data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given the parabola $C$: $x^2 = 2py (p > 0)$ and the line $2x-y+2=0$, they intersect at points $A$ and $B$. A vertical line is drawn from the midpoint of the line segment $AB$ to the $x$-axis, intersecting the parabola $C$ at point $Q$. If $\\overrightarrow{QA} \\cdot \\overrigh... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 30049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Many television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is $4:3$. What is the horizontal length (in inches) of a ``27-inch'' television screen?\n\n[asy]\nfill((0,... | math | {
"style": "rule",
"ground_truth": "21.6"
} | {
"split": "train",
"index": 30050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin x\\cos x-\\sqrt{3}\\cos^{2}x$.\n\n(1) Find the smallest positive period of $f(x)$;\n\n(2) Find the maximum and minimum values of $f(x)$ when $x\\in[0,\\frac{\\pi }{2}]$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "-\\sqrt{3}"
} | {
"split": "train",
"index": 30051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$, and $C$ are sets for which $n(A)+n(B)+n(C)=n(A\\cup B\\cup C)$ and $|A|=|B|=100$, then what is the minim... | math | {
"style": "rule",
"ground_truth": "97"
} | {
"split": "train",
"index": 30052,
"solution": "1. **Given Information and Basic Calculations**:\n - $|A| = |B| = 100$, so $n(A) = n(B) = 2^{100}$.\n - The formula for the number of subsets of a set $S$ is $n(S) = 2^{|S|}$.\n - Given $n(A) + n(B) + n(C) = n(A \\cup B \\cup C)$, substituting the values, we ... |
deepscaler | [
{
"role": "user",
"content": "A rectangle has its length increased by $30\\%$ and its width increased by $15\\%$. Determine the percentage increase in the area of the rectangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49.5\\%"
} | {
"split": "train",
"index": 30053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a regular 2017-gon, all diagonals are drawn. Peter randomly selects some $\\mathrm{N}$ diagonals. What is the smallest $N$ such that there are guaranteed to be two diagonals of the same length among the selected ones? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"split": "train",
"index": 30054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the triangle bounded by the $y$-axis and the lines $y-3x=-2$ and $3y+x=12$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{27}{5}"
} | {
"split": "train",
"index": 30055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle with a radius of 5, its center lies on the x-axis with an integer horizontal coordinate and is tangent to the line 4x + 3y - 29 = 0.\n\n(1) Find the equation of the circle;\n\n(2) If the line ax - y + 5 = 0 (a ≠ 0) intersects the circle at points A and B, does th... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 30056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains 4 red, 3 blue, and 6 yellow marbles. What is the probability, expressed as a decimal, of drawing one red marble followed by one blue marble without replacement? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.076923"
} | {
"split": "train",
"index": 30057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sin 150^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 30058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The vectors $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ satisfy $\\|\\mathbf{a}\\| = \\|\\mathbf{b}\\| = 1,$ $\\|\\mathbf{c}\\| = 2,$ and\n\\[\\mathbf{a} \\times (\\mathbf{a} \\times \\mathbf{c}) + \\mathbf{b} = \\mathbf{0}.\\]If $\\theta$ is the angle between $\\mathbf{a}$... | math | {
"style": "rule",
"ground_truth": "150^\\circ"
} | {
"split": "train",
"index": 30059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x) = \\sin \\left( \\frac{\\pi}{3}x \\right)$, and the set $A = \\{1, 2, 3, 4, 5, 6, 7, 8\\}$. Now, choose any two distinct elements $s$ and $t$ from set $A$. Find out the number of possible pairs $(s, t)$ such that $f(s)\\cdot f(t) = 0$. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 30060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\sin (\\omega x+\\frac{\\pi}{3})$ ($\\omega>0$) is shifted to the left by $\\frac{\\pi}{2}$ units to obtain the curve $C$. If $C$ is symmetric about the $y$-axis, determine the minimum value of $\\omega$. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 30061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Factor $58x^5-203x^{11}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-29x^5(7x^6-2)"
