data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Find all functions $f$ from the set $\\mathbb{R}$ of real numbers into $\\mathbb{R}$ which satisfy for all $x, y, z \\in \\mathbb{R}$ the identity \\[f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "f(x) = 0 \\text{ and } f(x) = x^2",
"style": "rule"
} | {
"index": "3700",
"split": "train"
} |
math | [
{
"content": "Two people are playing \"Easter egg battle.\" In front of them is a large basket of eggs. They randomly pick one egg each and hit them against each other. One of the eggs breaks, the defeated player takes a new egg, and the winner keeps their egg for the next round (the outcome of each round depen... | {
"ground_truth": "11/12",
"style": "rule"
} | {
"index": "3701",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = \\cos x \\cdot \\sin\\left(x + \\frac{\\pi}{3}\\right) - \\sqrt{3}\\cos^2x + \\frac{\\sqrt{3}}{4}$, where $x \\in \\mathbb{R}$.\n\n(1) Find the interval of monotonic increase for $f(x)$.\n\n(2) In an acute triangle $\\triangle ABC$, where the sides opposite angles $A$, $... | {
"ground_truth": "\\frac{3\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "3702",
"split": "train"
} |
math | [
{
"content": "In isosceles $\\vartriangle ABC, AB = AC, \\angle BAC$ is obtuse, and points $E$ and $F$ lie on sides $AB$ and $AC$ , respectively, so that $AE = 10, AF = 15$ . The area of $\\vartriangle AEF$ is $60$ , and the area of quadrilateral $BEFC$ is $102$ . Find $BC$ . Please output the ... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "3703",
"split": "train"
} |
math | [
{
"content": "In a zoo, there were 200 parrots. One day, they each made a statement in turn. Starting from the second parrot, all statements were: \"Among the previous statements, more than 70% are false.\" How many false statements did the parrots make in total? Please output the final answer within \\boxed{}.... | {
"ground_truth": "140",
"style": "rule"
} | {
"index": "3704",
"split": "train"
} |
math | [
{
"content": "In a right triangle, one of the acute angles $\\beta$ satisfies \\[\\tan \\frac{\\beta}{2} = \\frac{1}{\\sqrt[3]{3}}.\\] Let $\\phi$ be the angle between the median and the angle bisector drawn from this acute angle $\\beta$. Calculate $\\tan \\phi.$ Please output the final answer within \\boxed{}... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "3705",
"split": "train"
} |
math | [
{
"content": "If \\( x \\) and \\( y \\) are positive integers with \\( x>y \\) and \\( x+x y=391 \\), what is the value of \\( x+y \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "3706",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence $\\{a\\_n\\}$ with a common difference $d > 0$, and $a\\_2$, $a\\_5-1$, $a\\_{10}$ form a geometric sequence. If $a\\_1=5$, and $S\\_n$ represents the sum of the first $n$ terms of the sequence, find the minimum value of $\\frac{2S\\_n+n+32}{a\\_n+1}$. Please output th... | {
"ground_truth": "\\frac{20}{3}",
"style": "rule"
} | {
"index": "3707",
"split": "train"
} |
math | [
{
"content": "Given that $\\cos(75^\\circ + \\alpha) = \\frac{1}{3}$, where $\\alpha$ is an angle in the third quadrant, find the value of $\\cos(105^\\circ - \\alpha) + \\sin(\\alpha - 105^\\circ)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\sqrt{2} - 1}{3}",
"style": "rule"
} | {
"index": "3708",
"split": "train"
} |
math | [
{
"content": "In the subtraction shown, $K, L, M$, and $N$ are digits. What is the value of $K+L+M+N$?\\n$$\\begin{array}{r}6 K 0 L \\\\ -\\quad M 9 N 4 \\\\ \\hline 2011\\end{array}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "3709",
"split": "train"
} |
math | [
{
"content": "A company allocates 5 employees to 3 different departments, with each department being allocated at least one employee. Among them, employees A and B must be allocated to the same department. Calculate the number of different allocation methods. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "3710",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ and point $P$ in the same plane are given. Point $P$ is equidistant from $A$ and $B$, angle $APB$ is twice angle $ACB$, and $\\overline{AC}$ intersects $\\overline{BP}$ at point $D$. If $PB = 3$ and $PD= 2$, then $AD\\cdot CD =$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "3711",
"split": "train"
} |
math | [
{
"content": "Out of the digits 0 through 9, three digits are randomly chosen to form a three-digit number without repeating any digits. What is the probability that this number is not divisible by 3? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2/3",
