data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "For how many unordered sets $\\{a, b, c, d\\}$ of positive integers, none of which exceed 168, do there exist integers $w, x, y, z$ such that $(-1)^{w} a+(-1)^{x} b+(-1)^{y} c+(-1)^{z} d=168$? If your answer is $A$ and the correct answer is $C$, then your score on this problem ... | math | {
"style": "rule",
"ground_truth": "761474"
} | {
"split": "train",
"index": 29749,
"solution": "As an approximation, we assume $a, b, c, d$ are ordered to begin with (so we have to divide by 24 later) and add to 168 with a unique choice of signs; then, it suffices to count $e+f+g+h=168$ with each $e, f, g, h$ in $[-168,168]$ and then divide by 24 (we drop the... |
deepscaler | [
{
"role": "user",
"content": "A regular $n$-gon has $n$ diagonals, its perimeter is $p$, and the sum of the lengths of all the diagonals is $q$. What is $\\frac{p}{q} + \\frac{q}{p}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 29750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Without using a calculator, compute $1003^2-997^2-1001^2+999^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8000"
} | {
"split": "train",
"index": 29751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express as a common fraction in simplest form: $$\n\\sqrt{6\\frac{1}{4}}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 29752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 5y^2 = 6,$ as shown below. Find $r.$\n[asy]\nsize(7cm);\ndraw(scale(sqrt(6), sqrt(6)/sqrt(5))* unitcircle);\ndraw((0,-1.5)--(0,1.7),EndArrow);\ndraw((-3,0)--(3,0),EndArro... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt6}{5}"
} | {
"split": "train",
"index": 29753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains red and yellow balls. When 60 balls are taken out, exactly 56 of them are red. Thereafter, every time 18 balls are taken out, 14 of them are always red, until the last batch of 18 balls is taken out. If the total number of red balls in the bag is exactly four-fif... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 29754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the complex plane, the points corresponding to the complex number $1+ \\sqrt {3}i$ and $- \\sqrt {3}+i$ are $A$ and $B$, respectively, with $O$ as the coordinate origin. Calculate the measure of $\\angle AOB$. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\dfrac {\\pi}{2}"
} | {
"split": "train",
"index": 29755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b$ be integers chosen independently and uniformly at random from the set $\\{0,1,2, \\ldots, 80\\}$. Compute the expected value of the remainder when the binomial coefficient $\\binom{a}{b}=\\frac{a!}{b!(a-b)!}$ is divided by 3. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "\\frac{1816}{6561}"
} | {
"split": "train",
"index": 29756,
"solution": "By Lucas' Theorem we're looking at $\\prod_{i=1}^{4}\\binom{a_{i}}{b_{i}}$ where the $a_{i}$ and $b_{i}$ are the digits of $a$ and $b$ in base 3. If any $a_{i}<b_{i}$, then the product is zero modulo 3. Otherwise, the potential residues are $\\binom{2}{0}=1,\\binom... |
deepscaler | [
{
"role": "user",
"content": "How many zeros are there in the last digits of the following number $P = 11\\times12\\times ...\\times 88\\times 89$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 29757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers between 1 and 200 are multiples of both 3 and 5 but not of either 4 or 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 29758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are several sets of three different numbers whose sum is $15$ which can be chosen from $\\{ 1,2,3,4,5,6,7,8,9 \\}$. How many of these sets contain a $5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 29759,
"solution": "1. **Define the set and condition**: Let the three-element set be $\\{a, b, c\\}$, where $a, b, c$ are distinct elements from the set $\\{1, 2, 3, 4, 5, 6, 7, 8, 9\\}$. We are given that the sum of the elements in the set is $15$.\n\n2. **Condition on including $5$... |
deepscaler | [
{
"role": "user",
"content": "Consider the polynomials \\[f(x)=1-12x+3x^2-4x^3+5x^4\\] and \\[g(x)=3-2x-6x^3+9x^4.\\] Find $c$ such that the polynomial $f(x)+cg(x)$ has degree 3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{9}"
} | {
"split": "train",
"index": 29760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of \\(\\sum_{k=1}^{60} \\sum_{n=1}^{k} \\frac{n^{2}}{61-2 n}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-18910"
} | {
"split": "train",
"index": 29761,