} | {
"split": "train",
"index": 30062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "John ordered $4$ pairs of black socks and some additional pairs of blue socks. The price of the black socks per pair was twice that of the blue. When the order was filled, it was found that the number of pairs of the two colors had been interchanged. This increased the bill by ... | math | {
"style": "rule",
"ground_truth": "1:4"
} | {
"split": "train",
"index": 30063,
"solution": "1. **Define Variables:**\n Let $b$ be the number of pairs of blue socks John originally ordered. Assume the price per pair of blue socks is $x$. Therefore, the price per pair of black socks is $2x$ since it is twice that of the blue socks.\n\n2. **Set Up the Orig... |
deepscaler | [
{
"role": "user",
"content": "A certain school has $7$ members in its student art department (4 males and 3 females). Two members are to be selected to participate in the school's art performance event. \n$(1)$ Find the probability that only one female member is selected. \n$(2)$ Given that a male member,... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 30064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of finite roots that the equation\n\n$$\n\\left|x - a_{1}\\right| + \\ldots + |x - a_{50}| = \\left|x - b_{1}\\right| + \\ldots + |x - b_{50}|\n$$\n\ncan have, where $a_{1}, a_{2}, \\ldots, a_{50}, b_{1}, b_{2}, \\ldots, b_{50}$ are distinct numbers? ... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 30065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( ABC \\) be a triangle. The midpoints of the sides \\( BC \\), \\( AC \\), and \\( AB \\) are denoted by \\( D \\), \\( E \\), and \\( F \\) respectively.\n\nThe two medians \\( AD \\) and \\( BE \\) are perpendicular to each other and their lengths are \\(\\overline{AD}... | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"split": "train",
"index": 30066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g:[0,1]\\rightarrow \\mathbb{R}$ be a continuous function and let $f_{n}:[0,1]\\rightarrow \\mathbb{R}$ be a \nsequence of functions defined by $f_{0}(x)=g(x)$ and\n$$f_{n+1}(x)=\\frac{1}{x}\\int_{0}^{x}f_{n}(t)dt.$$\nDetermine $\\lim_{n\\to \\infty}f_{n}(x)$ for every $x\... | math | {
"style": "rule",
"ground_truth": "g(0)"
} | {
"split": "train",
"index": 30067,
"solution": "\nGiven a continuous function \\( g:[0,1] \\rightarrow \\mathbb{R} \\) and a sequence of functions \\( f_n:[0,1] \\rightarrow \\mathbb{R} \\) defined by \\( f_0(x) = g(x) \\) and\n\n\\[\nf_{n+1}(x) = \\frac{1}{x} \\int_{0}^{x} f_n(t) \\, dt,\n\\]\n\nour task is to ... |
deepscaler | [
{
"role": "user",
"content": "Let set $\\mathcal{A}$ be a 70-element subset of $\\{1,2,3,\\ldots,120\\}$, and let $S$ be the sum of the elements of $\\mathcal{A}$. Find the number of possible values of $S$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3501"
} | {
"split": "train",
"index": 30068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=f(x)$ that satisfies $f(-x)=-f(x)$ and $f(1+x)=f(1-x)$ for $x \\in [-1,1]$ with $f(x)=x^{3}$, find the value of $f(2015)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 30069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Before the district play, the Unicorns had won $45$% of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 30070,
"solution": "1. Let $y$ be the total number of games the Unicorns played before the district play, and $x$ be the number of games they won. According to the problem, they won 45% of these games, so we have the equation:\n \\[\n \\frac{x}{y} = 0.45\n \\]\n Multiplying bo... |
deepscaler | [
{
"role": "user",
"content": "If $10x+y=75$ and $10y+x=57$ for some positive integers $x$ and $y$, what is the value of $x+y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 30071,
"solution": "Since $10x+y=75$ and $10y+x=57$, then $(10x+y)+(10y+x)=75+57$ and so $11x+11y=132$. Dividing by 11, we get $x+y=12$. (We could have noticed initially that $(x, y)=(7,5)$ is a pair that satisfies the two equations, thence concluding that $x+y=12$.)"