"style": "rule"
} | {
"index": "3712",
"split": "train"
} |
math | [
{
"content": "Inside triangle \\(ABC\\), a point \\(O\\) is chosen such that \\(\\angle ABO = \\angle CAO\\), \\(\\angle BAO = \\angle BCO\\), and \\(\\angle BOC = 90^{\\circ}\\). Find the ratio \\(AC : OC\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "3713",
"split": "train"
} |
math | [
{
"content": "What is $2343_6+15325_6$? Express your answer in base $6$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22112_6",
"style": "rule"
} | {
"index": "3714",
"split": "train"
} |
math | [
{
"content": "Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "3715",
"split": "train"
} |
math | [
{
"content": "Points $P, Q, R, S, T, U, V,$ and $W$ lie, in that order, on line $\\overline{PW}$, dividing it into seven equal segments, each of length 1. Point $X$ is not on line $PW$. Points $Y$ and $Z$ lie on line segments $\\overline{XR}$ and $\\overline{XW}$ respectively. The line segments $\\overline{YQ},... | {
"ground_truth": "\\frac{7}{6}",
"style": "rule"
} | {
"index": "3716",
"split": "train"
} |
math | [
{
"content": "It is known that the numbers \\( x, y, z \\) form an arithmetic progression in the given order with a common difference \\( \\alpha = \\arccos \\left(-\\frac{1}{3}\\right) \\), and the numbers \\( \\frac{1}{\\cos x}, \\frac{3}{\\cos y}, \\frac{1}{\\cos z} \\) also form an arithmetic progression in... | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": "3717",
"split": "train"
} |
math | [
{
"content": "How many solutions in integers $x$ and $y$ does the inequality\n\n$$\n|x| + |y| < 10\n$$\n\nhave? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "181",
"style": "rule"
} | {
"index": "3718",
"split": "train"
} |
math | [
{
"content": "The sum of the numerical coefficients in the expansion of the binomial $(a+b)^6$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "3719",
"split": "train"
} |
math | [
{
"content": "In a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with a side length of 1, points $E$ and $F$ are located on $A A_{1}$ and $C C_{1}$ respectively, such that $A E = C_{1} F$. Determine the minimum area of the quadrilateral $E B F D_{1}$. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "\\frac{\\sqrt{6}}{2}",
"style": "rule"
} | {
"index": "3720",
"split": "train"
} |
math | [
{
"content": "Given the sample contains 5 individuals with values a, 0, 1, 2, 3, and the average value of the sample is 1, calculate the standard deviation of the sample. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "3721",
"split": "train"
} |
math | [
{
"content": "A positive integer cannot be divisible by 2 or 3, and there do not exist non-negative integers \\(a\\) and \\(b\\) such that \\(|2^a - 3^b| = n\\). Find the smallest value of \\(n\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "3722",
"split": "train"
} |
math | [
{
"content": "How many 3-term geometric sequences $a$ , $b$ , $c$ are there where $a$ , $b$ , and $c$ are positive integers with $a < b < c$ and $c = 8000$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "3723",
"split": "train"
} |
math | [
{
"content": "A total of $731$ objects are put into $n$ nonempty bags where $n$ is a positive integer. These bags can be distributed into $17$ red boxes and also into $43$ blue boxes so that each red and each blue box contain $43$ and $17$ objects, respectively. Find the minimum value of $n$ . Pl... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "3724",
"split": "train"
} |
math | [
{
"content": "My grandpa has 10 pieces of art, including 3 prints by Escher. If he hangs the pieces of art in a row in a random order, what is the probability that all three pieces by Escher will be placed consecutively? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{15}",
"style": "rule"
} | {
"index": "3725",
"split": "train"
} |
math | [
{
"content": "A palindromic number is a number that reads the same when the order of its digits is reversed. What is the difference between the largest and smallest five-digit palindromic numbers that are both multiples of 45? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9090",
"style": "rule"
} | {
"index": "3726",
"split": "train"
} |
math | [
{
