data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let \\( M = \\{1, 2, 3, \\ldots, 1995\\} \\). Subset \\( A \\) of \\( M \\) satisfies the condition: If \\( x \\in A \\), then \\( 15x \\notin A \\). What is the maximum number of elements in \\( A \\)? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "1870"
} | {
"split": "train",
"index": 17449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From a group of six students, how many different four-student committees can be chosen? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 17450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An \\( n \\)-digit number \\( x \\) has the following property: if the last digit of \\( x \\) is moved to the front, the result is \\( 2x \\). Find the smallest possible value of \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 17451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( [x] \\) denote the greatest integer not exceeding \\( x \\), e.g., \\( [\\pi]=3 \\), \\( [5.31]=5 \\), and \\( [2010]=2010 \\). Given \\( f(0)=0 \\) and \\( f(n)=f\\left(\\left[\\frac{n}{2}\\right]\\right)+n-2\\left[\\frac{n}{2}\\right] \\) for any positive integer \\( ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 17452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the value of $y$ is doubled and then this increased value is divided by 5, the result is 10. What is the value of $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 17453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\tan\\alpha= \\frac {1}{2}$ and $\\tan(\\alpha-\\beta)=- \\frac {2}{5}$, calculate the value of $\\tan(2\\alpha-\\beta)$. 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": 17454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a meeting is convened with 2 representatives from one company and 1 representative from each of the other 3 companies, find the probability that 3 randomly selected individuals to give speeches come from different companies. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "0.6"
} | {
"split": "train",
"index": 17455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( ABC \\) be any triangle. Let \\( D \\) and \\( E \\) be points on \\( AB \\) and \\( BC \\) respectively such that \\( AD = 7DB \\) and \\( BE = 10EC \\). Assume that \\( AE \\) and \\( CD \\) meet at a point \\( F \\). Determine \\( \\lfloor k \\rfloor \\), where \\( k... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"split": "train",
"index": 17456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The reality game show Survivor is played with 16 people divided into two tribes of 8. In the first episode, two people get homesick and quit. If every person has an equal chance of being one of the two quitters, and the probability that one person quits is independent of the pr... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{15}"
} | {
"split": "train",
"index": 17457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sasa wants to make a pair of butterfly wings for her Barbie doll. As shown in the picture, she first drew a trapezoid and then drew two diagonals, which divided the trapezoid into four triangles. She cut off the top and bottom triangles, and the remaining two triangles are exac... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 17458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An isosceles triangle has sides with lengths that are composite numbers, and the square of the sum of the lengths is a perfect square. What is the smallest possible value for the square of its perimeter? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 17459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number that leaves a remainder of 2 when divided by 3, 4, 6, and 8. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 17460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graphs of $y=3-x^2+x^3$ and $y=1+x^2+x^3$ intersect in multiple points. Find the maximum difference between the $y$-coordinates of these intersection points. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 17461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frus... | math | {
"style": "rule",
"ground_truth": "512"
} | {
"split": "train",
"index": 17462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Side $AB$ of triangle $ABC$ was divided into $n$ equal parts (dividing points $B_0 = A, B_1, B_2, ..., B_n = B$ ), and side $AC$ of this triangle was divided into $(n + 1)$ equal parts (dividing points $C_0 = A, C_1, C_2, ..., C_{n+1} = C$ ). Colored are the triangl... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 17463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A metallic weight has a mass of 25 kg and is an alloy of four metals. The first metal in this alloy is one and a half times more than the second; the mass of the second metal is related to the mass of the third as \\(3: 4\\), and the mass of the third metal to the mass of the f... | math | {
"style": "rule",
"ground_truth": "7.36"
} | {
"split": "train",
"index": 17464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the space vectors $\\overrightarrow{{e_1}}$ and $\\overrightarrow{{e_2}}$ satisfy $|\\overrightarrow{{e_1}}|=|2\\overrightarrow{{e_1}}+\\overrightarrow{{e_2}}|=3$, determine the maximum value of the projection of $\\overrightarrow{{e_1}}$ in the direction of $\\overrightarro... | math | {
"style": "rule",