} |
deepscaler | [
{
"role": "user",
"content": "Given a fixed integer \\( n \\) where \\( n \\geq 2 \\):\n\na) Determine the smallest constant \\( c \\) such that the inequality \\(\\sum_{1 \\leq i < j \\leq n} x_i x_j (x_i^2 + x_j^2) \\leq c \\left( \\sum_{i=1}^n x_i \\right)^4\\) holds for all nonnegative real numbers \\( ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 30072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $m$ and $n$ are positive integers such that $\\gcd(m,n) = 12$, then what is the smallest possible value of $\\gcd(10m,15n)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 30073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $1447$, $1005$ and $1231$ have something in common: each is a $4$-digit number beginning with $1$ that has exactly two identical digits. How many such numbers are there?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 30074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $x^3$ is divided by $x^2 + 5x + 1.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24x + 5"
} | {
"split": "train",
"index": 30075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\\overrightarrow {BA} \\cdot \\overrightarrow {AC} = 6$, $b-c=2$, and $\\tan A = -\\sqrt {15}$, find the length of the altitude drawn from $A$ to s... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{15}}{4}"
} | {
"split": "train",
"index": 30076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the two roots of the equation $x^{2}+3ax+3a+1=0$ where $a > 1$ are $\\tan \\alpha$ and $\\tan \\beta$, and $\\alpha, \\beta \\in \\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)$, find the value of $\\alpha + \\beta$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "-\\frac{3\\pi}{4}"
} | {
"split": "train",
"index": 30077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the side of a triangle, a point is taken such that an angle equals another angle. What is the smallest possible distance between the centers of the circles circumscribed around triangles, if \\( BC = 1 \\)? Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 30078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what value of $k$ is the line through the points $(3,2k+1)$ and $(8,4k-5)$ parallel to the $x$-axis? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 30079,
"solution": "A line segment joining two points is parallel to the $x$-axis exactly when the $y$-coordinates of the two points are equal. Here, this means that $2k+1=4k-5$ and so $6=2k$ or $k=3$. (We can check that when $k=3$, the coordinates of the points are $(3,7)$ and $(8,7)... |
deepscaler | [
{
"role": "user",
"content": "For how many integers \\(n\\) with \\(1 \\le n \\le 2020\\) is the product\n\\[\n \\prod_{k=0}^{n-1} \\left( \\left( 1 + e^{2 \\pi i k / n} \\right)^n + 1 \\right)\n\\]\nequal to zero? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "337"
} | {
"split": "train",
"index": 30080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Frank and Joe are playing ping pong. For each game, there is a $30 \\%$ chance that Frank wins and a $70 \\%$ chance Joe wins. During a match, they play games until someone wins a total of 21 games. What is the expected value of number of games played per match? Let's think ste... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 30081,
"solution": "The expected value of the ratio of Frank's to Joe's score is 3:7, so Frank is expected to win 9 games for each of Frank's 21. Thus the expected number of games in a match is 30."
} |
deepscaler | [
{
"role": "user",
"content": "The four faces of a tetrahedral die are labelled $0, 1, 2,$ and $3,$ and the die has the property that, when it is rolled, the die promptly vanishes, and a number of copies of itself appear equal to the number on the face the die landed on. For example, if it lands on the f... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 30082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three concentric circles have radii $5$ meters, $15$ meters, and $25$ meters respectively. Calculate the total distance a beetle travels, which starts at a point $P$ on the outer circle, moves inward along a radius to the middle circle, traces a one-third arc of the middle circ... | math | {
"style": "rule",
"ground_truth": "15\\pi + 70"
} | {
"split": "train",