"content": "A person can commute by train or car. If he takes the train to work in the morning, he takes the car in the afternoon; if he takes the train home in the afternoon, he takes the car in the morning. Over $x$ days, this person took the train 9 times, took the car in the morning 8 times, and took the ... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "3727",
"split": "train"
} |
math | [
{
"content": "Let \\( n = 1990 \\), then evaluate the expression \\(\\frac{1}{2^{n}}\\left(1 - 3 C_{n}^{2} + 3^{2} C_{n}^{4} - 3^{3} C_{n}^{6} + \\cdots + 3^{994} C_{n}^{1988} - 3^{995} C_{n}^{1990}\\right)\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "3728",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbb{R}$ denote the set of real numbers. Find all functions $f:\\mathbb{R}\\rightarrow\\mathbb{R}$ such that \n\\[f(xf(y)+y)+f(-f(x))=f(yf(x)-y)+y\\]\nfor all $x,y\\in\\mathbb{R}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "f(x) = x + 1",
"style": "rule"
} | {
"index": "3729",
"split": "train"
} |
math | [
{
"content": "289. A remarkable number. Find a number such that its fractional part, its integer part, and the number itself form a geometric progression. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1+\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "3730",
"split": "train"
} |
math | [
{
"content": "Determine the number of real number $a$ , such that for every $a$ , equation $x^3=ax+a+1$ has a root $x_0$ satisfying following conditions:\n(a) $x_0$ is an even integer;\n(b) $|x_0|<1000$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "999",
"style": "rule"
} | {
"index": "3731",
"split": "train"
} |
math | [
{
"content": "A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery a... | {
"ground_truth": "351",
"style": "rule"
} | {
"index": "3732",
"split": "train"
} |
math | [
{
"content": "The gas tank in Catherine's car is $\\frac{1}{8}$ full. When 30 litres of gas are added, the tank becomes $\\frac{3}{4}$ full. If the gas costs Catherine $\\$ 1.38$ per litre, how much will it cost her to fill the remaining quarter of the tank? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\$16.56",
"style": "rule"
} | {
"index": "3733",
"split": "train"
} |
math | [
{
"content": "Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1+\\sqrt{3}",
"style": "rule"
} | {
"index": "3734",
"split": "train"
} |
math | [
{
"content": "The sum of seven integers is $-1$. What is the maximum number of the seven integers that can be larger than $13$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "3735",
"split": "train"
} |
math | [
{
"content": "Given $$\\frac {1}{3} \\leq a \\leq 1$$, if the function $f(x) = ax^2 - 2x + 1$ has a domain of $[1, 3]$. \n(1) Find the minimum value of $f(x)$ in its domain (expressed in terms of $a$); \n(2) Let the maximum value of $f(x)$ in its domain be $M(a)$, and the minimum value be $N(a)$. Find the min... | {
"ground_truth": "\\frac {1}{2}",
"style": "rule"
} | {
"index": "3736",
"split": "train"
} |
math | [
{
"content": "On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "3737",
"split": "train"
} |
math | [
{
"content": "$x$ is a real number with the property that $x+\\tfrac1x = 3$. Let $S_m = x^m + \\tfrac{1}{x^m}$. Determine the value of $S_7$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "843",
"style": "rule"
} | {
"index": "3738",
"split": "train"
} |
math | [
{
"content": "How many paths are there from the starting point $C$ to the end point $D$, if every step must be up or to the right in a grid of 8 columns and 7 rows? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6435",
"style": "rule"
} | {
"index": "3739",
"split": "train"
} |
math | [
{
"content": "The hour and minute hands of a clock move continuously and at constant speeds. A moment of time $X$ is called interesting if there exists such a moment $Y$ (the moments $X$ and $Y$ do not necessarily have to be different), so that the hour hand at moment $Y$ will be where the minute hand is at mom... | {
"ground_truth": "143",
"style": "rule"
} | {
"index": "3740",
"split": "train"
} |
math | [
{
"content": "How many perfect cubes are between 100 and 900? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "3741",
"split": "train"
} |
math | [
{
"content": "What is the arithmetic mean of the integers from -4 through 5, inclusive? Express your answer as a decimal to the nearest tenth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.5",
"style": "rule"
} | {
"index": "3742",
"split": "train"
} |
math | [
{
