data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "$A$ and $B$ are two distinct points on the parabola $y=3-x^{2}$ that are symmetric with respect to the line $x+y=0$. Find $|AB|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$3 \\sqrt{2}$",
"style": "rule"
} | {
"index": "100",
"split": "train"
} |
math | [
{
"content": "Consider a three-person game involving the following three types of fair six-sided dice. - Dice of type $A$ have faces labelled $2,2,4,4,9,9$. - Dice of type $B$ have faces labelled $1,1,6,6,8,8$. - Dice of type $C$ have faces labelled $3,3,5,5,7,7$. All three players simultaneously choose a die (... | {
"ground_truth": "\\frac{8}{9}",
"style": "rule"
} | {
"index": "101",
"split": "train"
} |
math | [
{
"content": "How many natural five-digit numbers have the product of their digits equal to 2000? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "102",
"split": "train"
} |
math | [
{
"content": "What is the smallest whole number larger than the perimeter of any triangle with a side of length $7$ and a side of length $24$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "62",
"style": "rule"
} | {
"index": "103",
"split": "train"
} |
math | [
{
"content": "A standard deck of 54 playing cards (with four cards of each of thirteen ranks, as well as two Jokers) is shuffled randomly. Cards are drawn one at a time until the first queen is reached. What is the probability that the next card is also a queen? Please output the final answer within \\boxed{}."... | {
"ground_truth": "\\frac{2}{27}",
"style": "rule"
} | {
"index": "104",
"split": "train"
} |
math | [
{
"content": "Suppose that $a_1 = 1$ , and that for all $n \\ge 2$ , $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \\ldots + (n-1)a_1.$ Suppose furthermore that $b_n = a_1 + a_2 + \\ldots + a_n$ for all $n$ . If $b_1 + b_2 + b_3 + \\ldots + b_{2021} = a_k$ for some $k$ , find $k$ .\n\n*Proposed by Andrew Wu... | {
"ground_truth": "2022",
"style": "rule"
} | {
"index": "105",
"split": "train"
} |
math | [
{
"content": "Xiao Ming observed a faucet continuously dripping water due to damage. To investigate the waste caused by the water leakage, Xiao Ming placed a graduated cylinder under the faucet to collect water. He recorded the total amount of water in the cylinder every minute, but due to a delay in starting t... | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "106",
"split": "train"
} |
math | [
{
"content": "Find $\\left(\\sqrt{(\\sqrt3)^3}\\right)^4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "107",
"split": "train"
} |
math | [
{
"content": "$ABCD$ is a trapezoid with $AB \\parallel CD$, $AB=6$, and $CD=15$. If the area of $\\triangle AED=30$, what is the area of $\\triangle AEB?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "108",
"split": "train"
} |
math | [
{
"content": "Let $[x]$ be the greatest integer less than or equal to the real number $x$. Given the sequence $\\left\\{a_{n}\\right\\}$ which satisfies $a_{1}=\\frac{1}{2}, a_{n+1}=a_{n}^{2}+3 a_{n}+1$ for $n \\in N^{*}$, find the value of $\\left[\\sum_{k=1}^{2017} \\frac{a_{k}}{a_{k}+2}\\right]$. Please outp... | {
"ground_truth": "2015",
"style": "rule"
} | {
"index": "109",
"split": "train"
} |
math | [
{
"content": "Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.298",
"style": "rule"
} | {
"index": "110",
"split": "train"
} |
math | [
{
"content": "Given an isosceles triangle with side lengths of $4x-2$, $x+1$, and $15-6x$, its perimeter is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12.3",
"style": "rule"
} | {
"index": "111",
"split": "train"
} |
math | [
{
"content": "Using the digits 1, 2, 3, 4, 5, how many even three-digit numbers less than 500 can be formed if each digit can be used more than once? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "112",
"split": "train"
} |
math | [
{
"content": "How many ways, without taking order into consideration, can 2002 be expressed as the sum of 3 positive integers (for instance, $1000+1000+2$ and $1000+2+1000$ are considered to be the same way)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "334000",
"style": "rule"
} | {
"index": "113",
"split": "train"
} |
math | [
{
"content": "Given a skewed six-sided die is structured so that rolling an odd number is twice as likely as rolling an even number, calculate the probability that, after rolling the die twice, the sum of the numbers rolled is odd. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{9}",
"style": "rule"
} | {