"index": 30083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ and $q$ be positive integers such that \\[\\frac{3}{5} < \\frac{p}{q} < \\frac{5}{8}\\] and $q$ is as small as possible. What is $p+q$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 30084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a quadratic function $f(x) = ax^2 + bx + 1$ that satisfies $f(-1) = 0$, and when $x \\in \\mathbb{R}$, the range of $f(x)$ is $[0, +\\infty)$. \n(1) Find the expression for $f(x)$. \n(2) Let $g(x) = f(x) - 2kx$, where $k \\in \\mathbb{R}$. \n (i) If $g(x)$ is monoton... | math | {
"style": "rule",
"ground_truth": "k = 6"
} | {
"split": "train",
"index": 30085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A person has a three times higher probability of scoring a basket than missing it. Let random variable $X$ represent the number of scores in one shot. Then $P(X=1) = \\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"split": "train",
"index": 30086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $\\frac{1}{1001}$ is expressed as a decimal, what is the sum of the first 50 digits after the decimal point? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 30087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two unit vectors $a$ and $b$, and $|a-2b| = \\sqrt{7}$, then the angle between $a$ and $b$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 30088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $(a_1,a_2,\\ldots, a_{13})$ be a permutation of $(1, 2, \\ldots, 13)$ . Ayvak takes this permutation and makes a series of *moves*, each of which consists of choosing an integer $i$ from $1$ to $12$ , inclusive, and swapping the positions of $a_i$ and $a_{i+1}$ .... | math | {
"style": "rule",
"ground_truth": "13703"
} | {
"split": "train",
"index": 30089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle with integer length and width has a perimeter of 120 units. What is the number of square units in the least possible area, assuming at least one of the dimensions is a prime number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "116"
} | {
"split": "train",
"index": 30090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given any two positive real numbers $x$ and $y$, then $x \\diamond y$ is a positive real number defined in terms of $x$ and $y$ by some fixed rule. Suppose the operation $x \\diamond y$ satisfies the equations \\((x \\cdot y) \\diamond y=x(y \\diamond y)\\) and \\((x \\diamond ... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 30091,
"solution": "Note first that \\(x \\diamond 1=(x \\cdot 1) \\diamond 1=x \\cdot(1 \\diamond 1)=x \\cdot 1=x\\). Also, \\(x \\diamond x=(x \\diamond 1) \\diamond x=x \\diamond 1=x\\). Now, we have \\((x \\cdot y) \\diamond y=x \\cdot(y \\diamond y)=x \\cdot y\\). So \\(19 \\diam... |
deepscaler | [
{
"role": "user",
"content": "How many positive integer values of $x$ are solutions to the inequality $10 < -x + 13$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 30092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be an isosceles triangle with $AB=AC$ and incentre $I$ . If $AI=3$ and the distance from $I$ to $BC$ is $2$ , what is the square of length on $BC$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 30093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the integer $n$, $12 \\le n \\le 18$, such that \\[n \\equiv 9001 \\pmod{7}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 30094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of permutations $(a_1, a_2, a_3, a_4, a_5, a_6, a_7)$ of $(1,2,3,4,5,6,7)$ that satisfy\n\\[\\frac{a_1 + 1}{2} \\cdot \\frac{a_2 + 2}{2} \\cdot \\frac{a_3 + 3}{2} \\cdot \\frac{a_4 + 4}{2} \\cdot \\frac{a_5 + 5}{2} \\cdot \\frac{a_6 + 6}{2} \\cdot \\frac{a_7 + 7... | math | {
"style": "rule",
"ground_truth": "5039"
} | {
"split": "train",
"index": 30095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all real numbers \\( x \\) that satisfy the equation\n$$\n\\frac{x-2020}{1} + \\frac{x-2019}{2} + \\cdots + \\frac{x-2000}{21} = \\frac{x-1}{2020} + \\frac{x-2}{2019} + \\cdots + \\frac{x-21}{2000},\n$$\nand simplify your answer(s) as much as possible. Justify your solutio... | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"split": "train",
"index": 30096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1, 2, 3, 5"
} | {
"split": "train",
"index": 30097,
"solution": "\nTo solve this problem, let's denote the four different natural numbers as \\( a \\), \\( b \\), \\( c \\), and \\( d \\). The condition given is that for any pair of these numbers, say \\( (a, b) \\), the expression:\n\n\\[\nab + cd\n\\]\n\nmust be a prime number... |
deepscaler | [
{