"ground_truth": "-\\frac{3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 17465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each integer $n$ greater than 1, let $G(n)$ be the number of solutions of the equation $\\sin x = \\sin (n+1)x$ on the interval $[0, 2\\pi]$. Calculate $\\sum_{n=2}^{100} G(n)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10296"
} | {
"split": "train",
"index": 17466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Set \\( S \\) satisfies the following conditions:\n1. The elements of \\( S \\) are positive integers not exceeding 100.\n2. For any \\( a, b \\in S \\) where \\( a \\neq b \\), there exists \\( c \\in S \\) different from \\( a \\) and \\( b \\) such that \\(\\gcd(a + b, c) = ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 17467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of the expression \\( x^{2} + y^{2} \\) if \\( |x-y| \\leq 2 \\) and \\( |3x + y| \\leq 6 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 17468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are some identical square pieces of paper. If a part of them is paired up to form rectangles with a length twice their width, the total perimeter of all the newly formed rectangles is equal to the total perimeter of the remaining squares. Additionally, the total perimeter... | math | {
"style": "rule",
"ground_truth": "280"
} | {
"split": "train",
"index": 17469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of the expression for \\( a, b > 0 \\):\n\n$$\n\\frac{|4a - 10b| + |2(a - b\\sqrt{3}) - 5(a\\sqrt{3} + b)|}{\\sqrt{a^2 + b^2}}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{87}"
} | {
"split": "train",
"index": 17470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\frac{1}{(-5^2)^3} \\cdot (-5)^8 \\cdot \\sqrt{5}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5^{5/2}"
} | {
"split": "train",
"index": 17471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Grace has $\\$4.80$ in U.S. coins. She has the same number of dimes and pennies. What is the greatest number of dimes she could have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 17472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the average of all the integer values of $M$ such that $\\frac{M}{70}$ is strictly between $\\frac{2}{5}$ and $\\frac{3}{10}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24.5"
} | {
"split": "train",
"index": 17473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Of the 50 students in the drama club, 36 take mathematics, 27 take physics and 20 students take both mathematics and physics. How many drama club students take neither mathematics nor physics? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 17474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a sequence satisfies $x_0=0$ and $|x_k|=|x_{k-1}+3|$ for all integers $k\\ge 1,$ find the minimum possible value of $|x_1+x_2+\\cdots+x_{2006}|.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 17475,
"solution": "Playing around with a couple numbers, we see that we can generate the sequence $0, 3, -6, 3, -6, \\cdots$, and we can also generate the sequence $3, 6, 9, 12, \\cdots$ after each $-6$ value. Thus, we will apply this to try and find some bounds. We can test if the f... |
deepscaler | [
{
"role": "user",
"content": "A seven-digit number has the following properties: the hundreds digit is twice the ten millions digit, the tens digit is twice the hundred thousands digit, the units digit is twice the ten thousands digit, the thousands digit is 0, and it must be divisible by a five-digit numbe... | math | {
"style": "rule",
"ground_truth": "10002"
} | {
"split": "train",
"index": 17476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nA printer received an annual order to print 10,000 posters each month. Each month's poster should have the month's name printed on it. Thus, he needs to print 10,000 posters with the word \"JANUARY\", 10,000 posters with the word \"FEBRUARY\", 10,000 posters with the word \"M... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 17477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Maria baked 60 cakes, of which one-third contained strawberries, half contained blueberries, three-fifths contained raspberries, and one-tenth contained coconut flakes. What is the largest possible number of cakes that had none of these ingredients? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 17478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express as a fraction in lowest terms: $0.\\overline{23} + 0.\\overline{056} + 0.\\overline{004}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{28917}{98901}"
} | {
"split": "train",
"index": 17479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the probability of failing on the first attempt and succeeding on the second attempt to guess the last digit of a phone number is requested, calculate the probability of this event. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"split": "train",
"index": 17480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be constants, and suppose that the inequality \\[\\frac{(x-a)(x-b)}{x-c} \\le 0\\]is true if and only if either $x < -4$ or $|x-25| \\le 1.$ Given that $a < b,$ find the value of $a + 2b + 3c.$ Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 17481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the perimeter of a square is 28, what is the side length of the square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 17482,
"solution": "Since a square has four equal sides, the side length of a square equals one-quarter of the perimeter of the square. Thus, the side length of a square with perimeter 28 is $28 \\div 4 = 7$."
