truth": "4+2\\sqrt3"
} | {
"index": 33554,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The parabola $y=ax^2+bx+c$ has vertex $(p,p)$ and $y$-intercept $(0,-p)$, where $p\\ne 0$. What is $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 33555,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The parabola $y^2 = 12x$ and the circle $x^2 + y^2 - 4x - 6y = 0$ intersect at two points $C$ and $D$. Find the distance $CD$."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{5}"
} | {
"index": 33556,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The parabola $y^2 = 8x$ and the circle $x^2 + y^2 - 2x - 4y = 0$ intersect at two points $A$ and $B.$ Find the distance $AB.$"
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{5}"
} | {
"index": 33557,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The parabola \\(C_{1}: x^{2}=2 p y\\) has a focus at \\(F\\). The hyperbola \\(C_{2}: \\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1\\) has foci \\(F_{1}\\) and \\(F_{2}\\). Point \\(P\\) is a common point of the two curves in the first quadrant. If the points \\(P\\), \\(F\\), an... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 33558,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The parabola with equation $y=ax^2+bx+c$ is graphed below:\n\n[asy]\nxaxis(-3,7);\n\nyaxis(-5,32);\n\nreal g(real x)\n\n{\n\nreturn 4(x-2)^2-4;\n}\n\ndraw(graph(g,-1,5));\ndot((2,-4));\nlabel(\"Vertex: $(2,-4)$\", (2,-4), SE);\ndot((4,12));\nlabel(\"$(4,12)$\", (4,12), E);\n[/a... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 33559,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The parabola with the equation \\( y = 25 - x^2 \\) intersects the \\( x \\)-axis at points \\( A \\) and \\( B \\).\n\n(a) Determine the length of \\( AB \\).\n\n(b) Rectangle \\( ABCD \\) is formed such that \\( C \\) and \\( D \\) are below the \\( x \\)-axis and \\( BD = 26... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 33560,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The parabolas $y = (x + 1)^2$ and $x + 4 = (y - 3)^2$ intersect at four points $(x_1,y_1),$ $(x_2,y_2),$ $(x_3,y_3),$ and $(x_4,y_4).$ Find\n\\[x_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 33561,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The parabolas $y = (x + 1)^2$ and $x + 4 = (y - 3)^2$ intersect at four points. All four points lie on a circle of radius $r.$ Find $r^2.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{2}"
} | {
"index": 33562,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The parabolas $y = (x - 2)^2$ and $x + 6 = (y - 2)^2$ intersect at four points $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)$. Find\n\\[\nx_1 + x_2 + x_3 + x_4 + y_1 + y_2 + y_3 + y_4.\n\\]"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 33563,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The parabolas $y=ax^2 - 2$ and $y=4 - bx^2$ intersect the coordinate axes in exactly four points, and these four points are the vertices of a kite of area $12$. What is $a+b$?"
}
] | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"index": 33564,
"solution": "\n#### Step 1: Understand the problem\nWe are given two parabolas, \\(y = ax^2 - 2\\) and \\(y = 4 - bx^2\\), and we know they intersect the coordinate axes in exactly four points. These points form the vertices of a kite with an area of 12. We need to find the sum \\(a + b\\).\n\n###... |
DeepScaleR | [
{
"role": "user",
"content": "The parabolas defined by the equations $y=2x^2-4x+4$ and $y=-x^2-2x+4$ intersect at points $(a,b)$ and $(c,d)$, where $c\\ge a$. What is $c-a$? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 33565,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The parabolas with equations \\( y = x^2 - 2x - 3 \\) and \\( y = -x^2 + 4x + c \\) intersect at points \\( A \\) and \\( B \\). Determine the value of \\( c \\) so that the sum of the \\( x \\)-coordinate and \\( y \\)-coordinate of the midpoint of \\( AB \\) is 2017."
}
] | math | {
"style": "rule",
"ground_truth": "4031"
} | {
"index": 33566,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The parametric equation of curve $C_{1}$ is $\\begin{cases} x=2+2\\cos \\alpha \\\\ y=2\\sin \\alpha \\end{cases}$ ($\\alpha$ is the parameter), with the origin $O$ as the pole and the positive $x$-axis as the polar axis, a polar coordinate system is established. The curve $C_{... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 33567,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The parametric equation of line $l$ is \n$$\n\\begin{cases} \nx= \\frac { \\sqrt {2}}{2}t \\\\ \ny= \\frac { \\sqrt {2}}{2}t+4 \\sqrt {2}\n\\end{cases}\n$$\n(where $t$ is the parameter), and the polar equation of circle $c$ is $\\rho=2\\cos(\\theta+ \\frac{\\pi}{4})$. Tangent l... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{6}"
} | {
"index": 33568,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The partial fraction decomposition of\n\\[\\frac{x^2 - 19}{x^3 - 2x^2 - 5x + 6}\\]is\n\\[\\frac{A}{x - 1} + \\frac{B}{x + 2} + \\frac{C}{x - 3}.\\]Find the product $ABC.$"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 33569,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The password consists of four different digits whose sum equals 27. How many possible passwords are there?"
