data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 28549,
"solution": "1. **Define the problem in terms of variables:**\n Let the dimensions of the rectangular pan be $m$ (length) and $n$ (width). The total number of brownies is $mn$.\n\n2. **Determine the number of interior and perimeter pieces:**\n - The interior pieces are form... |
deepscaler | [
{
"role": "user",
"content": "How many different positive values of $x$ will make this statement true: there are exactly $2$ positive two-digit multiples of $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 28550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the function $f(x)=\\sin \\omega x+\\sqrt{3}\\cos \\omega x$ $(x\\in \\mathbb{R})$, and $f(\\alpha)=-2,f(\\beta)=0$, with the minimum value of $|\\alpha -\\beta|$ being $\\frac{3\\pi}{4}$, determine the value of the positive number $\\omega$. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 28551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( p, q, r, s, t, u \\) be positive real numbers such that \\( p + q + r + s + t + u = 10 \\). Find the minimum value of\n\\[ \\frac{1}{p} + \\frac{9}{q} + \\frac{4}{r} + \\frac{16}{s} + \\frac{25}{t} + \\frac{36}{u}. \\] Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "44.1"
} | {
"split": "train",
"index": 28552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $$\\sin\\theta= \\frac {3}{5}$$ and $$\\frac {5\\pi}{2}<\\theta<3\\pi$$, then $$\\sin \\frac {\\theta}{2}$$ equals \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac {3 \\sqrt {10}}{10}"
} | {
"split": "train",
"index": 28553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x$ and $y$ are real numbers that satisfy the two equations $3x+2y=6$ and $9x^2+4y^2=468$. What is the value of $xy$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-36"
} | {
"split": "train",
"index": 28554,
"solution": "Since $3 x+2 y=6$, then $(3 x+2 y)^{2}=6^{2}$ or $9 x^{2}+12 x y+4 y^{2}=36$. Since $9 x^{2}+4 y^{2}=468$, then $12 x y=\\left(9 x^{2}+12 x y+4 y^{2}\\right)-\\left(9 x^{2}+4 y^{2}\\right)=36-468=-432$ and so $x y=\\frac{-432}{12}=-36$."
} |
deepscaler | [
{
"role": "user",
"content": "Call a polygon normal if it can be inscribed in a unit circle. How many non-congruent normal polygons are there such that the square of each side length is a positive integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 28555,
"solution": "The side lengths of the polygon can only be from the set $\\{1, \\sqrt{2}, \\sqrt{3}, 2\\}$, which take up $60^{\\circ}, 90^{\\circ}, 120^{\\circ}, 180^{\\circ}$ of the circle respectively. By working modulo 60 degrees we see that $\\sqrt{2}$ must be used an even n... |
deepscaler | [
{
"role": "user",
"content": "Let $r$ and $s$ be the solutions to $2x^2 - 3x = 11.$ Compute the value of $(4r^3 - 4s^3)(r - s)^{-1}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 28556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a complex number $z=3+bi\\left(b=R\\right)$, and $\\left(1+3i\\right)\\cdot z$ is an imaginary number.<br/>$(1)$ Find the complex number $z$;<br/>$(2)$ If $ω=\\frac{z}{{2+i}}$, find the complex number $\\omega$ and its modulus $|\\omega|$. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 28557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C_{1}$ is $\\rho \\cos \\theta = 4$. \n$(1)$ Let $M$ be a moving point on the curve $C_{1}$, point $P... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{3}"
} | {
"split": "train",
"index": 28558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Dewa writes down a list of four integers. He calculates the average of each group of three of the four integers. These averages are $32,39,40,44$. What is the largest of the four integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 28559,
"solution": "Suppose that Dewa's four numbers are $w, x, y, z$. The averages of the four possible groups of three of these are $\\frac{w+x+y}{3}, \\frac{w+x+z}{3}, \\frac{w+y+z}{3}, \\frac{x+y+z}{3}$. These averages are equal to $32,39,40,44$, in some order. The sums of the gro... |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1$ $(a > b > 0)$, the symmetric point $Q$ of the right focus $F(c, 0)$ with respect to the line $y = \\dfrac{b}{c}x$ is on the ellipse. Find the eccentricity of the ellipse. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 28560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest integer $b > 3$ for which the base $b$ number $23_b$ is a perfect square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 28561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there of using diagonals to divide a regular 6-sided polygon into triangles such that at least one side of each triangle is a side of the original polygon and that each vertex of each triangle is a vertex of the original polygon? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 28562,
"solution": "The number of ways of triangulating a convex $(n+2)$-sided polygon is $\\binom{2 n}{n} \\frac{1}{n+1}$, which is 14 in this case. However, there are two triangulations of a hexagon which produce one triangle sharing no sides with the original polygon, so the answer... |
deepscaler | [
{
"role": "user",
"content": "A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\\sqrt{n}$, where $m$ and $n$ are positive integers... | math | {
"style": "rule",
"ground_truth": "803"
} | {
"split": "train",
"index": 28563,
"solution": "Notation is shown on diagram. \\[AM = MB = c = 12, AC = BC = b = 20,\\] \\[DA = DB = DC = a = 25.\\] \\[CM = x + y = \\sqrt{b^2-c^2} = 16,\\] \\[x^2 - y^2 = CD^2 – DM^2 = CD^2 – (BD^2 – BM^2) = c^2 = 144,\\] \\[x – y = \\frac{x^2 – y^2}{x+y} = \\frac {c^2} {16} = 9... |
deepscaler | [
{