"content": "A box contains 4 cards, each with one of the following functions defined on \\\\(R\\\\): \\\\(f_{1}(x)={x}^{3}\\\\), \\\\(f_{2}(x)=|x|\\\\), \\\\(f_{3}(x)=\\sin x\\\\), \\\\(f_{4}(x)=\\cos x\\\\). Now, if we randomly pick 2 cards from the box and multiply the functions on the cards to get a new fu... | {
"ground_truth": "\\dfrac{2}{3}",
"style": "rule"
} | {
"index": "3743",
"split": "train"
} |
math | [
{
"content": "Juan, Carlos and Manu take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Manu will win? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{7}",
"style": "rule"
} | {
"index": "3744",
"split": "train"
} |
math | [
{
"content": "Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "95",
"style": "rule"
} | {
"index": "3745",
"split": "train"
} |
math | [
{
"content": "Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\\circ$ angle with $AB$. At the same time Allie leav... | {
"ground_truth": "160",
"style": "rule"
} | {
"index": "3746",
"split": "train"
} |
math | [
{
"content": "Consider all 1000-element subsets of the set $\\{1, 2, 3, ... , 2015\\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.\nHint\n Use the Hockey Stick Ident... | {
"ground_truth": "431",
"style": "rule"
} | {
"index": "3747",
"split": "train"
} |
math | [
{
"content": "If the sum of the first $3n$ positive integers is $150$ more than the sum of the first $n$ positive integers, then the sum of the first $4n$ positive integers is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "3748",
"split": "train"
} |
math | [
{
"content": "Given that \\( m \\) and \\( n \\) are two distinct positive integers and the last four digits of \\( 2019^{m} \\) and \\( 2019^{n} \\) are the same, find the minimum value of \\( m+n \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "502",
"style": "rule"
} | {
"index": "3749",
"split": "train"
} |
math | [
{
"content": "The number of books issued from the library to readers constitutes $\\frac{1}{16}$ of the number of books on the shelves. After transferring 2000 books from the library to the reading room, the number of books absent from the shelves became $\\frac{1}{15}$ of the number of books remaining on the s... | {
"ground_truth": "544000",
"style": "rule"
} | {
"index": "3750",
"split": "train"
} |
math | [
{
"content": "Out of 500 participants in a remote math olympiad, exactly 30 did not like the problem conditions, exactly 40 did not like the organization of the event, and exactly 50 did not like the method used to determine the winners. A participant is called \"significantly dissatisfied\" if they were dissat... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "3751",
"split": "train"
} |
math | [
{
"content": "Let \\( A(2,0) \\) be a fixed point on the plane, \\( P\\left(\\sin \\left(2 t-60^{\\circ}\\right), \\cos \\left(2 t-60^{\\circ}\\right)\\right) \\) be a moving point. When \\( t \\) changes from \\( 15^{\\circ} \\) to \\( 45^{\\circ} \\), the area swept by the line segment \\( AP \\) is ______. P... | {
"ground_truth": "\\frac{\\pi}{6}",
"style": "rule"
} | {
"index": "3752",
"split": "train"
} |
math | [
{
"content": "A steak initially at a temperature of 5°C is put into an oven. After 15 minutes, its temperature reaches 45°C. After another 15 minutes, its temperature is 77°C. The oven maintains a constant temperature. The steak changes temperature at a rate proportional to the difference between its temperatur... | {
"ground_truth": "205",
"style": "rule"
} | {
"index": "3753",
"split": "train"
} |
math | [
{
"content": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$, and for $x \\geqslant 0$, $f(x) = \\begin{cases} \\log_{\\frac{1}{2}}(x+1), & 0 \\leqslant x < 1 \\\\ 1-|x-3|, & x \\geqslant 1 \\end{cases}$, determine the sum of all zeros of the function $y = f(x) + \\frac{1}{2}$. Please output the ... | {
"ground_truth": "\\sqrt{2} - 1",
"style": "rule"
} | {
"index": "3754",
"split": "train"
} |
math | [
{
"content": "Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Claudia have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "3755",
"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": "1716",
"style": "rule"
} | {
"index": "3756",
"split": "train"
} |
math | [
{
"content": "In triangle $PQR,$ $S$ is on $\\overline{PQ}$ such that $PS:SQ = 4:1,$ and $T$ is on $\\overline{QR}$ such that $QT:TR = 4:1.$ If lines $ST$ and $PR$ intersect at $U,$ then find $\\frac{ST}{TU}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "3757",
"split": "train"
} |
math | [
{