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the integers from $-30$ to $50$, inclusive? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "810"
} | {
"split": "train",
"index": 17483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the cube whose vertices are the eight points $(x, y, z)$ for which each of $x, y$, and $z$ is either 0 or 1 . How many ways are there to color its vertices black or white such that, for any vertex, if all of its neighbors are the same color then it is also that color? ... | math | {
"style": "rule",
"ground_truth": "118"
} | {
"split": "train",
"index": 17484,
"solution": "Divide the 8 vertices of the cube into two sets $A$ and $B$ such that each set contains 4 vertices, any two of which are diagonally adjacent across a face of the cube. We do casework based on the number of vertices of each color in set $A$. - Case 1: 4 black. Then ... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest four-digit palindrome that is divisible by 4? (A palindrome is a number that reads the same forwards and backwards, like 61216.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2112"
} | {
"split": "train",
"index": 17485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point $P$ is on curve $C_1: y^2 = 8x$ and point $Q$ is on curve $C_2: (x-2)^2 + y^2 = 1$. If $O$ is the coordinate origin, find the maximum value of $\\frac{|OP|}{|PQ|}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{7}}{7}"
} | {
"split": "train",
"index": 17486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square sheet of paper $ABCD$ is folded straight in such a way that point $B$ hits to the midpoint of side $CD$ . In what ratio does the fold line divide side $BC$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5/3"
} | {
"split": "train",
"index": 17487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Polyhedron $ABCDEFG$ has six faces. Face $ABCD$ is a square with $AB = 12;$ face $ABFG$ is a trapezoid with $\\overline{AB}$ parallel to $\\overline{GF},$ $BF = AG = 8,$ and $GF = 6;$ and face $CDE$ has $CE = DE = 14.$ The other three faces are $ADEG, BCEF,$ and $EFG.$ The dist... | math | {
"style": "rule",
"ground_truth": "163"
} | {
"split": "train",
"index": 17488,
"solution": "We let $A$ be the origin, or $(0,0,0)$, $B = (0,0,12)$, and $D = (12,0,0)$. Draw the perpendiculars from F and G to AB, and let their intersections be X and Y, respectively. By symmetry, $FX = GY = \\frac{12-6}2 = 3$, so $G = (a,b,3)$, where a and b are variables.\... |
deepscaler | [
{
"role": "user",
"content": "Through two vertices of an equilateral triangle \\(ABC\\) with an area of \\(21 \\sqrt{3} \\ \\text{cm}^2\\), a circle is drawn such that two sides of the triangle are tangent to the circle. Find the radius of this circle. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{7}"
} | {
"split": "train",
"index": 17489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Angelica wants to choose a three-digit code for her suitcase lock. To make it easier to remember, Angelica wants all the digits in her code to be in non-decreasing order. How many different possible codes does Angelica have to choose from? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "220"
} | {
"split": "train",
"index": 17490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB = 13$, $BC = 15$, and $CA = 14$. Point $D$ is on $\\overline{BC}$ with $CD = 6$. Point $E$ is on $\\overline{BC}$ such that $\\angle BAE = \\angle CAD$. Find $BE.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2535}{463}"
} | {
"split": "train",
"index": 17491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, there are points $P_{1}, P_{2}, \\ldots, P_{n-1}, P_{n}, \\ldots (n \\in \\mathbb{N}^{*})$. Let the coordinates of point $P_{n}$ be $(n, a_{n})$, where $a_{n}= \\frac {2}{n} (n \\in \\mathbb{N}^{*})$. The line passing through points $P_{n}$ a... | math | {
"style": "rule",
"ground_truth": "\\frac {125}{6}"
} | {
"split": "train",
"index": 17492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\(a\\) and \\(b\\) are real numbers, and the equation \\( x^{4} + a x^{3} + b x^{2} + a x + 1 = 0 \\) has at least one real root, find the minimum value of \\(a^{2} + b^{2}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4/5"
} | {
"split": "train",
"index": 17493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f: \\mathbb{Z}^{2} \\rightarrow \\mathbb{Z}$ satisfies - $f(x, 0)=f(0, y)=0$, and - $f(x, y)=f(x-1, y)+f(x, y-1)+x+y$ for all nonnegative integers $x$ and $y$. Find $f(6,12)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "77500"
} | {
"split": "train",
"index": 17494,
"solution": "We claim $f(x, y)=\\binom{x+y+2}{x+1}-(x+y+2)$. Indeed, the hypothesis holds true for our base cases $f(x, 0)$ and $f(0, y)$, and moreover, $f(x-1, y)+f(x, y-1)+x+y=\\binom{x+y+1}{x}+\\binom{x+y+1}{x+1}-2(x+y+1)+x+y=\\binom{x+y+2}{x+1}-(x+y+2)$. Thus, the final ans... |