"role": "user",
"content": "On a rectangular sheet of paper, a picture in the shape of a \"cross\" is drawn from two rectangles \\(ABCD\\) and \\(EFGH\\), whose sides are parallel to the edges of the sheet. It is known that \\(AB = 9\\), \\(BC = 5\\), \\(EF = 3\\), and \\(FG = 10\\). Find the area of the ... | math | {
"style": "rule",
"ground_truth": "52.5"
} | {
"split": "train",
"index": 28564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( M = \\{1, 2, \\cdots, 17\\} \\). If there exist four distinct numbers \\( a, b, c, d \\in M \\) such that \\( a + b \\equiv c + d \\pmod{17} \\), then \\( \\{a, b\\} \\) and \\( \\{c, d\\} \\) are called a balanced pair of the set \\( M \\). Find the number of balanced ... | math | {
"style": "rule",
"ground_truth": "476"
} | {
"split": "train",
"index": 28565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A teacher received letters on Monday through Friday with counts of $10$, $6$, $8$, $5$, $6$ respectively. Calculate the variance (${s^{2}} =$) of this data set. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3.2"
} | {
"split": "train",
"index": 28566,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive four-digit integers are divisible by $8$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1125"
} | {
"split": "train",
"index": 28567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon's center and vertices together make 7 points. Calculate the number of triangles that can be formed using any 3 of these points as vertices. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 28568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A shop advertises everything is \"half price in today's sale.\" In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage off the original price? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 28569,
"solution": "1. **Calculate the initial discount:** The shop advertises everything at \"half price in today's sale.\" This means that the price is reduced to half of the original price. Mathematically, this can be expressed as:\n \\[\n \\text{Sale Price} = 50\\% \\times \\t... |
deepscaler | [
{
"role": "user",
"content": "Calculate: $\\sqrt{27} - 2\\cos 30^{\\circ} + \\left(\\frac{1}{2}\\right)^{-2} - |1 - \\sqrt{3}|$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} + 5"
} | {
"split": "train",
"index": 28570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The radius of a sphere is $p$ units and the radius of a hemisphere is $2p$ units. What is the ratio of the volume of the sphere to the volume of the hemisphere? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 28571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that \\[\\operatorname{lcm}(1024,2016)=\\operatorname{lcm}(1024,2016,x_1,x_2,\\ldots,x_n),\\] with $x_1$ , $x_2$ , $\\cdots$ , $x_n$ are distinct postive integers. Find the maximum value of $n$ .\n\n*Proposed by Le Duc Minh* Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 28572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of $\\frac{1}{2 + \\frac{1}{3 + \\frac{1}{4}}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{30}"
} | {
"split": "train",
"index": 28573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The *equatorial algebra* is defined as the real numbers equipped with the three binary operations $\\natural$ , $\\sharp$ , $\\flat$ such that for all $x, y\\in \\mathbb{R}$ , we have \\[x\\mathbin\\natural y = x + y,\\quad x\\mathbin\\sharp y = \\max\\{x, y\\},\\quad x\\m... | math | {
"style": "rule",
"ground_truth": "419"
} | {
"split": "train",
"index": 28574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 8 units has its center at $(20, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a... | math | {
"style": "rule",
"ground_truth": "\\frac{60}{11}"
} | {
"split": "train",
"index": 28575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the line $x=\\dfrac{\\pi }{6}$ is the axis of symmetry of the graph of the function $f\\left(x\\right)=\\sin \\left(2x+\\varphi \\right)\\left(|\\varphi | \\lt \\dfrac{\\pi }{2}\\right)$, determine the horizontal shift required to transform the graph of the function ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\pi}{12}"
} | {
"split": "train",
"index": 28576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are positive real numbers satisfying $a + 2b + 3c = 6$, find the maximum value of $\\sqrt{a + 1} + \\sqrt{2b + 1} + \\sqrt{3c + 1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"split": "train",
"index": 28577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of $543_7$, $65_7$, and $6_7$ in base $7$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "650_7"
} | {
"split": "train",
"index": 28578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let P be any point on the curve $y=x^2-\\ln x$. Find the minimum distance from point P to the line $y=x-4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 28579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct four-digit positive integers have a digit product equal to 18? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 28580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A class leader is planning to invite graduates from the class of 2016 to give speeches. Out of 8 people, labeled A, B, ..., H, the leader wants to select 4 to speak. The conditions are: (1) at least one of A and B must participate; (2) if both A and B participate, there must be... | math | {
"style": "rule",
"ground_truth": "1080"
} | {
"split": "train",
"index": 28581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of the product \\(\\cos \\frac{\\pi}{15} \\cos \\frac{2 \\pi}{15} \\cos \\frac{3 \\pi}{15} \\cdots \\cos \\frac{7 \\pi}{15}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{128}"