"index": "114",
"split": "train"
} |
math | [
{
"content": "Consider the quadratic equation $2x^2 - 5x + m = 0$. Find the value of $m$ such that the sum of the roots of the equation is maximized while ensuring that the roots are real and rational. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{25}{8}",
"style": "rule"
} | {
"index": "115",
"split": "train"
} |
math | [
{
"content": "A triangle has area $30$, one side of length $10$, and the median to that side of length $9$. Let $\\theta$ be the acute angle formed by that side and the median. What is $\\sin{\\theta}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "116",
"split": "train"
} |
math | [
{
"content": "What is the largest $2$-digit prime factor of the integer $n = {300 \\choose 150}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "89",
"style": "rule"
} | {
"index": "117",
"split": "train"
} |
math | [
{
"content": "For how many values of $a$ is it true that the line $y=x+a$ passes through the vertex of parabola $y=x^2+a^2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "118",
"split": "train"
} |
math | [
{
"content": "Determine the distance in feet between the 5th red light and the 23rd red light, where the lights are hung on a string 8 inches apart in the pattern of 3 red lights followed by 4 green lights. Recall that 1 foot is equal to 12 inches. Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "119",
"split": "train"
} |
math | [
{
"content": "Given the parabola $y^{2}=4x$, let $AB$ and $CD$ be two chords perpendicular to each other and passing through its focus. Find the value of $\\frac{1}{|AB|}+\\frac{1}{|CD|}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "120",
"split": "train"
} |
math | [
{
"content": "Solve the equation \\[-x^2 = \\frac{3x+1}{x+3}.\\]Enter all solutions, separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "121",
"split": "train"
} |
math | [
{
"content": "In a math lesson, each gnome needs to find a three-digit number without zero digits, divisible by 3, such that when 297 is added to it, the resulting number consists of the same digits but in reverse order. What is the minimum number of gnomes that must be in the lesson so that among the numbers t... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "122",
"split": "train"
} |
math | [
{
"content": "When a bucket is two-thirds full of water, the bucket and water weigh $a$ kilograms. When the bucket is one-half full of water the total weight is $b$ kilograms. In terms of $a$ and $b$, what is the total weight in kilograms when the bucket is full of water? Please output the final answer within \... | {
"ground_truth": "3a - 2b",
"style": "rule"
} | {
"index": "123",
"split": "train"
} |
math | [
{
"content": "Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2028",
"style": "rule"
} | {
"index": "124",
"split": "train"
} |
math | [
{
"content": "Rectangle $EFGH$ has sides $\\overline {EF}$ of length 5 and $\\overline {FG}$ of length 4. Divide $\\overline {EF}$ into 196 congruent segments with points $E=R_0, R_1, \\ldots, R_{196}=F$, and divide $\\overline {FG}$ into 196 congruent segments with points $F=S_0, S_1, \\ldots, S_{196}=G$. For ... | {
"ground_truth": "195 \\sqrt{41}",
"style": "rule"
} | {
"index": "125",
"split": "train"
} |
math | [
{
"content": "Given that $\\triangle ABC$ is an acute triangle, vector $\\overrightarrow{m} = (\\cos(A + \\frac{\\pi}{3}), \\sin(A + \\frac{\\pi}{3}))$, $\\overrightarrow{n} = (\\cos B, \\sin B)$, and $\\overrightarrow{m} \\perp \\overrightarrow{n}$.\n\n(Ⅰ) Find the value of $A-B$; \n(Ⅱ) If $\\cos B = \\frac{3... | {
"ground_truth": "4\\sqrt{3} + 3",
"style": "rule"
} | {
"index": "126",
"split": "train"
} |
math | [
{
"content": "Given that $a > 0$, if $f(g(a)) = 18$, where $f(x) = x^2 + 10$ and $g(x) = x^2 - 6$, what is the value of $a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2\\sqrt{2} + 6}",
"style": "rule"
} | {
"index": "127",
"split": "train"
} |
math | [
{
"content": "Solve the following equations: \n2x + 62 = 248; x - 12.7 = 2.7; x ÷ 5 = 0.16; 7x + 2x = 6.3. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.7",
"style": "rule"
} | {
"index": "128",
"split": "train"
} |
math | [
{
"content": "There are 3 math teams in the area, with 5, 7, and 8 students respectively. Each team has two co-captains. If I randomly select a team, and then randomly select two members of that team to give a copy of $\\emph{Introduction to Geometry}$, what is the probability that both of the people who receiv... | {