"content": "Let $A = (-3, 0),$ $B=(-2,1),$ $C=(2,1),$ and $D=(3,0).$ Suppose that point $P$ satisfies \\[PA + PD = PB + PC = 8.\\]Then the $y-$coordinate of $P,$ when simplified, can be expressed in the form $\\frac{-a + b \\sqrt{c}}{d},$ where $a,$ $b,$ $c,$ $d$ are positive integers. Find $a + b + c + d.$ ... | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "3758",
"split": "train"
} |
math | [
{
"content": "A rectangle with a diagonal of length $x$ is twice as long as it is wide. What is the area of the rectangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{5}x^2",
"style": "rule"
} | {
"index": "3759",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$ that satisfies $a_{n+1} = a_n - a_{n-1}$ ($n \\in N^*, n \\geqslant 2$), with $a_1 = 2018$ and $a_2 = 2017$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\\{a_n\\}$. The value of $S_{100}$ is ______. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": "3760",
"split": "train"
} |
math | [
{
"content": "The sides of a regular polygon of $n$ sides, $n>4$, are extended to form a star. The number of degrees at each point of the star is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{(n-2)180}{n}",
"style": "rule"
} | {
"index": "3761",
"split": "train"
} |
math | [
{
"content": "Two points $A, B$ are randomly chosen on a circle with radius $100.$ For a positive integer $x$ , denote $P(x)$ as the probability that the length of $AB$ is less than $x$ . Find the minimum possible integer value of $x$ such that $\\text{P}(x) > \\frac{2}{3}$ . Please output the fina... | {
"ground_truth": "174",
"style": "rule"
} | {
"index": "3762",
"split": "train"
} |
math | [
{
"content": "Compute\n$$\\sum_{k=1}^{2000} k(\\lceil \\log_{2}{k}\\rceil- \\lfloor\\log_{2}{k} \\rfloor).$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1998953",
"style": "rule"
} | {
"index": "3763",
"split": "train"
} |
math | [
{
"content": "An equilateral triangle is inscribed in a circle. A smaller equilateral triangle has one vertex coinciding with a vertex of the larger triangle and another vertex on the midpoint of a side of the larger triangle. What percent of the area of the larger triangle is the area of the smaller triangle? ... | {
"ground_truth": "25\\%",
"style": "rule"
} | {
"index": "3764",
"split": "train"
} |
math | [
{
"content": "A store arranges a decorative tower of balls where the top level has 2 balls and each lower level has 3 more balls than the level above. The display uses 225 balls. What is the number of levels in the tower? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "3765",
"split": "train"
} |
math | [
{
"content": "Compute\n\\[\\cos^6 0^\\circ + \\cos^6 1^\\circ + \\cos^6 2^\\circ + \\dots + \\cos^6 90^\\circ.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{229}{8}",
"style": "rule"
} | {
"index": "3766",
"split": "train"
} |
math | [
{
"content": "A circle with radius 1 is tangent to a circle with radius 3 at point \\( C \\). A line passing through point \\( C \\) intersects the smaller circle at point \\( A \\) and the larger circle at point \\( B \\). Find \\( AC \\), given that \\( AB = 2\\sqrt{5} \\). Please output the final answer with... | {
"ground_truth": "\\frac{\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "3767",
"split": "train"
} |
math | [
{
"content": "Simplify $21 \\cdot \\frac{8}{15} \\cdot \\frac{1}{14}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": "3768",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a triangle with $AB=9$ , $BC=10$ , $CA=11$ , and orthocenter $H$ . Suppose point $D$ is placed on $\\overline{BC}$ such that $AH=HD$ . Compute $AD$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{102}",
"style": "rule"
} | {
"index": "3769",
"split": "train"
} |
math | [
{
"content": "A rectangle with dimensions $8 \\times 2 \\sqrt{2}$ and a circle with a radius of 2 have a common center. Find the area of their overlapping region. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\pi + 4",
"style": "rule"
} | {
"index": "3770",
"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{}.",
... | {
"ground_truth": "-13",
"style": "rule"
} | {
"index": "3771",
"split": "train"
} |
math | [
{
"content": "The polynomial $P(x)$ is a monic, quartic polynomial with real coefficients, and two of its roots are $\\cos \\theta + i \\sin \\theta$ and $\\sin \\theta + i \\cos \\theta,$ where $0 < \\theta < \\frac{\\pi}{4}.$ When the four roots of $P(x)$ are plotted in the complex plane, they form a quadril... | {
"ground_truth": "1 + \\sqrt{3}",