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 33570,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The pattern of Pascal's triangle is illustrated in the diagram shown. What is the fourth element in Row 15 of Pascal's triangle? $$\n\\begin{array}{ccccccccccccc}\\vspace{0.1in}\n\\textrm{Row 0}: & \\qquad & & & & & 1 & & & & & & \\\\ \\vspace{0.1in}\n\\textrm{Row 1}: & \\qqua... | math | {
"style": "rule",
"ground_truth": "455"
} | {
"index": 33571,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The pensioners on one of the planets of Alpha Centauri enjoy spending their free time solving numeric puzzles: they choose natural numbers from a given range $[A, B]$ such that the sum of any two chosen numbers is not divisible by a certain number $N$. Last week, the newspaper ... | math | {
"style": "rule",
"ground_truth": "356"
} | {
"index": 33572,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The pentagon \\( A B C D E \\) is inscribed around a circle.\n\nThe angles \\( \\angle A B C \\), \\( \\angle B A E \\), and \\( \\angle C D E \\) each measure \\( 104^\\circ \\). Find \\( \\angle A D B \\). Provide the answer in degrees (only the number, without units)."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 33573,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The pentagon \\(PQRST\\) is divided into four triangles with equal perimeters. The triangle \\(PQR\\) is equilateral. \\(PTU\\), \\(SUT\\), and \\(RSU\\) are congruent isosceles triangles. What is the ratio of the perimeter of the pentagon \\(PQRST\\) to the perimeter of the tr... | math | {
"style": "rule",
"ground_truth": "5:3"
} | {
"index": 33574,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The percent that $M$ is greater than $N$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{100(M-N)}{N}"
} | {
"index": 33575,
"solution": "1. **Identify the Increase**: The difference $M - N$ represents the amount by which $M$ is greater than $N$.\n\n2. **Convert to a Fraction**: To find out by what percent $M$ is greater than $N$, we need to express the increase as a fraction of $N$. This fraction is given by:\n \\[\n... |
DeepScaleR | [
{
"role": "user",
"content": "The percentage of seventh-grade students participating in the gymnastics section is between 2.9% and 3.1%. Determine the smallest possible number of students in this class."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 33576,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of $\\triangle ABC$ is equal to the perimeter of rectangle $DEFG$. What is the area of $\\triangle ABC$?"
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"index": 33577,
"solution": "The perimeter of $\\triangle ABC$ is equal to $(3x+4)+(3x+4)+2x=8x+8$. The perimeter of rectangle $DEFG$ is equal to $2 \\times (2x-2)+2 \\times (3x-1)=4x-4+6x-2=10x-6$. Since these perimeters are equal, we have $10x-6=8x+8$ which gives $2x=14$ and so $x=7$. Thus, $\\triangle ABC$ has... |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of a particular square and the circumference of a particular circle are equal. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 33578,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of a rectangle exceeds the perimeter of an equilateral triangle by 1950 cm. The length of each side of the rectangle exceeds the length of each side of the triangle by \\( d \\ \\text{cm} \\). All sides of the triangle are equal, and the rectangle is such that its... | math | {
"style": "rule",
"ground_truth": "650"
} | {
"index": 33579,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of a rectangle is 24 inches. What is the number of square inches in the maximum possible area for this rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 33580,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of a rectangle is 30 inches. One side of the rectangle is fixed at 7 inches. What is the number of square inches in the maximum possible area for this rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 33581,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of a rectangle is 48. What is the largest possible area of the rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 33582,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of a rectangle is 56 meters. The ratio of its length to its width is 4:3. What is the length in meters of a diagonal of the rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 33583,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of a rectangular garden is 60 feet. If the length of the field is twice the width, what is the area of the field, in square feet?"
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 33584,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of a semicircle with an area of ______ square meters is 15.42 meters."
}
] | math | {
"style": "rule",
"ground_truth": "14.13"
} | {
"index": 33585,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of a square with side length $x$ units is equal to the circumference of a circle with radius 2 units. What is the value of $x$? Express your answer as a decimal to the nearest hundredth."