"ground_truth": "\\dfrac{11}{180}",
"style": "rule"
} | {
"index": "129",
"split": "train"
} |
math | [
{
"content": "Compute the number of ordered pairs of integers $(x,y)$ with $1\\le x<y\\le 200$ such that $i^x+i^y$ is a real number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4950",
"style": "rule"
} | {
"index": "130",
"split": "train"
} |
math | [
{
"content": "James is standing at the point $(0,1)$ on the coordinate plane and wants to eat a hamburger. For each integer $n \\geq 0$, the point $(n, 0)$ has a hamburger with $n$ patties. There is also a wall at $y=2.1$ which James cannot cross. In each move, James can go either up, right, or down 1 unit as l... | {
"ground_truth": "\\frac{7}{3}",
"style": "rule"
} | {
"index": "131",
"split": "train"
} |
math | [
{
"content": "What is the number of units in the distance between $(2,5)$ and $(-6,-1)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "132",
"split": "train"
} |
math | [
{
"content": "While one lion cub, who is 6 minutes away from the water hole, heads there, another, having already quenched its thirst, heads back along the same road 1.5 times faster than the first. At the same time, a turtle starts towards the water hole along the same road, being 32 minutes away from it. At s... | {
"ground_truth": "2.4",
"style": "rule"
} | {
"index": "133",
"split": "train"
} |
math | [
{
"content": "A circle has its center at $(2,0)$ with a radius of 2, and another circle has its center at $(5,0)$ with a radius of 1. A line is tangent to both circles in the first quadrant. The $y$-intercept of this line is closest to: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$2 \\sqrt{2}$",
"style": "rule"
} | {
"index": "134",
"split": "train"
} |
math | [
{
"content": "In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?\n[... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "135",
"split": "train"
} |
math | [
{
"content": "What is the sum of all the odd divisors of $360$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "78",
"style": "rule"
} | {
"index": "136",
"split": "train"
} |
math | [
{
"content": "There exist $s$ unique nonnegative integers $m_1 > m_2 > \\cdots > m_s$ and $s$ unique integers $b_k$ ($1\\le k\\le s$) with each $b_k$ either $1$ or $- 1$ such that\\[b_13^{m_1} + b_23^{m_2} + \\cdots + b_s3^{m_s} = 2012.\\]Find $m_1 + m_2 + \\cdots + m_s$. Please output the final answer within \... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "137",
"split": "train"
} |
math | [
{
"content": "Find the equation of the directrix of the parabola $y = 8x^2 + 2.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "y = \\frac{63}{32}",
"style": "rule"
} | {
"index": "138",
"split": "train"
} |
math | [
{
"content": "Multiply the sum of $158.23$ and $47.869$ by $2$, then round your answer to the nearest tenth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "412.2",
"style": "rule"
} | {
"index": "139",
"split": "train"
} |
math | [
{
"content": "Shauna takes five tests, each worth a maximum of $100$ points. Her scores on the first three tests are $76$, $94$, and $87$. In order to average $81$ for all five tests, what is the lowest score she could earn on one of the other two tests? Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "140",
"split": "train"
} |
math | [
{
"content": "Find all odd natural numbers greater than 500 but less than 1000, each of which has the property that the sum of the last digits of all its divisors (including 1 and the number itself) is equal to 33. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "729",
"style": "rule"
} | {
"index": "141",
"split": "train"
} |
math | [
{
"content": "Calculate the total surface area of two hemispheres of radius 8 cm each, joined at their bases to form a complete sphere. Assume that one hemisphere is made of a reflective material that doubles the effective surface area for purposes of calculation. Express your answer in terms of $\\pi$. Please ... | {
"ground_truth": "384\\pi",
"style": "rule"
} | {
"index": "142",
"split": "train"
} |
math | [
{
"content": "What is the number of positive integers $p$ for which $-1<\\sqrt{p}-\\sqrt{100}<1$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "143",
"split": "train"
} |
math | [
{
"content": "A positive integer with 3 digits $\\overline{ABC}$ is $Lusophon$ if $\\overline{ABC}+\\overline{CBA}$ is a perfect square. Find all $Lusophon$ numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "110,143,242,341,440,164,263,362,461,560,198,297,396,495,594,693,792,891,990",