"style": "rule"
} | {
"index": "3772",
"split": "train"
} |
math | [
{
"content": "On the game show $\\text{\\emph{Wheel of Fraction}}$, you see the following spinner. Given that each region is the same area, what is the probability that you will earn exactly $\\$1700$ in your first three spins? Express your answer as a common fraction. [asy]\nimport olympiad; import geometry; i... | {
"ground_truth": "\\frac{6}{125}",
"style": "rule"
} | {
"index": "3773",
"split": "train"
} |
math | [
{
"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). Please output the final an... | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "3774",
"split": "train"
} |
math | [
{
"content": "Find the area bounded by the graph of $y = \\arcsin(\\cos x)$ and the $x$-axis on the interval $\\frac{\\pi}{4} \\le x \\le \\frac{9\\pi}{4}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\pi^2",
"style": "rule"
} | {
"index": "3775",
"split": "train"
} |
math | [
{
"content": "In a math competition, 5 problems were assigned. There were no two contestants who solved exactly the same problems. However, for any problem that is disregarded, for each contestant there is another contestant who solved the same set of the remaining 4 problems. How many contestants participated ... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "3776",
"split": "train"
} |
math | [
{
"content": "$n$ coins are simultaneously flipped. The probability that two or fewer of them show tails is $\\frac{1}{4}$. Find $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "n = 5",
"style": "rule"
} | {
"index": "3777",
"split": "train"
} |
math | [
{
"content": "If $x$ is a positive integer, what is the value of $x$ for the equation $(x!-(x-3)!) \\div 23 = 1$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "3778",
"split": "train"
} |
math | [
{
"content": "Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to the vertex. If the sum of the numbers on the vertices is equal to $1386$, then what is the sum of the numbers written on the faces? Please output the fina... | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "3779",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\frac{ax^{2}+bx+c}{e^{x}} (a > 0)$ whose derivative $y=f′(x)$ has two zeros at $-3$ and $0$.\n\n1. Find the monotonic intervals of $f(x)$;\n2. If the minimum value of $f(x)$ is $-e^{3}$, find the maximum value of $f(x)$ on the interval $[-5,+\\infty)$. Please output the f... | {
"ground_truth": "5e^{5}",
"style": "rule"
} | {
"index": "3780",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, with $AB=3$, $AC=4$, $BC=5$, let $I$ be the incenter of $\\triangle ABC$ and $P$ be a point inside $\\triangle IBC$ (including the boundary). If $\\overrightarrow{AP}=\\lambda \\overrightarrow{AB} + \\mu \\overrightarrow{AC}$ (where $\\lambda, \\mu \\in \\mathbf{R}$), find the... | {
"ground_truth": "7/12",
"style": "rule"
} | {
"index": "3781",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has an inradius of $5$ and a circumradius of $16$. If $2\\cos{B} = \\cos{A} + \\cos{C}$, then the area of triangle $ABC$ can be expressed as $\\frac{a\\sqrt{b}}{c}$, where $a, b,$ and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by th... | {
"ground_truth": "141",
"style": "rule"
} | {
"index": "3782",
"split": "train"
} |
math | [
{
"content": "A boss plans a business meeting at Starbucks with the two engineers below him. However, he fails to set a time, and all three arrive at Starbucks at a random time between 2:00 and 4:00 p.m. When the boss shows up, if both engineers are not already there, he storms out and cancels the meeting. Each... | {
"ground_truth": "\\frac{7}{24}",
"style": "rule"
} | {
"index": "3783",
"split": "train"
} |
math | [
{
"content": "Experts and Viewers play \"What? Where? When?\" until one side wins six rounds. The probability of Experts winning a single round is 0.6, and there are no ties. Currently, the Experts are losing with a score of 3 to 4. Find the probability that the Experts will eventually win. Please output the fi... | {
"ground_truth": "0.4752",
"style": "rule"
} | {
"index": "3784",
"split": "train"
} |
math | [
{
"content": "Calculate the lengths of the arcs of curves given by the equations in the rectangular coordinate system.\n\n$$\ny=\\ln x, \\sqrt{3} \\leq x \\leq \\sqrt{15}\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2} \\ln \\frac{9}{5} + 2",
"style": "rule"
} | {
"index": "3785",
"split": "train"
} |
math | [