"solution": "Change the order of summation and simplify the inner sum: \\(\\sum_{k=1}^{60} \\sum_{n=1}^{k} \\frac{n^{2}}{61-2 n} =\\sum_{n=1}^{60} \\sum_{k=n}^{60} \\frac{n^{2}}{61-2 n} =\\sum_{n=1}^{60} \\frac{n^{2}(61-n)}{61-2 n}\\). Then, we rearrange the sum to add the term... |
deepscaler | [
{
"role": "user",
"content": "In recent years, the awareness of traffic safety among citizens has gradually increased, leading to a greater demand for helmets. A certain store purchased two types of helmets, type A and type B. It is known that they bought 20 type A helmets and 30 type B helmets, spending a ... | math | {
"style": "rule",
"ground_truth": "1976"
} | {
"split": "train",
"index": 29762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $x$ in the plane figure shown?\n\n[asy]\npair A;\ndraw(dir(40)--A); draw(dir(200)--A); draw(dir(300)--A);\nlabel(\"$160^{\\circ}$\",A,dir(120)); label(\"$x^{\\circ}$\",A,dir(250)); label(\"$x^{\\circ}$\",A,dir(350));\n[/asy] Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 29763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four whole numbers, when added three at a time, give the sums $180, 197, 208$ and $222$. What is the largest of the four numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 29764,
"solution": "Let the four whole numbers be $a$, $b$, $c$, and $d$. According to the problem, the sums of these numbers taken three at a time are given as follows:\n- $a + b + c = 180$\n- $a + b + d = 197$\n- $a + c + d = 208$\n- $b + c + d = 222$\n\n1. **Calculate the sum of al... |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $n$, an associated non-negative integer $f(n)$ is defined to satisfy the following three rules:\ni) $f(a b)=f(a)+f(b)$.\n\nii) $f(n)=0$ if $n$ is a prime greater than 10.\n\niii) $f(1)<f(243)<f(2)<11$.\n\nGiven that $f(2106)<11$, determine the value of... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 29765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles with centers $A$ and $B$ and radii 2 and 1, respectively, are tangent to each other. Point $C$ lies on a line that is tangent to each of the circles and is at a distance of $\\frac{3 \\sqrt{3}}{2 \\sqrt{2}}$ from the midpoint of segment $AB$. Find the area $S$ of tr... | math | {
"style": "rule",
"ground_truth": "\\frac{15 \\sqrt{2}}{8}"
} | {
"split": "train",
"index": 29766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kolya was supposed to square a certain natural number for his homework. Instead, he mistakenly doubled the number and got a two-digit number, written with the same digits as the square of the number but in reverse order. What should be the correct answer? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 29767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the sides $AB$ and $CD$ of rectangle $ABCD$, points $E$ and $F$ are marked such that $AFCE$ forms a rhombus. It is known that $AB = 16$ and $BC = 12$. Find $EF$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 29768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arrange the letters a, a, b, b, c, c into three rows and two columns, such that in each row and each column, the letters are different. How many different arrangements are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 29769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Schools A and B are having a sports competition with three events. In each event, the winner gets 10 points and the loser gets 0 points, with no draws. The school with the higher total score after the three events wins the championship. It is known that the probabilities of sch... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 29770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A robot has 6 modules, each needing a base and a cap attached sequentially. In how many different orders can the robot attach its bases and caps, assuming that, on each module, the base must be attached before the cap?\nA) $6! \\cdot 2^6$\nB) $(6!)^2$\nC) $12!$\nD) $\\frac{12!}... | math | {
"style": "rule",
"ground_truth": "\\frac{12!}{2^6}"
} | {
"split": "train",
"index": 29771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A Yule log is shaped like a right cylinder with height $10$ and diameter $5$ . Freya cuts it parallel to its bases into $9$ right cylindrical slices. After Freya cut it, the combined surface area of the slices of the Yule log increased by $a\\pi$ . Compute $a$ . Let's th... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 29772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert the point $\\left( 8, \\frac{7 \\pi}{6} \\right)$ in polar coordinates to rectangular coordinates. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-4 \\sqrt{3},-4)"
} | {
"split": "train",