"style": "rule"
} | {
"index": "144",
"split": "train"
} |
math | [
{
"content": "If the arithmetic mean of $a$ and $b$ is double their geometric mean, with $a>b>0$, then a possible value for the ratio $a/b$, to the nearest integer, is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "145",
"split": "train"
} |
math | [
{
"content": "Given the ellipse $\\frac{{x}^{2}}{3{{m}^{2}}}+\\frac{{{y}^{2}}}{5{{n}^{2}}}=1$ and the hyperbola $\\frac{{{x}^{2}}}{2{{m}^{2}}}-\\frac{{{y}^{2}}}{3{{n}^{2}}}=1$ share a common focus, find the eccentricity of the hyperbola ( ). Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "\\frac{\\sqrt{19}}{4}",
"style": "rule"
} | {
"index": "146",
"split": "train"
} |
math | [
{
"content": "The equation\n\n$$\n(x-1) \\times \\ldots \\times(x-2016) = (x-1) \\times \\ldots \\times(x-2016)\n$$\n\nis written on the board. We want to erase certain linear factors so that the remaining equation has no real solutions. Determine the smallest number of linear factors that need to be erased to ... | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": "147",
"split": "train"
} |
math | [
{
"content": "Given $f(x) = x^2$ and $g(x) = |x - 1|$, let $f_1(x) = g(f(x))$, $f_{n+1}(x) = g(f_n(x))$, calculate the number of solutions to the equation $f_{2015}(x) = 1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2017",
"style": "rule"
} | {
"index": "148",
"split": "train"
} |
math | [
{
"content": "Given that $θ$ is an angle in the second quadrant and $\\tan(\\begin{matrix}θ+ \\frac{π}{4}\\end{matrix}) = \\frac{1}{2}$, find the value of $\\sin(θ) + \\cos(θ)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{\\sqrt{10}}{5}",
"style": "rule"
} | {
"index": "149",
"split": "train"
} |
math | [
{
"content": "The time it takes for person A to make 90 parts is the same as the time it takes for person B to make 120 parts. It is also known that A and B together make 35 parts per hour. Determine how many parts per hour A and B each make. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "150",
"split": "train"
} |
math | [
{
"content": "Increase Grisha's yield by 40% and Vasya's yield by 20%.\n\nGrisha, the most astute among them, calculated that in the first case their total yield would increase by 1 kg; in the second case, it would decrease by 0.5 kg; in the third case, it would increase by 4 kg. What was the total yield of the... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "151",
"split": "train"
} |
math | [
{
"content": "Person A and Person B start from points $A$ and $B$ simultaneously and move towards each other. It is known that the speed ratio of Person A to Person B is 6:5. When they meet, they are 5 kilometers from the midpoint between $A$ and $B$. How many kilometers away is Person B from point $A$ when Per... | {
"ground_truth": "5/3",
"style": "rule"
} | {
"index": "152",
"split": "train"
} |
math | [
{
"content": "On a balance scale, $3$ green balls balance $6$ blue balls, $2$ yellow balls balance $5$ blue balls, and $6$ blue balls balance $4$ white balls. How many blue balls are needed to balance $4$ green, $2$ yellow and $2$ white balls? Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "153",
"split": "train"
} |
math | [
{
"content": "Determine how many \"super prime dates\" occurred in 2007, where a \"super prime date\" is defined as a date where both the month and day are prime numbers, and additionally, the day is less than or equal to the typical maximum number of days in the respective prime month. Please output the final ... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "154",
"split": "train"
} |
math | [
{
"content": "Given triangle \\( \\triangle ABC \\) with \\( Q \\) as the midpoint of \\( BC \\), \\( P \\) on \\( AC \\) such that \\( CP = 3PA \\), and \\( R \\) on \\( AB \\) such that \\( S_{\\triangle PQR} = 2 S_{\\triangle RBQ} \\). If \\( S_{\\triangle ABC} = 300 \\), find \\( S_{\\triangle PQR} \\). Ple... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "155",
"split": "train"
} |
math | [
{
"content": "Dr. Math's four-digit house number $WXYZ$ contains no zeroes and can be split into two different two-digit primes ``$WX$'' and ``$YZ$'' where the digits $W$, $X$, $Y$, and $Z$ are not necessarily distinct. If each of the two-digit primes is less than 50, how many such house numbers are possible? P... | {
"ground_truth": "110",
"style": "rule"
} | {
"index": "156",
"split": "train"
} |
math | [
{