"role": "user",
"content": "A five-digit number is called irreducible if it cannot be expressed as a product of two three-digit numbers.\n\nWhat is the greatest number of consecutive irreducible five-digit numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 30098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On an island, there are red, yellow, green, and blue chameleons.\n\n- On a cloudy day, either one red chameleon changes its color to yellow, or one green chameleon changes its color to blue.\n- On a sunny day, either one red chameleon changes its color to green, or one yellow c... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 30099,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,3), (3,3), and (3,0). What is the probability that $x+y < 5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{17}{18}"
} | {
"split": "train",
"index": 30100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Math City has ten streets, none of which are parallel, and some of which can intersect more than once due to their curved nature. There are two curved streets which each make an additional intersection with three other streets. Calculate the maximum number of police officers ne... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 30101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Team A and Team B have a table tennis team match. Each team has three players, and each player plays once. Team A's three players are \\( A_{1}, A_{2}, A_{3} \\) and Team B's three players are \\( B_{1}, B_{2}, B_{3} \\). The winning probability of \\( A_{i} \\) against \\( B_{... | math | {
"style": "rule",
"ground_truth": "91/60"
} | {
"split": "train",
"index": 30102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of rooks that can be placed on a $300 \\times 300$ chessboard such that each rook attacks at most one other rook? (A rook attacks all the squares it can reach according to chess rules without passing through other pieces.) Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 30103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four normal students, A, B, C, and D, are to be assigned to work at three schools, School A, School B, and School C, with at least one person at each school. It is known that A is assigned to School A. What is the probability that B is assigned to School B? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{12}"
} | {
"split": "train",
"index": 30104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $M$ , $N$ , $P$ are selected on sides $\\overline{AB}$ , $\\overline{AC}$ , $\\overline{BC}$ , respectively, of triangle $ABC$ . Find the area of triangle $MNP$ given that $AM=MB=BP=15$ and $AN=NC=CP=25$ .\n\n*Proposed by Evan Chen* Let's think step by step... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 30105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find\n\\[\\cos \\left( 6 \\arccos \\frac{1}{3} \\right).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{329}{729}"
} | {
"split": "train",
"index": 30106,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\\overline{AC}$ of isosceles right triangle $ABC$ with leg length $2$, and $\\angle CAD = 30^{\\circ}$, find $\\sin(2\\angle BAD)$. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 30107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\\%$. Find the maximum discount rate that can be offered. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "12\\%"
} | {
"split": "train",
"index": 30108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many five-digit numbers divisible by 3 are there that include the digit 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12504"
} | {
"split": "train",
"index": 30109,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a geometric sequence $\\{b_n\\}$ with a common ratio of $4$, if $T_n$ is the product of the first $n$ terms of the sequence $\\{b_n\\}$, then $\\frac{T_{20}}{T_{10}}$, $\\frac{T_{30}}{T_{20}}$, $\\frac{T_{40}}{T_{30}}$ also form a geometric sequence with a common ratio of $... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 30110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the real solution(s) to the equation $(x+y)^{2}=(x+1)(y-1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-1,1)"
} | {
"split": "train",
"index": 30111,
"solution": "Set $p=x+1$ and $q=y-1$, then we get $(p+q)^{2}=pq$, which simplifies to $p^{2}+pq+q^{2}=0$. Then we have $\\left(p+\\frac{q}{2}\\right)^{2}+\\frac{3q^{2}}{4}$, and so $p=q=0$. Thus $(x, y)=(-1,1)$."