"index": 29773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many pairs of positive integers $(a, b)$ with $a \\leq b$ satisfy $\\frac{1}{a} + \\frac{1}{b} = \\frac{1}{6}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 29774,
"solution": "$\\frac{1}{a} + \\frac{1}{b} = \\frac{1}{6} \\Rightarrow \\frac{a+b}{ab} = \\frac{1}{6} \\Rightarrow ab = 6a + 6b \\Rightarrow ab - 6a - 6b = 0$. Factoring yields $(a-6)(b-6) = 36$. Because $a$ and $b$ are positive integers, only the factor pairs of 36 are possible... |
deepscaler | [
{
"role": "user",
"content": "A wooden cube has edges of length $3$ meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. The entire surface area including the inside, in square meters, is Let's thin... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 29775,
"solution": "1. **Calculate the original surface area of the cube**: \n The cube has 6 faces, and each face is a square with side length 3 meters. The area of one face is therefore $3^2 = 9$ square meters. Thus, the total surface area of the cube before any modifications is:\... |
deepscaler | [
{
"role": "user",
"content": "In a bag, there are 10 red balls, 8 white balls, and 7 black balls. The balls are randomly drawn one by one. To ensure that a red ball is drawn for the first time on the $k$-th draw or before, the minimum value of $k$ is ______. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 29776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given three composite numbers \\( A, B, C \\) that are pairwise coprime and \\( A \\times B \\times C = 11011 \\times 28 \\). What is the maximum value of \\( A + B + C \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1626"
} | {
"split": "train",
"index": 29777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The triangle $\\triangle ABC$ is equilateral, and the point $P$ is such that $PA = 3 \\, \\text{cm}$, $PB = 4 \\, \\text{cm}$, and $PC = 5 \\, \\text{cm}$. Calculate the length of the sides of the triangle $\\triangle ABC$. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\sqrt{25 + 12 \\sqrt{3}}"
} | {
"split": "train",
"index": 29778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rationalize the denominator of $\\frac{2}{3\\sqrt{5} + 2\\sqrt{11}}$ and write your answer in the form $\\displaystyle \\frac{A\\sqrt{B} + C\\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is $A+B+C+D+E$? Let's t... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 29779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find maximal positive integer $p$ such that $5^7$ is sum of $p$ consecutive positive integers Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 29780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of parts into which the coordinate plane \\(xOy\\) can be divided by the graphs of 100 quadratic polynomials of the form \n\\[ y = a_{n} x^{2} + b_{n} x + c_{n} \\quad (n=1, 2, \\ldots, 100) ? \\] Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "10001"
} | {
"split": "train",
"index": 29781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains $12$ ping-pong balls, of which $9$ are new and $3$ are old. Three balls are randomly drawn from the box for use, and then returned to the box. Let $X$ denote the number of old balls in the box after this process. What is the value of $P(X = 4)$? Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\frac{27}{220}"
} | {
"split": "train",
"index": 29782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ellipse $x^2 + 9y^2 = 9$ and the hyperbola $x^2 - m(y+3)^2 = 1$ are tangent. Compute $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"split": "train",
"index": 29783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the smallest square that will contain a circle of radius 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "196"
} | {
"split": "train",
"index": 29784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A person flips a coin, where the probability of heads up and tails up is $\\frac{1}{2}$ each. Construct a sequence $\\left\\{a_{n}\\right\\}$ such that\n$$\na_{n}=\\left\\{\n\\begin{array}{ll}\n1, & \\text{if the } n \\text{th flip is heads;} \\\\\n-1, & \\text{if the } n \\tex... | math | {
"style": "rule",
"ground_truth": "13/128"
} | {
"split": "train",
"index": 29785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that a real number $x$ satisfies \\[\\sqrt{49-x^2}-\\sqrt{25-x^2}=3.\\]What is the value of $\\sqrt{49-x^2}+\\sqrt{25-x^2}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 29786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function \\( f(x) \\) is defined on the set of real numbers, and satisfies the equations \\( f(2+x) = f(2-x) \\) and \\( f(7+x) = f(7-x) \\) for all real numbers \\( x \\). Let \\( x = 0 \\) be a root of \\( f(x) = 0 \\). Denote the number of roots of \\( f(x) = 0 \\) in th... | math | {