}
] | math | {
"style": "rule",
"ground_truth": "3.14"
} | {
"index": 33586,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of a square with side length $y$ inches is equal to the circumference of a circle with radius 5 centimeters. If 1 inch equals 2.54 centimeters, what is the value of $y$ in inches? Express your answer as a decimal to the nearest hundredth."
}
] | math | {
"style": "rule",
"ground_truth": "3.09"
} | {
"index": 33587,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of a triangle is 30, and all sides are different integers. There are a total of triangles."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33588,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of an equilateral triangle exceeds the perimeter of a square by $1989 \\text{ cm}$. The length of each side of the triangle exceeds the length of each side of the square by $d \\text{ cm}$. The square has perimeter greater than 0. How many positive integers are NO... | math | {
"style": "rule",
"ground_truth": "663"
} | {
"index": 33589,
"solution": "1. **Set up the equations based on the problem statement:**\n Let $t$ be the length of each side of the equilateral triangle, and $s$ be the length of each side of the square. The perimeter of the triangle is $3t$ and the perimeter of the square is $4s$. According to the problem, th... |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of an isosceles right triangle is $2p$. Its area is:"
}
] | math | {
"style": "rule",
"ground_truth": "$(3-2\\sqrt{2})p^2$"
} | {
"index": 33590,
"solution": "1. **Identify the properties of the triangle**: We are given an isosceles right triangle. In such a triangle, the two legs are equal, and the hypotenuse is $\\sqrt{2}$ times the length of a leg. Let the length of each leg be $x$.\n\n2. **Write the equation for the perimeter**: The per... |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of one square is $3$ times the perimeter of another square. The area of the larger square is how many times the area of the smaller square?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 33591,
"solution": "1. **Define Variables:**\n Let $a$ be the side length of the larger square and $b$ be the side length of the smaller square.\n\n2. **Relate Perimeters:**\n The perimeter of a square is given by $4$ times the side length. If the perimeter of the larger square is $3$ times the perim... |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of quadrilateral PQRS, made from two similar right-angled triangles PQR and PRS, is given that the length of PQ is 3, the length of QR is 4, and ∠PRQ = ∠PSR. Find the perimeter of PQRS."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 33592,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of the polygon shown is"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 33593,
"solution": "To solve this problem, we need to determine the perimeter of the polygon. The solution suggests visualizing the polygon as part of a rectangle and then using the properties of the rectangle to find the perimeter of the polygon.\n\n1. **Visualize the Polygon as Part of a Rectangle**:\n... |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of the quadrilateral formed by the four vertices of the ellipse $C: \\frac {x^{2}}{4}+ \\frac {y^{2}}{16}=1$ is equal to _____."
}
] | math | {
"style": "rule",
"ground_truth": "8 \\sqrt {5}"
} | {
"index": 33594,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of the triangle formed by the line $\\frac{x}{3} + \\frac{y}{4} = 1$ and the two coordinate axes is $14$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 33595,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of the triangles that make up rectangle \\(ABCD\\) is 180 cm. \\(BK = KC = AE = ED\\), \\(AK = KD = 17 \\) cm. Find the perimeter of a rectangle, one of whose sides is twice as long as \\(AB\\), and the other side is equal to \\(BC\\)."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 33596,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of triangle $APM$ is $152$, and the angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\\overline{AP}$ is drawn so that it is tangent to $\\overline{AM}$ and $\\overline{PM}$. Given that $OP=m/n$ where $m$ and $n$ are relatively prime positi... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 33597,
"solution": "Let the circle intersect $\\overline{PM}$ at $B$. Then note $\\triangle OPB$ and $\\triangle MPA$ are similar. Also note that $AM = BM$ by power of a point. Using the fact that the ratio of corresponding sides in similar triangles is equal to the ratio of their perimeters, we have \\[... |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of triangle $APM$ is $180$, and angle $PAM$ is a right angle. A circle of radius $20$ with center $O$ on line $\\overline{AP}$ is drawn such that it is tangent to $\\overline{AM}$ and $\\overline{PM}$. Let $\\overline{AM} = 2\\overline{PM}$. Find the length of $OP... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 33598,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The perimeter of triangle $BQN$ is $180$, and the angle $QBN$ is a right angle. A circle of radius $15$ with center $O$ on $\\overline{BQ}$ is drawn such that it is tangent to $\\overline{BN}$ and $\\overline{QN}$. Given that $OQ=p/q$ where $p$ and $q$ are relatively prime posi... | math | {
"style": "rule",
"ground_truth": "79"
} | {
"index": 33599,
"solution": ""
} |
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