{
"content": "Given that $\\cos (\\alpha - \\frac{\\pi }{3}) - \\cos \\alpha = \\frac{1}{3}$, find the value of $\\sin (\\alpha - \\frac{\\pi }{6})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "3786",
"split": "train"
} |
math | [
{
"content": "If $g(x) = 3x^2 + 4$ and $h(x) = -2x^3 + 2$, what is the value of $g(h(2))$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "592",
"style": "rule"
} | {
"index": "3787",
"split": "train"
} |
math | [
{
"content": "Tom is searching for the $6$ books he needs in a random pile of $30$ books. What is the expected number of books must he examine before finding all $6$ books he needs? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14.7",
"style": "rule"
} | {
"index": "3788",
"split": "train"
} |
math | [
{
"content": "Consider three coins where two are fair and a third coin lands on heads with a probability of $\\frac{3}{5}$. Alice flips the three coins, and then Bob flips the same three coins. Let $\\frac{p}{q}$ be the probability that Alice and Bob get the same number of heads, where $p$ and $q$ are coprime i... | {
"ground_truth": "263",
"style": "rule"
} | {
"index": "3789",
"split": "train"
} |
math | [
{
"content": "How many different right-angled triangles exist, one of the legs of which is \\(\\sqrt{2016}\\), and the other leg and hypotenuse are expressed in natural numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "3790",
"split": "train"
} |
math | [
{
"content": "Given that vectors $\\overrightarrow{α}$ and $\\overrightarrow{β}$ are two mutually perpendicular unit vectors in a plane, and $(5\\overrightarrow{α} - 2\\overrightarrow{γ}) \\cdot (12\\overrightarrow{β} - 2\\overrightarrow{γ}) = 0$, find the maximum value of $|\\overrightarrow{γ}|$. Please output... | {
"ground_truth": "\\frac{13}{2}",
"style": "rule"
} | {
"index": "3791",
"split": "train"
} |
math | [
{
"content": "Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50<f(7)<60$, $70<f(8)<80$, $5000k<f(100)<5000(k+1)$ for some integer $k$. What is $k$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "3792",
"split": "train"
} |
math | [
{
"content": "The function \\[f(x) = \\left\\{ \\begin{aligned} 2x + 1 & \\quad \\text{ if } x < 3 \\\\ x^2 & \\quad \\text{ if } x \\ge 3 \\end{aligned} \\right.\\] has an inverse $f^{-1}.$ Compute the value of $f^{-1}(-3) + f^{-1}(0) + \\dots + f^{-1}(4) + f^{-1}(9).$ Please output the final answer within \\b... | {
"ground_truth": "3.5",
"style": "rule"
} | {
"index": "3793",
"split": "train"
} |
math | [
{
"content": "Evaluate the expression $$\\frac{\\sin 10°}{1 - \\sqrt{3}\\tan 10°}.$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "3794",
"split": "train"
} |
math | [
{
"content": "Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 8.$ Find the maximum value of\n\\[\\sqrt{3x + 2} + \\sqrt{3y + 2} + \\sqrt{3z + 2}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\sqrt{10}",
"style": "rule"
} | {
"index": "3795",
"split": "train"
} |
math | [
{
"content": "The seventh and tenth terms of a geometric sequence are $7$ and $21$, respectively. What is the $13$th term of this progression? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "63",
"style": "rule"
} | {
"index": "3796",
"split": "train"
} |
math | [
{
"content": "A linear function \\( f(x) \\) is given. It is known that the distance between the points of intersection of the graphs \\( y = x^{2} \\) and \\( y = f(x) \\) is \\( 2 \\sqrt{3} \\), and the distance between the points of intersection of the graphs \\( y = x^{2}-2 \\) and \\( y = f(x)+1 \\) is \\(... | {
"ground_truth": "2 \\sqrt{11}",
"style": "rule"
} | {
"index": "3797",
"split": "train"
} |
math | [
{
"content": "A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of the polynomial\n$x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2$\n are all real can be written in the form $\\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find... | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "3798",
"split": "train"
} |
math | [
{
"content": "A clothing retailer offered a discount of $\\frac{1}{4}$ on all jackets tagged at a specific price. If the cost of the jackets was $\\frac{2}{3}$ of the price they were actually sold for and considering this price included a sales tax of $\\frac{1}{10}$, what would be the ratio of the cost to the ... | {
"ground_truth": "\\frac{11}{30}",
"style": "rule"
} | {
"index": "3799",
"split": "train"
} |
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