deepscaler | [
{
"role": "user",
"content": "Eight identical cubes with of size $1 \\times 1 \\times 1$ each have the numbers $1$ through $6$ written on their faces with the number $1$ written on the face opposite number $2$ , number $3$ written on the face opposite number $5$ , and number $4$ written on the... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 17495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are eleven positive integers $n$ such that there exists a convex polygon with $n$ sides whose angles, in degrees, are unequal integers that are in arithmetic progression. Find the sum of these values of $n$. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 17496,
"solution": "The sum of the angles of an $n$-gon is $(n-2) 180$, so the average angle measure is $(n-2) 180 / n$. The common difference in this arithmetic progression is at least 1 , so the difference between the largest and smallest angles is at least $n-1$. So the largest ang... |
deepscaler | [
{
"role": "user",
"content": "If $\frac{1}{2n} + \frac{1}{4n} = \frac{3}{12}$, what is the value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 17497,
"solution": "Since $\frac{1}{2n} + \frac{1}{4n} = \frac{2}{4n} + \frac{1}{4n} = \frac{3}{4n}$, then the given equation becomes $\frac{3}{4n} = \frac{3}{12}$ or $4n = 12$. Thus, $n = 3$."
} |
deepscaler | [
{
"role": "user",
"content": "In a particular state, the design of vehicle license plates was changed from an old format to a new one. Under the old scheme, each license plate consisted of two letters followed by three digits (e.g., AB123). The new scheme is made up of four letters followed by two digits (e... | math | {
"style": "rule",
"ground_truth": "\\frac{26^2}{10}"
} | {
"split": "train",
"index": 17498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a Queen and the second card is a $\\diamondsuit$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{289}{15068}"
} | {
"split": "train",
"index": 17499,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the real numbers $a$, $b$, $c$, $d$ that satisfy $$\\frac {a-2e^{a}}{b}= \\frac {2-c}{d-1}=1$$, where $e$ is the base of the natural logarithm, find the minimum value of $(a-c)^2+(b-d)^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{2}"
} | {
"split": "train",
"index": 17500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The three numbers $5, a, b$ have an average (mean) of 33. What is the average of $a$ and $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 17501,
"solution": "Since $5, a, b$ have an average of 33, then $\\frac{5+a+b}{3}=33$. Multiplying by 3, we obtain $5+a+b=3 \\times 33=99$, which means that $a+b=94$. The average of $a$ and $b$ is thus equal to $\\frac{a+b}{2}=\\frac{94}{2}=47$."
} |
deepscaler | [
{
"role": "user",
"content": "A soccer team has 16 members. We need to select a starting lineup including a goalkeeper, a defender, a midfielder, and two forwards. However, only 3 players can play as a goalkeeper, 5 can play as a defender, 8 can play as a midfielder, and 4 players can play as forwards. In h... | math | {
"style": "rule",
"ground_truth": "1440"
} | {
"split": "train",
"index": 17502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $a_0 + a_1 + a_2 + \\cdots + a_6$ given that $(2-x)^7 = a_0 + a_1(1+x)^2 + \\cdots + a_7(1+x)^7$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "129"
} | {
"split": "train",
"index": 17503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Aerith timed herself solving a contest and noted the time both as days:hours:minutes:seconds and in seconds. For example, if she spent 1,000,000 seconds, she recorded it as 11:13:46:40 and 1,000,000 seconds. Bob subtracts these numbers, ignoring punctuation. In this case, he co... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 17504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A high school basketball team has 12 players, including a set of twins, John and James. In how many ways can we choose a starting lineup of 5 players if exactly one of the twins must be in the lineup? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "660"
} | {
"split": "train",
"index": 17505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A community plans to organize three activities, \"Book Club,\" \"Fun Sports,\" and \"Environmental Theme Painting,\" to enrich the lives of residents. A total of 120 people have signed up for the activities, with each resident participating in at most two activities. It is know... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 17506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $PQR$ has positive integer side lengths with $PQ=PR$. Let $J$ be the intersection of the bisectors of $\\angle Q$ and $\\angle R$. Suppose $QJ=10$. Find the smallest possible perimeter of $\\triangle PQR$. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 17507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A$ , $B$ , $C$ , $D$ , and $E$ are on the same plane such that $A,E,C$ lie on a line in that order, $B,E,D$ lie on a line in that order, $AE = 1$ , $BE = 4$ , $CE = 3$ , $DE = 2$ , and $\\angle AEB = 60^\\circ$ . Let $AB$ and $CD$ intersect at $P$ . ... | math | {