"content": "A nine-digit integer is formed by repeating a positive three-digit integer three times. For example, 123,123,123 or 456,456,456 are integers of this form. What is the greatest common divisor of all nine-digit integers of this form? Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "1001001",
"style": "rule"
} | {
"index": "157",
"split": "train"
} |
math | [
{
"content": "What is the smallest integer greater than 10 such that the sum of the digits in its base 17 representation is equal to the sum of the digits in its base 10 representation? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "153",
"style": "rule"
} | {
"index": "158",
"split": "train"
} |
math | [
{
"content": "For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?\n$\\textbf{(A) }110\\qquad\\textbf{(B) }191\\qquad\\textbf{(C) }261\\qquad\\textbf{(D) }325\\qquad\\textbf... | {
"ground_truth": "325",
"style": "rule"
} | {
"index": "159",
"split": "train"
} |
math | [
{
"content": "Given the digits 1, 2, 3, 4, 5, 6 to form a six-digit number (without repeating any digit), requiring that any two adjacent digits have different parity, and 1 and 2 are adjacent, determine the number of such six-digit numbers. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "160",
"split": "train"
} |
math | [
{
"content": "A starship enters an extraordinary meteor shower. Some of the meteors travel along a straight line at the same speed, equally spaced. Another group of meteors travels similarly along another straight line, parallel to the first, with the same speed but in the opposite direction, also equally space... | {
"ground_truth": "9.1",
"style": "rule"
} | {
"index": "161",
"split": "train"
} |
math | [
{
"content": "In right triangle $DEF$, $DE=15$, $DF=9$ and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7.5",
"style": "rule"
} | {
"index": "162",
"split": "train"
} |
math | [
{
"content": "What two digits need to be added to the right of the number 2013 to make the resulting six-digit number divisible by 101? Find all possible answers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "94",
"style": "rule"
} | {
"index": "163",
"split": "train"
} |
math | [
{
"content": "Given \\(a, b, c \\in (0, 1]\\) and \\(\\lambda \\) is a real number such that\n\\[ \\frac{\\sqrt{3}}{\\sqrt{a+b+c}} \\geqslant 1+\\lambda(1-a)(1-b)(1-c), \\]\nfind the maximum value of \\(\\lambda\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{64}{27}",
"style": "rule"
} | {
"index": "164",
"split": "train"
} |
math | [
{
"content": "The digits 1, 2, 3, 4, and 5 were used, each one only once, to write a certain five-digit number \\(abcde\\) such that \\(abc\\) is divisible by 4, \\(bcd\\) is divisible by 5, and \\(cde\\) is divisible by 3. Find this number. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "12453",
"style": "rule"
} | {
"index": "165",
"split": "train"
} |
math | [
{
"content": "On a particular street in Waterloo, there are exactly 14 houses, each numbered with an integer between 500 and 599, inclusive. The 14 house numbers form an arithmetic sequence in which 7 terms are even and 7 terms are odd. One of the houses is numbered 555 and none of the remaining 13 numbers has ... | {
"ground_truth": "506",
"style": "rule"
} | {
"index": "166",
"split": "train"
} |
math | [
{
"content": "Rectangle \\(ABCD\\) is divided into four parts by \\(CE\\) and \\(DF\\). It is known that the areas of three of these parts are \\(5\\), \\(16\\), and \\(20\\) square centimeters, respectively. What is the area of quadrilateral \\(ADOE\\) in square centimeters? Please output the final answer with... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "167",
"split": "train"
} |
math | [
{
"content": "A regular hexagon has one side along the diameter of a semicircle, and the two opposite vertices on the semicircle. Find the area of the hexagon if the diameter of the semicircle is 1. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 \\sqrt{3} / 26",
"style": "rule"
} | {
"index": "168",
"split": "train"
} |
math | [
{
"content": "What was Tony's average speed, in miles per hour, during the 3-hour period when his odometer increased from 12321 to the next higher palindrome? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "33.33",
"style": "rule"
} | {
"index": "169",
"split": "train"
} |
math | [
{
"content": "Given positive numbers \\(a,b\\) satisfying \\(2ab=\\dfrac{2a-b}{2a+3b},\\) then the maximum value of \\(b\\) is \\_\\_\\_\\_\\_ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{3}",
"style": "rule"
} | {