"style": "rule",
"ground_truth": "401"
} | {
"split": "train",
"index": 29787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit numbers contain one even digit and three odd digits, with no repeated digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1140"
} | {
"split": "train",
"index": 29788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer. Determine the size of the largest subset of $\\{ - n, - n + 1, \\ldots , n - 1, n\\}$ which does not contain three elements $a, b, c$ (not necessarily distinct) satisfying $a + b + c = 0$ . Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\left\\lceil \\frac{n}{2} \\right\\rceil"
} | {
"split": "train",
"index": 29789,
"solution": "Let $S$ be a subset of $\\{-n,-n+1,\\dots,n-1,n\\}$ of largest size satisfying $a+b+c\\neq 0$ for all $a,b,c\\in S$ . First, observe that $0\\notin S$ . Next note that $|S|\\geq \\lceil n/2\\rceil$ , by observing that the set of all the odd numbers in $\\{-n,-n+1,\... |
deepscaler | [
{
"role": "user",
"content": "Chester traveled from Hualien to Lukang in Changhua to participate in the Hua Luogeng Gold Cup Math Competition. Before leaving, his father checked the car’s odometer, which displayed a palindromic number of 69,696 kilometers (a palindromic number reads the same forward and bac... | math | {
"style": "rule",
"ground_truth": "82.2"
} | {
"split": "train",
"index": 29790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance? Let's think ste... | math | {
"style": "rule",
"ground_truth": "\\frac{147}{1024}"
} | {
"split": "train",
"index": 29791,
"solution": "To solve this problem, we need to consider the different ways two cubes can be painted such that they are identical in appearance after possible rotations. We will calculate the number of ways to paint each cube and then find the probability that they match.\n\n###... |
deepscaler | [
{
"role": "user",
"content": "Two equilateral triangles with perimeters of 12 and 15 are positioned such that their sides are respectively parallel. Find the perimeter of the resulting hexagon. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 29792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two \\(1 \\times 1\\) squares are removed from a \\(5 \\times 5\\) grid as shown. Determine the total number of squares of various sizes on the grid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 29793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Trapezoid $PQRS$ has vertices $P(-3, 3)$, $Q(3, 3)$, $R(5, -1)$, and $S(-5, -1)$. If a point is selected at random from the region determined by the trapezoid, what is the probability that the point is above the $x$-axis? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 29794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation $3x^{2}-4=-2x$, find the quadratic coefficient, linear coefficient, and constant term. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 29795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A doctor told Mikael to take a pill every 75 minutes. He took his first pill at 11:05. At what time did he take his fourth pill? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14:50"
} | {
"split": "train",
"index": 29796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $l,$ $m,$ and $n$ be real numbers, and let $A,$ $B,$ $C$ be points such that the midpoint of $\\overline{BC}$ is $(l,0,0),$ the midpoint of $\\overline{AC}$ is $(0,m,0),$ and the midpoint of $\\overline{AB}$ is $(0,0,n).$ Find\n\\[\\frac{AB^2 + AC^2 + BC^2}{l^2 + m^2 + n^2... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 29797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integers $n$ between 1 and 11 (inclusive) is $\\frac{n}{12}$ a repeating decimal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 29798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ana has an iron material of mass $20.2$ kg. She asks Bilyana to make $n$ weights to be used in a classical weighning scale with two plates. Bilyana agrees under the condition that each of the $n$ weights is at least $10$ g. Determine the smallest possible value of $n$ ... | math | {
"style": "rule",
"ground_truth": "2020"
} | {
"split": "train",
"index": 29799,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given six thin wooden sticks, the two longer ones are $\\sqrt{3}a$ and $\\sqrt{2}a$, while the remaining four are of length $a$. If they are used to form a triangular prism, find the cosine of the angle between the lines containing the two longer edges. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{3}"