"style": "rule",
"ground_truth": "967"
} | {
"split": "train",
"index": 17508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers \\( a, b, c, d \\) belong to the interval \\([-4 ; 4]\\). Find the maximum value of the expression \\( a + 2b + c + 2d - ab - bc - cd - da \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 17509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Use the Horner's method to compute the value of the polynomial $f(x)=0.5x^{5}+4x^{4}-3x^{2}+x-1$ when $x=3$, and determine the first operation to perform. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5.5"
} | {
"split": "train",
"index": 17510,
"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": 17511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the set $I = \\{1, 2, 3, 4, 5\\}$. Choose two non-empty subsets $A$ and $B$ from $I$. How many different ways are there to choose $A$ and $B$ such that the smallest number in $B$ is greater than the largest number in $A$? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 17512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $1+2\\cdot3-4+5.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 17513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "John has recorded completion times, in seconds, of 100, 108, 112, 104, and 110 for running a 100-meter dash. After another race, he realized his median time dropped to 106 seconds. What was his time, in seconds, for the latest race? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "104"
} | {
"split": "train",
"index": 17514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The entire surface of a cube with dimensions $13 \\times 13 \\times 13$ was painted red, and then this cube was cut into $1 \\times 1 \\times 1$ cubes. All faces of the $1 \\times 1 \\times 1$ cubes that were not painted red were painted blue. By what factor is the total area o... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 17515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a three-digit number \\( n = \\overline{abc} \\) is a perfect square and its digit sum \\( a+b+c \\) is also a perfect square, then \\( n \\) is called a super square number. Let the set of super square numbers be \\( A \\), and let the sum of the elements in set \\( A \\) b... | math | {
"style": "rule",
"ground_truth": "384"
} | {
"split": "train",
"index": 17516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest multiple of 7 that is greater than -50? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-49"
} | {
"split": "train",
"index": 17517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate \\( \\frac{18}{4.9 \\times 106} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{18}{519.4}"
} | {
"split": "train",
"index": 17518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The nine horizontal and nine vertical lines on an $8\\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 17519,
"solution": "To determine the two horizontal sides of a rectangle, we have to pick two of the horizontal lines of the checkerboard, or ${9\\choose 2} = 36$. Similarly, there are ${9\\choose 2}$ ways to pick the vertical sides, giving us $r = 1296$ rectangles.\n For $s$, there a... |
deepscaler | [
{
"role": "user",
"content": "Let \\( S = \\{1, 2, 3, \\ldots, 100\\} \\). Find the smallest positive integer \\( n \\) such that every \\( n \\)-element subset of \\( S \\) contains 4 pairwise coprime numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 17520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the expression \\[2 - (-3) - 4 - (-5) - 6 - (-7) \\times 2,\\] calculate its value. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-14"
} | {
"split": "train",
"index": 17521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $33$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $33$ vertices equals $99$. The midpoints of the sides of $P_1$ form a second $33$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $33$-gon, $P_3$. Find the sum of t... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 17522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a permutation $p = (a_1,a_2,\\ldots,a_9)$ of the digits $1,2,\\ldots,9$, let $s(p)$ denote the sum of the three $3$-digit numbers $a_1a_2a_3$, $a_4a_5a_6$, and $a_7a_8a_9$. Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$... | math | {
"style": "rule",
"ground_truth": "162"
} | {
"split": "train",