"index": "170",
"split": "train"
} |
math | [
{
"content": "Find the maximum value of the expression \\( x^{2} + y^{2} \\) if \\( |x-y| \\leq 2 \\) and \\( |3x + y| \\leq 6 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "171",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), \\(BK\\) is the median, \\(BE\\) is the angle bisector, and \\(AD\\) is the altitude. Find the length of side \\(AC\\) if it is known that lines \\(BK\\) and \\(BE\\) divide segment \\(AD\\) into three equal parts and the length of \\(AB\\) is 4. Please output the final answ... | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": "172",
"split": "train"
} |
math | [
{
"content": "A rectangular garden 60 feet long and 15 feet wide is enclosed by a fence. To utilize the same fence but change the shape, the garden is altered to an equilateral triangle. By how many square feet does this change the area of the garden? Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "182.53",
"style": "rule"
} | {
"index": "173",
"split": "train"
} |
math | [
{
"content": "Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of... | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "174",
"split": "train"
} |
math | [
{
"content": "Find the integer $n$, $-180 < n < 180$, such that $\\tan n^\\circ = \\tan 1500^\\circ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "175",
"split": "train"
} |
math | [
{
"content": "Brand Z juice claims, \"We offer 30% more juice than Brand W at a price that is 15% less.\" What is the ratio of the unit price of Brand Z juice to the unit price of Brand W juice? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{17}{26}",
"style": "rule"
} | {
"index": "176",
"split": "train"
} |
math | [
{
"content": "Let $S_n$ denote the sum of the first $n$ terms of an arithmetic sequence with common difference 3. If $\\frac{S_{3n}}{S_n}$ is a constant that does not depend on $n,$ for all positive integers $n,$ then find the first term. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "177",
"split": "train"
} |
math | [
{
"content": "The area of a triangle \\(ABC\\) is \\(\\displaystyle 40 \\text{ cm}^2\\). Points \\(D, E\\) and \\(F\\) are on sides \\(AB, BC\\) and \\(CA\\) respectively. If \\(AD = 3 \\text{ cm}, DB = 5 \\text{ cm}\\), and the area of triangle \\(ABE\\) is equal to the area of quadrilateral \\(DBEF\\), find t... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "178",
"split": "train"
} |
math | [
{
"content": "Among the numbers from 1 to 1000, how many are divisible by 4 and do not contain the digit 4 in their representation? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "162",
"style": "rule"
} | {
"index": "179",
"split": "train"
} |
math | [
{
"content": "On Jessie's 10th birthday, in 2010, her mother said, \"My age is now five times your age.\" In what year will Jessie's mother be able to say, \"My age is now 2.5 times your age,\" on Jessie's birthday? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2027",
"style": "rule"
} | {
"index": "180",
"split": "train"
} |
math | [
{
"content": "A curve is described parametrically by\n\\[(x,y) = (2 \\cos t - \\sin t, 4 \\sin t).\\]The graph of the curve can be expressed in the form\n\\[ax^2 + bxy + cy^2 = 1.\\]Enter the ordered triple $(a,b,c).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left( \\frac{1}{4}, \\frac{1}{8}, \\frac{5}{64} \\right)",
"style": "rule"
} | {
"index": "181",
"split": "train"
} |
math | [
{
"content": "How many real solutions are there for $x$ in the following equation: $$(x - 5x + 12)^2 + 1 = -|x|$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "182",
"split": "train"
} |
math | [
{
"content": "The circumference of a particular circle is 18 cm. In square centimeters, what is the area of the circle? Express your answer as a common fraction in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{81}{\\pi}",
"style": "rule"
} | {
"index": "183",
"split": "train"
} |
math | [
{
"content": "Let $x$ and $y$ be positive real numbers. Find the minimum value of\n\\[\\frac{\\sqrt{(x^2 + y^2)(3x^2 + y^2)}}{xy}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1 + \\sqrt{3}",
"style": "rule"
} | {
"index": "184",
"split": "train"
} |
math | [
{
"content": "The numbers 2, 3, 5, 7, 11, 13, 17, 19 are arranged in a multiplication table, with four along the top and the other four down the left. The multiplication table is completed and the sum of the sixteen entries is tabulated. What is the largest possible sum of the sixteen entries?\n\n\\[\n\\begin{a... | {