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers $n,$ $1 \\le n \\le 2600,$ for which the polynomial $x^2 + x - n$ can be factored as the product of two linear factors with integer coefficients. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 30112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $xy = b$ and $\\frac{1}{x^2} + \\frac{1}{y^2} = a$, then $(x + y)^2$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "b(ab + 2)"
} | {
"split": "train",
"index": 30113,
"solution": "Given the equations:\n1. \\( xy = b \\)\n2. \\( \\frac{1}{x^2} + \\frac{1}{y^2} = a \\)\n\nWe need to find \\( (x+y)^2 \\).\n\nFirst, let's express \\( \\frac{1}{x^2} + \\frac{1}{y^2} \\) in terms of \\( x \\) and \\( y \\):\n\\[ \\frac{1}{x^2} + \\frac{1}{y^2} = \... |
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": 30114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The faces of a cubical die are marked with the numbers $1$, $2$, $3$, $3$, $4$, and $5$. Another die is marked with $2$, $3$, $4$, $6$, $7$, and $9$. What is the probability that the sum of the top two numbers will be $6$, $8$, or $10$?\nA) $\\frac{8}{36}$\nB) $\\frac{11}{36}$\... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{36}"
} | {
"split": "train",
"index": 30115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(2\\sin \\omega x,2\\cos \\omega x)$ and $\\overrightarrow{b}=(\\sqrt{3}\\cos\\omega x,-\\cos\\omega x)$, where the function $f(x)=\\overrightarrow{a}\\cdot\\overrightarrow{b}$ has a minimum positive period of $6\\pi$. Find the value of the re... | math | {
"style": "rule",
"ground_truth": "-\\frac{16}{65}"
} | {
"split": "train",
"index": 30116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A parametric graph is defined by:\n\\[\nx = \\cos t + \\frac{t}{3}, \\quad y = \\sin t.\n\\]\nDetermine the number of times the graph intersects itself between \\(x = 3\\) and \\(x = 45\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 30117,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( m > n \\geqslant 1 \\), find the smallest value of \\( m + n \\) such that\n\\[ 1000 \\mid 1978^{m} - 1978^{n} . \\ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 30118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of positive integers $x$, $y$ and $z$ equals 2004. What is the minimum possible value of the sum $x + y + z$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "174"
} | {
"split": "train",
"index": 30119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find \\(\\cos 2 \\alpha\\), given that \\(2 \\operatorname{ctg}^{2} \\alpha+7 \\operatorname{ctg} \\alpha+3=0\\) and the value of \\(\\alpha\\) satisfies the inequalities: \na) \\(\\frac{3 \\pi}{2}<\\alpha<\\frac{7 \\pi}{4}\\);\nb) \\(\\frac{7 \\pi}{4}<\\alpha<2 \\pi\\). Let's ... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 30120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pedestrian traffic light allows pedestrians to cross the street for one minute and prohibits crossing for two minutes. Find the average waiting time for a pedestrian who approaches the intersection. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 30121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a type of four-digit number where the sum of any two adjacent digits is no greater than 2. When these numbers are arranged in ascending order, what is the second to last number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2011"
} | {
"split": "train",
"index": 30122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If circle $$C_{1}: x^{2}+y^{2}+ax=0$$ and circle $$C_{2}: x^{2}+y^{2}+2ax+ytanθ=0$$ are both symmetric about the line $2x-y-1=0$, then $sinθcosθ=$ \\_\\_\\_\\_\\_\\_ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{5}"
} | {
"split": "train",
"index": 30123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Observing the equations:<br/>$1\\times 3+1=4=2^{2}$;<br/>$2\\times 4+1=9=3^{2}$;<br/>$3\\times 5+1=16=4^{2}$;<br/>$4\\times 6+1=25=5^{2}$;<br/>$\\ldots $<br/>$(1)7\\times 9+1=( $______)$^{2}$;<br/>$(2)$ Using the pattern you discovered, calculate: $(1+\\frac{1}{1×3})×(1+\\frac{... | math | {
"style": "rule",
"ground_truth": "\\frac{199}{100}"
} | {
"split": "train",
"index": 30124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all real numbers $x$ for which the median of the numbers $4,6,8,17,$ and $x$ is equal to the mean of those five numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 30125,