"index": 17523,
"solution": "To minimize $s(p)$, the numbers $1$, $2$, and $3$ (which sum to $6$) must be in the hundreds places. For the units digit of $s(p)$ to be $0$, the numbers in the ones places must have a sum of either $10$ or $20$. However, since the tens digit contributes more to th... |
deepscaler | [
{
"role": "user",
"content": "Three workers can complete a certain task. The second and third worker together can complete it twice as fast as the first worker; the first and third worker together can complete it three times faster than the second worker. How many times faster can the first and second worke... | math | {
"style": "rule",
"ground_truth": "7/5"
} | {
"split": "train",
"index": 17524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $18\\left(\\frac{200}{3} + \\frac{50}{6} + \\frac{16}{18} + 2\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1402"
} | {
"split": "train",
"index": 17525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The side of the base of a regular quadrilateral pyramid \\( \\operatorname{ABCDP} \\) (with \\( P \\) as the apex) is \\( 4 \\sqrt{2} \\), and the angle between adjacent lateral faces is \\( 120^{\\circ} \\). Find the area of the cross-section of the pyramid by a plane passing ... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{6}"
} | {
"split": "train",
"index": 17526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\\ldots,r_{2007}$. A polynomial $P$ of degree $2007$ has the property that\n\\[P\\left(r_j+\\dfrac{1}{r_j}\\right)=0\\]for $j=1,\\ldots,2007$. Determine the value of $\\frac{P(1)}{P(-1)}$. Let's think step b... | math | {
"style": "rule",
"ground_truth": "\\frac{289}{259}"
} | {
"split": "train",
"index": 17527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The three roots of the cubic $ 30 x^3 \\minus{} 50x^2 \\plus{} 22x \\minus{} 1$ are distinct real numbers between $ 0$ and $ 1$ . For every nonnegative integer $ n$ , let $ s_n$ be the sum of the $ n$ th powers of these three roots. What is the value of the infinite ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 17528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Candice starts driving home from work at 5:00 PM. Starting at exactly 5:01 PM, and every minute after that, Candice encounters a new speed limit sign and slows down by 1 mph. Candice's speed, in miles per hour, is always a positive integer. Candice drives for \\(2/3\\) of a mil... | math | {
"style": "rule",
"ground_truth": "5:05(PM)"
} | {
"split": "train",
"index": 17529,
"solution": "Suppose that Candice starts driving at \\(n\\) miles per hour. Then she slows down and drives \\((n-1)\\) mph, \\((n-2)\\) mph, and so on, with her last speed being \\((m+1)\\) mph. Then the total distance traveled is \\(\\frac{1}{60}\\left(\\frac{n(n+1)}{2}-\\frac... |
deepscaler | [
{
"role": "user",
"content": "According to the notice from the Ministry of Industry and Information Technology on the comprehensive promotion of China's characteristic enterprise new apprenticeship system and the strengthening of skills training, our region clearly promotes the new apprenticeship system tra... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{10}"
} | {
"split": "train",
"index": 17530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which type of conic section is described by the equation \\[(x+5)^2 = (4y-3)^2 - 140?\\]Enter \"C\" for circle, \"P\" for parabola, \"E\" for ellipse, \"H\" for hyperbola, and \"N\" for none of the above. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "(\\text{H})"
} | {
"split": "train",
"index": 17531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A kite-shaped field is planted uniformly with wheat. The sides of the kite are 120 m and 80 m, with angles between the unequal sides being \\(120^\\circ\\) and the other two angles being \\(60^\\circ\\) each. At harvest, the wheat at any point in the field is brought to the nea... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 17532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\frac{x^2}{16} + \\frac{y^2}{b^2} = 1$, a line passing through its left focus intersects the ellipse at points $A$ and $B$, and the maximum value of $|AF_{2}| + |BF_{2}|$ is $10$. Find the eccentricity of the ellipse. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 17533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a square table with 2015 rows and columns, positive numbers are placed. The product of the numbers in each row and in each column is equal to 2, and the product of the numbers in any 3x3 square is equal to 1. What number is in the center of the table? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "2^{-2017}"
} | {
"split": "train",