} | {
"split": "train",
"index": 29800,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x \\in (1,5)$, find the minimum value of the function $y= \\frac{2}{x-1}+ \\frac{1}{5-x}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3+2 \\sqrt{2}}{4}"
} | {
"split": "train",
"index": 29801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pablo is solving a quadratic equation and notices that the ink has smeared over the coefficient of $x$, making it unreadable. He recalls that the equation has two distinct negative, integer solutions. He needs to calculate the sum of all possible coefficients that were under th... | math | {
"style": "rule",
"ground_truth": "-60"
} | {
"split": "train",
"index": 29802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a right circular cone with three mutually perpendicular side edges, each with a length of $\\sqrt{3}$, determine the surface area of the circumscribed sphere. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9\\pi"
} | {
"split": "train",
"index": 29803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then find the value of $\\frac{{{a^2}-{b^2}}}{{{a^2}b-a{b^2}}}÷(1+\\frac{{{a^2}+{b^2}}}{2ab})$, where $a=\\sqrt{3}-\\sqrt{11}$ and $b=\\sqrt{3}+\\sqrt{11}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 29804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all integers $n$, not necessarily positive, for which there exist positive integers $a, b, c$ satisfying $a^{n}+b^{n}=c^{n}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pm 1, \\pm 2"
} | {
"split": "train",
"index": 29805,
"solution": "By Fermat's Last Theorem, we know $n<3$. Suppose $n \\leq-3$. Then $a^{n}+b^{n}=c^{n} \\Longrightarrow(b c)^{-n}+$ $(a c)^{-n}=(a b)^{-n}$, but since $-n \\geq 3$, this is also impossible by Fermat's Last Theorem. As a result, $|n|<3$. Furthermore, $n \\neq 0$, as ... |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x) = \\frac{1}{3^x + \\sqrt{3}}$, find the value of $f(-12) + f(-11) + f(-10) + ... + f(0) + ... + f(11) + f(12) + f(13)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 29806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the sum of all integral values of $c$ such that $c \\leq 18$ for which the equation $y = x^2 - 5x - c$ has exactly two rational roots. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 29807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The second and fifth terms of an arithmetic sequence are 17 and 19, respectively. What is the eighth term? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 29808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any real number \\( x \\), let \\( f(x) \\) be the minimum of the values \\( 4x + 1 \\), \\( x + 2 \\), and \\( -2x + 4 \\). What is the maximum value of \\( f(x) \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"split": "train",
"index": 29809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 100 houses in a row on a street. A painter comes and paints every house red. Then, another painter comes and paints every third house (starting with house number 3) blue. Another painter comes and paints every fifth house red (even if it is already red), then another ... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 29810,
"solution": "House $n$ ends up red if and only if the largest odd divisor of $n$ is of the form $4 k+1$. We have 25 values of $n=4 k+1 ; 13$ values of $n=2(4 k+1)$ (given by $k=0,1,2, \\ldots, 12$ ); 7 values of $n=4(4 k+1)(k=0,1, \\ldots, 6) ; 3$ values of $n=8(4 k+1)(k=0,1,2)... |
deepscaler | [
{
"role": "user",
"content": "Let $ABCDE$ be a convex pentagon, and let $G_A, G_B, G_C, G_D, G_E$ denote the centroids of triangles $BCDE, ACDE, ABDE, ABCE, ABCD$, respectively. Find the ratio $\\frac{[G_A G_B G_C G_D G_E]}{[ABCDE]}$. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 29811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ A$ , $ B$ be the number of digits of $ 2^{1998}$ and $ 5^{1998}$ in decimal system. $ A \\plus B \\equal{} ?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1999"
} | {
"split": "train",
"index": 29812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five socks, colored blue, brown, black, red, and purple are in a drawer. In how many different ways can we choose three socks from the drawer if the order of the socks does not matter? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 29813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Count the number of sequences $1 \\leq a_{1} \\leq a_{2} \\leq \\cdots \\leq a_{5}$ of integers with $a_{i} \\leq i$ for all $i$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 29814,
"solution": "$C$ (number of terms) $=C(5)=42$."