"ground_truth": "1482",
"style": "rule"
} | {
"index": "185",
"split": "train"
} |
math | [
{
"content": "What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\\frac{8}{15} < \\frac{n}{n + k} < \\frac{7}{13}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "112",
"style": "rule"
} | {
"index": "186",
"split": "train"
} |
math | [
{
"content": "Simplify the following expression:\n\n$$5x + 6 - x + 12$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4x + 18",
"style": "rule"
} | {
"index": "187",
"split": "train"
} |
math | [
{
"content": "Let $\\mathcal{F}$ be the set of all functions $f : (0,\\infty)\\to (0,\\infty)$ such that $f(3x) \\geq f( f(2x) )+x$ for all $x$ . Find the largest $A$ such that $f(x) \\geq A x$ for all $f\\in\\mathcal{F}$ and all $x$ . Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": "188",
"split": "train"
} |
math | [
{
"content": "The Euler family has four girls aged $6$, $6$, $9$, and $11$, and two boys aged $13$ and $16$. What is the mean (average) age of the children? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{61}{6}",
"style": "rule"
} | {
"index": "189",
"split": "train"
} |
math | [
{
"content": "Let $\\mathcal{S}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\\overline{abc}$ where $a, b, c$ are distinct digits. Find the sum of the elements of $\\mathcal{S}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "360",
"style": "rule"
} | {
"index": "190",
"split": "train"
} |
math | [
{
"content": "If the two real roots of the equation (lgx)<sup>2</sup>\\-lgx+lg2•lg5=0 with respect to x are m and n, then 2<sup>m+n</sup>\\=\\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "128",
"style": "rule"
} | {
"index": "191",
"split": "train"
} |
math | [
{
"content": "Given the lines $l_{1}$: $x+\\left(m-3\\right)y+m=0$ and $l_{2}$: $mx-2y+4=0$. \n$(1)$ If line $l_{1}$ is perpendicular to line $l_{2}$, find the value of $m$. \n$(2)$ If line $l_{1}$ is parallel to line $l_{2}$, find the distance between $l_{1}$ and $l_{2}$. Please output the final answer withi... | {
"ground_truth": "\\frac{3\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "192",
"split": "train"
} |
math | [
{
"content": "Given an integer sequence \\(\\{a_i\\}\\) defined as follows:\n\\[ a_i = \\begin{cases} \ni, & \\text{if } 1 \\leq i \\leq 5; \\\\\na_1 a_2 \\cdots a_{i-1} - 1, & \\text{if } i > 5.\n\\end{cases} \\]\n\nFind the value of \\(\\sum_{i=1}^{2019} a_i^2 - a_1 a_2 \\cdots a_{2019}\\). Please output the ... | {
"ground_truth": "1949",
"style": "rule"
} | {
"index": "193",
"split": "train"
} |
math | [
{
"content": "Given two similar triangles $\\triangle ABC\\sim\\triangle FGH$, where $BC = 24 \\text{ cm}$ and $FG = 15 \\text{ cm}$. If the length of $AC$ is $18 \\text{ cm}$, find the length of $GH$. Express your answer as a decimal to the nearest tenth. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "11.3",
"style": "rule"
} | {
"index": "194",
"split": "train"
} |
math | [
{
"content": "Given a triangle \\(ABC\\) where \\(AB = AC\\) and \\(\\angle A = 80^\\circ\\). Inside triangle \\(ABC\\) is a point \\(M\\) such that \\(\\angle MBC = 30^\\circ\\) and \\(\\angle MCB = 10^\\circ\\). Find \\(\\angle AMC\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "195",
"split": "train"
} |
math | [
{
"content": "Find all values of \\(a\\) such that the roots \\(x_1, x_2, x_3\\) of the polynomial \\(x^3 - 6x^2 + ax + a\\) satisfy \\((x_1 - 3)^2 + (x_2 - 3)^3 + (x_3 - 3)^3 = 0\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-9",
"style": "rule"
} | {
"index": "196",
"split": "train"
} |
math | [
{
"content": "In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. What is the digit $A$? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "197",
"split": "train"
} |
math | [
{
"content": "A radio system consisting of 1000 components (each with a failure rate of $\\lambda_{i} = 10^{-6}$ failures/hour) has been tested and accepted by the customer. Determine the probability of the system operating without failure over the interval $t_{1} < (t = t_{1} + \\Delta t) < t_{2}$, where $\\De... | {
"ground_truth": "0.367879",
"style": "rule"
} | {
"index": "198",
"split": "train"
} |
math | [
{
"content": "A triangular array of numbers has a first row consisting of the odd integers $1,3,5,\\ldots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries ... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "199",
"split": "train"
} |
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