"index": 17534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p(x)$ be defined on $2 \\le x \\le 10$ such that $$p(x) = \\begin{cases} x + 1 &\\quad \\lfloor x \\rfloor\\text{ is prime} \\\\ p(y) + (x + 1 - \\lfloor x \\rfloor) &\\quad \\text{otherwise} \\end{cases}$$ where $y$ is the greatest prime factor of $\\lfloor x\\rfloor.$ Ex... | math | {
"style": "rule",
"ground_truth": "[3,7] \\cup [8,9)"
} | {
"split": "train",
"index": 17535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The quadratic \\( x^2 + 1800x + 1800 \\) can be written in the form \\( (x+b)^2 + c \\), where \\( b \\) and \\( c \\) are constants. What is \\( \\frac{c}{b} \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-898"
} | {
"split": "train",
"index": 17536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\frac{1}{2}x^2 - 2ax + b\\ln(x) + 2a^2$ achieves an extremum of $\\frac{1}{2}$ at $x = 1$, find the value of $a+b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 17537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two different integers from 1 through 20 inclusive are chosen at random. What is the probability that both numbers are prime? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{14}{95}"
} | {
"split": "train",
"index": 17538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Mr. Jacob's class, $12$ of the $20$ students received a 'B' on the latest exam. If the same proportion of students received a 'B' in Mrs. Cecilia's latest exam, and Mrs. Cecilia originally had $30$ students, but $6$ were absent during the exam, how many students present for ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 17539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute \\[\n\\left\\lfloor \\frac{2007! + 2004!}{2006! + 2005!}\\right\\rfloor.\n\\](Note that $\\lfloor x \\rfloor$ denotes the greatest integer less than or equal to $x$.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2006"
} | {
"split": "train",
"index": 17540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABCDEFGH$ shown below is a cube. Find $\\sin \\angle GAC$.\n\n[asy]\n\nimport three;\n\ntriple A,B,C,D,EE,F,G,H;\n\nA = (0,0,0);\n\nB = (1,0,0);\n\nC = (1,1,0);\n\nD= (0,1,0);\n\nEE = (0,0,1);\n\nF = B+EE;\n\nG = C + EE;\n\nH = D + EE;\n\ndraw(B--C--D);\n\ndraw(B--A--D,dashed)... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 17541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is ... | math | {
"style": "rule",
"ground_truth": "132"
} | {
"split": "train",
"index": 17542,
"solution": "\nWe are given that Amy, Beth, and Jo listen to four different songs and discuss which ones they like, with the conditions that no song is liked by all three and for each pair of girls, there is at least one song liked by those two but disliked by the third. We nee... |
deepscaler | [
{
"role": "user",
"content": "If $a = \\log_8 225$ and $b = \\log_2 15$, then $a$, in terms of $b$, is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2b}{3}"
} | {
"split": "train",
"index": 17543,
"solution": "1. **Express $a$ and $b$ using the change-of-base formula:**\n \\[ a = \\log_8 225 = \\frac{\\log_2 225}{\\log_2 8} \\]\n \\[ b = \\log_2 15 \\]\n\n2. **Simplify the expressions:**\n \\[ \\log_2 8 = 3 \\quad \\text{(since } 2^3 = 8\\text{)} \\]\n Thus,\n ... |
deepscaler | [
{
"role": "user",
"content": "For a certain complex number $c$, the polynomial\n\\[P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\\]has exactly 4 distinct roots. What is $|c|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"split": "train",
"index": 17544,
"solution": "1. **Factorize the given polynomial**: \n The polynomial \\( P(x) \\) is given as:\n \\[\n P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\n \\]\n We start by factorizing each quadratic term using the quadratic formula \\( x = \\frac{-b \\pm \\sqrt{b^2... |
deepscaler | [
{
"role": "user",
"content": "How many complex numbers $z$ such that $\\left| z \\right| < 30$ satisfy the equation\n\\[\n e^z = \\frac{z - 1}{z + 1} \\, ?\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 17545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T = \\{3^0, 3^1, 3^2, \\ldots, 3^{10}\\}$. Consider all possible positive differences of pairs of elements of $T$. Let $N$ be the sum of all these differences. Find $N$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "783492"
} | {
"split": "train",
"index": 17546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The problem involves finding the value of the expressions $\\lg 2 + \\lg 5$ and $4(-100)^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "400000000"
} | {
"split": "train",
"index": 17547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all possible positive integer values of $b$ such that the quadratic equation $2x^2 + 5x + b = 0$ has rational roots. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 17548,
"solution": ""
} |
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