} |
deepscaler | [
{
"role": "user",
"content": "For non-negative real numbers $x_1, x_2, \\ldots, x_n$ which satisfy $x_1 + x_2 + \\cdots + x_n = 1$, find the largest possible value of $\\sum_{j = 1}^{n} (x_j^{4} - x_j^{5})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 29815,
"solution": "\nLet \\( x_1, x_2, \\ldots, x_n \\) be non-negative real numbers such that \\( x_1 + x_2 + \\cdots + x_n = 1 \\). We aim to find the largest possible value of \\( \\sum_{j=1}^n (x_j^4 - x_j^5) \\).\n\nTo solve this, we use the method of smoothing. We start by cons... |
deepscaler | [
{
"role": "user",
"content": "In rectangle \\(ABCD\\), \\(BE = 5\\), \\(EC = 4\\), \\(CF = 4\\), and \\(FD = 1\\), as shown in the diagram. What is the area of triangle \\(\\triangle AEF\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42.5"
} | {
"split": "train",
"index": 29816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the remainder when $(x^4-1)(x^2-1)$ is divided by $1+x+x^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 29817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "1. Calculate $\\log _{2.5}6.25+ \\lg 0.01+ \\ln \\sqrt {e}-2\\,^{1+\\log _{2}3}$\n2. Given $\\tan \\alpha=-3$, and $\\alpha$ is an angle in the second quadrant, find $\\sin \\alpha$ and $\\cos \\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "- \\frac { \\sqrt {10}}{10}"
} | {
"split": "train",
"index": 29818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the integer\\[N = 9 + 99 + 999 + 9999 + \\cdots + \\underbrace{99\\ldots 99}_\\text{321 digits}.\\]Find the sum of the digits of $N$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "342"
} | {
"split": "train",
"index": 29819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the inequality\n\\[\\dfrac{x+1}{x+2}>\\dfrac{3x+4}{2x+9}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( -\\frac{9}{2} , -2 \\right) \\cup \\left( \\frac{1 - \\sqrt{5}}{2}, \\frac{1 + \\sqrt{5}}{2} \\right)"
} | {
"split": "train",
"index": 29820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given functions $f(x)$ and $g(x)$, where the graph of $g(x)$ is symmetric about $x=1$, and $f(x) - g(x) = 1$, $f(x+1) + g(2-x) = 1$, and $g(1) = 3$, find $\\sum_{i=1}^{23}f(x)=$____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 29821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $\\frac{a}{a+2}-\\frac{a+3}{{a}^{2}-4}\\div \\frac{2a+6}{2{a}^{2}-8a+8}$, where $a=|-6|-(\\frac{1}{2})^{-1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 29822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A mineralogist is hosting a competition to guess the age of an ancient mineral sample. The age is provided by the digits 2, 2, 3, 3, 5, and 9, with the condition that the age must start with an odd number. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 29823,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle $XYZ$, $\\angle XYZ = \\angle YXZ$. If $XZ=8$ and $YZ=11$, what is the perimeter of $\\triangle XYZ$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 29824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest integer that must divide the product of any $5$ consecutive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 29825,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the right parallelopiped $ABCDA^{\\prime}B^{\\prime}C^{\\prime}D^{\\prime}$ , with $AB=12\\sqrt{3}$ cm and $AA^{\\prime}=18$ cm, we consider the points $P\\in AA^{\\prime}$ and $N\\in A^{\\prime}B^{\\prime}$ such that $A^{\\prime}N=3B^{\\prime}N$ . Determine the le... | math | {
"style": "rule",
"ground_truth": "27/2"
} | {
"split": "train",
"index": 29826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight distinct integers are picked at random from $\\{1,2,3,\\ldots,15\\}$. What is the probability that, among those selected, the third smallest is $5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{72}{307}"
} | {
"split": "train",