"solution": "1. **Calculate the Mean:**\n The mean of the numbers $4, 6, 8, 17,$ and $x$ is given by:\n \\[\n \\text{Mean} = \\frac{4+6+8+17+x}{5} = \\frac{35+x}{5}\n \\]\n\n2. **Identify Possible Medians:**\n The median of these numbers depends on the value of $x$. T... |
deepscaler | [
{
"role": "user",
"content": "What is the instantaneous velocity of the robot at the moment $t=2$ given the robot's motion equation $s = t + \\frac{3}{t}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{4}"
} | {
"split": "train",
"index": 30126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:\n\n 1. The largest isn't first, but it is in one of the first three places. \n 2. The smallest isn't last, but it is in one of the last three places. \n 3. The median isn't first or... | math | {
"style": "rule",
"ground_truth": "6.5"
} | {
"split": "train",
"index": 30127,
"solution": "1. **Identify the largest and smallest numbers**: The largest number in the set $\\{-2, 4, 6, 9, 12\\}$ is $12$, and the smallest is $-2$.\n\n2. **Apply rule 1**: The largest number, $12$, cannot be first but must be in one of the first three places. Thus, $12$ can... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f$ mapping from set $M$ to set $N$, where $M=\\{a, b, c\\}$ and $N=\\{-3, -2, -1, 0, 1, 2, 3\\}$, how many mappings $f$ satisfy the condition $f(a) + f(b) + f(c) = 0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 30128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the point \\( P \\) lies in the plane of the right triangle \\( \\triangle ABC \\) with \\( \\angle BAC = 90^\\circ \\), and \\( \\angle CAP \\) is an acute angle. Also given are the conditions:\n\\[ |\\overrightarrow{AP}| = 2, \\quad \\overrightarrow{AP} \\cdot \\overrig... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 30129,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x_1, x_2, \\ldots, x_n$ be integers, satisfying:\n(1) $-1 \\leq x_i \\leq 2$, for $i=1, 2, \\ldots, n$;\n(2) $x_1 + x_2 + \\ldots + x_n = 19$;\n(3) $x_1^2 + x_2^2 + \\ldots + x_n^2 = 99$.\nFind the maximum and minimum values of $x_1^3 + x_2^3 + \\ldots + x_n^3$. Let's thin... | math | {
"style": "rule",
"ground_truth": "133"
} | {
"split": "train",
"index": 30130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Company $W$'s product $p$ produced by line $D$ follows a normal distribution $N(80, 0.25)$ in terms of size. $400$ randomly selected products were tested from the current production line, and the size distribution is summarized in the table below:\n\n| Product Size ($mm$) | $[7... | math | {
"style": "rule",
"ground_truth": "\\frac{57}{4}"
} | {
"split": "train",
"index": 30131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that the number $\\sqrt{2700} - 37$ can be expressed in the form $(\\sqrt a - b)^3,$ where $a$ and $b$ are positive integers. Find $a+b.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 30132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The instantaneous rate of change of carbon-14 content is $-\\frac{\\ln2}{20}$ (becquerel/year) given that at $t=5730$. Using the formula $M(t) = M_0 \\cdot 2^{-\\frac{t}{5730}}$, determine $M(2865)$. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "573\\sqrt{2}/2"
} | {
"split": "train",
"index": 30133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the binomial coefficient of only the sixth term in the expansion of $(\\sqrt{x} - \\frac{2}{x^{2}})^{n}$ is the largest, then the constant term in the expansion is _______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 30134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For natural numbers _m_ greater than or equal to 2, the _n_-th power of _m_ can be decomposed as follows: \n2<sup>2</sup> = 1 + 3, 3<sup>2</sup> = 1 + 3 + 5, 4<sup>2</sup> = 1 + 3 + 5 + 7… \n2<sup>3</sup> = 3 + 5, 3<sup>3</sup> = 7 + 9 + 11… \n2<sup>4</sup> = 7 + 9… \nAccor... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 30135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\cos \\left( \\arcsin \\frac{2}{3} \\right).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{3}"
} | {
"split": "train",