"index": 29827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given quadrilateral $ABCD$ with diagonals $AC$ and $BD$ intersecting at $O$, $BO=4$, $OD=5$, $AO=9$, $OC=2$, and $AB=7$, find the length of $AD$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{166}"
} | {
"split": "train",
"index": 29828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A nonzero polynomial with rational coefficients has all of the numbers \\[1+\\sqrt{2}, \\; 2+\\sqrt{3}, \\;3+\\sqrt{4},\\; \\dots, \\;1000+\\sqrt{1001}\\]as roots. What is the smallest possible degree of such a polynomial? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "1970"
} | {
"split": "train",
"index": 29829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive divisors $d$ of $15!=15 \\cdot 14 \\cdots 2 \\cdot 1$ such that $\\operatorname{gcd}(d, 60)=5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 29830,
"solution": "Since $\\operatorname{gcd}(d, 60)=5$, we know that $d=5^{i} d^{\\prime}$ for some integer $i>0$ and some integer $d^{\\prime}$ which is relatively prime to 60. Consequently, $d^{\\prime}$ is a divisor of $(15!) / 5$; eliminating common factors with 60 gives that $d... |
deepscaler | [
{
"role": "user",
"content": "It is currently $3\\!:\\!00\\!:\\!00 \\text{ p.m.}$ What time will it be in $6666$ seconds? (Enter the time in the format \"HH:MM:SS\", without including \"am\" or \"pm\".) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\!:\\!51\\!:\\!06 \\text{ p.m.}"
} | {
"split": "train",
"index": 29831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The angle $A$ at the vertex of the isosceles triangle $ABC$ is $100^{\\circ}$. On the ray $AB$, a segment $AM$ is laid off, equal to the base $BC$. Find the measure of the angle $BCM$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 29832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $A=3x^{2}-x+2y-4xy$, $B=x^{2}-2x-y+xy-5$.\n\n$(1)$ Find $A-3B$.\n\n$(2)$ If $(x+y-\\frac{4}{5})^{2}+|xy+1|=0$, find the value of $A-3B$.\n\n$(3)$ If the value of $A-3B$ is independent of $y$, find the value of $x$. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{7}"
} | {
"split": "train",
"index": 29833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers are there in $\\{0,1, 2,..., 2014\\}$ such that $C^x_{2014} \\ge C^{999}{2014}$ ?\n\nNote: $C^{m}_{n}$ stands for $\\binom {m}{n}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 29834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute \\[\\lfloor \\sqrt{1} \\rfloor + \\lfloor \\sqrt{2} \\rfloor + \\lfloor \\sqrt{3} \\rfloor + \\cdots + \\lfloor \\sqrt{25} \\rfloor.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 29835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A food factory has made 4 different exquisite cards. Each bag of food produced by the factory randomly contains one card. If all 4 different cards are collected, a prize can be won. Xiaoming buys 6 bags of this food at once. What is the probability that Xiaoming will win the pr... | math | {
"style": "rule",
"ground_truth": "195/512"
} | {
"split": "train",
"index": 29836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the positive difference of the solutions of $\\dfrac{s^2 - 4s - 22}{s + 3} = 3s + 8$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{27}{2}"
} | {
"split": "train",
"index": 29837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence: $\\frac{2}{3}, \\frac{2}{9}, \\frac{4}{9}, \\frac{6}{9}, \\frac{8}{9}, \\frac{2}{27}, \\frac{4}{27}, \\cdots$, $\\frac{26}{27}, \\cdots, \\frac{2}{3^{n}}, \\frac{4}{3^{n}}, \\cdots, \\frac{3^{n}-1}{3^{n}}, \\cdots$. Find the position of $\\frac{2018}{2187}$ ... | math | {
"style": "rule",
"ground_truth": "1552"
} | {
"split": "train",
"index": 29838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the six digits 0, 1, 2, 3, 4, 5,\n(1) How many distinct three-digit numbers can be formed?\n(2) How many distinct three-digit odd numbers can be formed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 29839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the dart board shown in the figure below, the outer circle has radius $6$ and the inner circle has radius $3$. Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the are... | math | {
"style": "rule",
"ground_truth": "\\frac{35}{72}"
} | {
"split": "train",
"index": 29840,
"solution": "1. **Identify the regions and their point values**: The dartboard is divided into six regions by three radii and two concentric circles. The inner circle is divided into three regions each with a radius of $3$, and the outer ring (between the inner circle and the o... |
deepscaler | [
{
"role": "user",
"content": "It is known that the optimal amount of a certain material to be added is between 100g and 1100g. If the 0.618 method is used to arrange the experiment and the first and second trials are at points $x_1$ and $x_2$ ($x_1 > x_2$), then when $x_2$ is considered the better point, th... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 29841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each interger $n\\geq 4$, we consider the $m$ subsets $A_1, A_2,\\dots, A_m$ of $\\{1, 2, 3,\\dots, n\\}$, such that\n$A_1$ has exactly one element, $A_2$ has exactly two elements,...., $A_m$ has exactly $m$ elements and none of these subsets is contained in any other set. ... | math | {
"style": "rule",
"ground_truth": "\\left\\lfloor \\frac{-1 + \\sqrt{1 + 8n}}{2} \\right\\rfloor"
} | {
"split": "train",
"index": 29842,
"solution": "\nTo find the maximum value of \\( m \\), we need to evaluate the constraints given by the problem, specifically that none of the subsets \\( A_1, A_2, \\ldots, A_m \\) is contained in any other subset. Each subset \\( A_i \\) has exactly \\( i \\) elements.\n\nLet... |
deepscaler | [
{
"role": "user",
"content": "A chord intercepted on the circle $x^{2}+y^{2}=4$ by the line $\\begin{cases} x=2-\\frac{1}{2}t \\\\ y=-1+\\frac{1}{2}t \\end{cases} (t\\text{ is the parameter})$ has a length of $\\_\\_\\_\\_\\_\\_\\_.$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\sqrt{14}"
} | {
"split": "train",
"index": 29843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a line $y = \\frac{\\sqrt{3}}{3}x$ and a circle $C$ with its center on the positive x-axis and a radius of 2 intersects the line at points $A$ and $B$ such that $|AB|=2\\sqrt{3}$.\n\n(1) Given a point $P(-1, \\sqrt{7})$, and $Q$ is any point on circle $C$, find the maximu... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 29844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non-empty subsets of $\\{ 1 , 2, 3, 4, 5, 6, 7, 8 \\}$ consist entirely of odd numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 29845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagonals of rectangle $PQRS$ intersect at point $X$. If $PS = 10$ and $RS=24$, then what is $\\cos \\angle PXS$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{119}{169}"
} | {
"split": "train",
"index": 29846,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The surface area of the circumscribed sphere of cube \\( K_1 \\) is twice the surface area of the inscribed sphere of cube \\( K_2 \\). Let \\( V_1 \\) denote the volume of the inscribed sphere of cube \\( K_1 \\), and \\( V_2 \\) denote the volume of the circumscribed sphere o... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2}}{27}"
} | {
"split": "train",
"index": 29847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a company of 100 children, some children are friends (friendship is always mutual). It is known that if any one child is excluded, the remaining 99 children can be divided into 33 groups of three such that in each group all three children are mutual friends. Find the minimum... | math | {
"style": "rule",
"ground_truth": "198"
} | {
"split": "train",
"index": 29848,
"solution": ""
} |
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