"index": 30136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Phoenix hiked the Rocky Path Trail last week. It took four days to complete the trip. The first two days she hiked a total of 22 miles. The second and third days she averaged 13 miles per day. The last two days she hiked a total of 30 miles. The total hike for the first and thi... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 30137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\sqrt{3}\\sin x\\cos x+2\\cos^{2}x-1$.\n\n(I) Find the axis of symmetry and the center of symmetry of $f(x)$;\n\n(II) Find the maximum and minimum values of $f(x)$ on the interval $\\left[-\\frac{\\pi }{6}, \\frac{\\pi }{4}\\right]$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 30138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the following sequence of sets of natural numbers. The first set \\( I_{0} \\) consists of two ones, 1,1. Then, between these numbers, we insert their sum \\( 1+1=2 \\); we obtain the set \\( I_{1}: 1,2,1 \\). Next, between each pair of numbers in \\( I_{1} \\) we inse... | math | {
"style": "rule",
"ground_truth": "1972"
} | {
"split": "train",
"index": 30139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40\\pi"
} | {
"split": "train",
"index": 30140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of the distinct solutions to the equation $|x-|2x+1||=3$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 30141,
"solution": "To solve the equation $|x - |2x + 1|| = 3$, we need to consider the different cases that arise from the absolute values.\n\n#### Step 1: Analyze the inner absolute value\nThe expression $|2x + 1|$ can be simplified by considering two cases:\n- **Case 1:** $2x + 1 \... |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ where each term is positive, and $a_1a_3=4$, $a_7a_9=25$, find $a_5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"split": "train",
"index": 30142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. Given that $C = \\frac{2\\pi}{3}$ and $a = 6$: \n(Ⅰ) If $c = 14$, find the value of $\\sin A$; \n(Ⅱ) If the area of $\\triangle ABC$ is $3\\sqrt{3}$, find the va... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{13}"
} | {
"split": "train",
"index": 30143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define two binary operations on real numbers where $a \\otimes b = \\frac{a+b}{a-b}$ and $b \\oplus a = \\frac{b-a}{b+a}$. Compute the value of $(8\\otimes 6) \\oplus 2$.\nA) $\\frac{5}{9}$ \nB) $\\frac{7}{9}$ \nC) $\\frac{12}{9}$ \nD) $\\frac{1}{9}$ \nE) $\\frac{14}{9}$ L... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"split": "train",
"index": 30144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A polynomial $P$ has four roots, $\\frac{1}{4}, \\frac{1}{2}, 2,4$. The product of the roots is 1, and $P(1)=1$. Find $P(0)$. 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": 30145,
"solution": "A polynomial $Q$ with $n$ roots, $x_{1}, \\ldots, x_{n}$, and $Q\\left(x_{0}\\right)=1$ is given by $Q(x)=\\frac{\\left(x-x_{1}\\right)\\left(x-x_{2}\\right) \\cdots\\left(x-x_{n}\\right)}{\\left(x_{0}-x_{1}\\right)\\left(x_{0}-x_{2}\\right) \\cdots\\left(x_{0}-x_{... |
deepscaler | [
{
"role": "user",
"content": "Determine $x^2+y^2+z^2+w^2$ if\n$\\frac{x^2}{2^2-1}+\\frac{y^2}{2^2-3^2}+\\frac{z^2}{2^2-5^2}+\\frac{w^2}{2^2-7^2}=1$\n$\\frac{x^2}{4^2-1}+\\frac{y^2}{4^2-3^2}+\\frac{z^2}{4^2-5^2}+\\frac{w^2}{4^2-7^2}=1$\n$\\frac{x^2}{6^2-1}+\\frac{y^2}{6^2-3^2}+\\frac{z^2}{6^2-5^2}+\\frac{w^2... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 30146,
"solution": "As in Solution 1, we have\n$(t-1)(t-9)(t-25)(t-49)-x^2(t-9)(t-25)(t-49)-y^2(t-1)(t-25)(t-49)$ $-z^2(t-1)(t-9)(t-49)-w^2(t-1)(t-9)(t-25)$\n$=(t-4)(t-16)(t-36)(t-64)$\n Now the coefficient of $t^3$ on both sides must be equal. So instead of expanding it fully, we wil... |
deepscaler | [
{
"role": "user",
"content": "Three points are chosen inside a unit cube uniformly and independently at random. What is the probability that there exists a cube with side length $\\frac{1}{2}$ and edges parallel to those of the unit cube that contains all three points? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 30147,
"solution": "Let the unit cube be placed on a $x y z$-coordinate system, with edges parallel to the $x, y, z$ axes. Suppose the three points are labeled $A, B, C$. If there exists a cube with side length $\\frac{1}{2}$ and edges parallel to the edges of the unit cube that conta... |
deepscaler | [
{
"role": "user",
"content": "Find the number of second-degree polynomials $f(x)$ with integer coefficients and integer zeros for which $f(0)=2010$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "163"
} | {
"split": "train",
"index": 30148,
"solution": "We use Burnside's Lemma. The set being acted upon is the set of integer triples $(a,r,s)$ such that $ars=2010$. Because $r$ and $s$ are indistinguishable, the permutation group consists of the identity and the permutation that switches $r$ and $s$. In cycle notatio... |
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