} | {
"split": "train",
"index": 28582,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The union of sets \\( A \\) and \\( B \\), \\( A \\cup B = \\{a_1, a_2, a_3\\} \\), and \\( A \\neq B \\). When \\( (A, B) \\) and \\( (B, A) \\) are considered as different pairs, how many such pairs \\( (A, B) \\) are there? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 28583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C:\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a > b > 0)$ with its right focus $F$ lying on the line $2x-y-2=0$, where $A$ and $B$ are the left and right vertices of $C$, and $|AF|=3|BF|$.<br/>$(1)$ Find the standard equation of $C$;<br/>$(2)$ A line $l$ passing throu... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"split": "train",
"index": 28584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer $ n$ so that the polynomial $ P(X)\\equal{}\\sqrt3\\cdot X^{n\\plus{}1}\\minus{}X^n\\minus{}1$ has at least one root of modulus $ 1$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 28585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\\triangle ABC$, if $\\cos B= \\frac {4}{5}$, $a=5$, and the area of $\\triangle ABC$ is $12$, find the value of $\\frac {a+c}{\\sin A+\\sin C}$. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\frac {25}{3}"
} | {
"split": "train",
"index": 28586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a hypothetical scenario, a small country is planning an international event in 2023. Let \\( A \\), \\( B \\), and \\( C \\) be distinct positive integers such that their product \\( A \\cdot B \\cdot C = 2023 \\). Determine the largest possible value of the sum \\( A + B + ... | math | {
"style": "rule",
"ground_truth": "297"
} | {
"split": "train",
"index": 28587,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line passing through one focus F_1 of the ellipse 4x^{2}+2y^{2}=1 intersects the ellipse at points A and B. Then, points A, B, and the other focus F_2 of the ellipse form ∆ABF_2. Calculate the perimeter of ∆ABF_2. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {2}"
} | {
"split": "train",
"index": 28588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x)=\\frac{ax+b}{cx+d}, abcd\\not=0$ and $f(f(x))=x$ for all $x$ in the domain of $f$, what is the value of $a+d$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 28589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line containing the points $(9, 1)$ and $(5, 5)$ intersects the $x$ -axis at what point? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(10, 0)"
} | {
"split": "train",
"index": 28590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=x^{2}+ax+4$, if for any $x \\in (0,2]$, $f(x) \\leqslant 6$ always holds, then find the maximum value of the real number $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 28591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the function $f(x) = 2x^2 - 4x + 9$. Evaluate $2f(3) + 3f(-3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "147"
} | {
"split": "train",
"index": 28592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the plane rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\\left\\{{\\begin{array}{l}{x=2+3\\cos\\alpha}\\\\{y=3\\sin\\alpha}\\end{array}}\\right.$ ($\\alpha$ is the parameter). Taking the coordinate origin $O$ as the pole and the non-negative... | math | {
"style": "rule",
"ground_truth": "\\frac{{3\\sqrt{5}}}{5}"
} | {
"split": "train",
"index": 28593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Someone observed that $6! = 8 \\cdot 9 \\cdot 10$. Find the largest positive integer $n^{}_{}$ for which $n^{}_{}!$ can be expressed as the product of $n - 3_{}^{}$ consecutive positive integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 28594,
"solution": "Let the largest of the $n-3$ consecutive positive integers be $k$. Clearly $k$ cannot be less than or equal to $n$, else the product of $n-3$ consecutive positive integers will be less than $n!$.\n Key observation: Now for $n$ to be maximum the smallest number (or ... |
deepscaler | [
{
"role": "user",
"content": "The equation $\\sin^2 x + \\sin^2 2x + \\sin^2 3x + \\sin^2 4x = 2$ can be reduced to the equivalent equation\n\\[\\cos ax \\cos bx \\cos cx = 0,\\]for some positive integers $a,$ $b,$ and $c.$ Find $a + b + c.$ Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 28595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the numbers from the set $\\{1,\\ 2,\\ 3,\\ldots,\\ 100\\}$ have a perfect square factor other than one? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 28596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " $15\\times\u0002 36$ -checkerboard is covered with square tiles. There are two kinds of tiles, with side $7$ or $5.$ \nTiles are supposed to cover whole squares of the board and be non-overlapping. \nWhat is the maximum number of squares to be covered? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "540"
} | {
"split": "train",
"index": 28597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "My school's math club has 6 boys and 8 girls. I need to select a team to send to the state math competition. We want 6 people on the team. In how many ways can I select the team to have 3 boys and 3 girls? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "1120"
} | {
"split": "train",
"index": 28598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Steve has an isosceles triangle with base 8 inches and height 10 inches. He wants to cut it into eight pieces that have equal areas, as shown below. To the nearest hundredth of an inch what is the number of inches in the greatest perimeter among the eight pieces? [asy]\nsize(15... | math | {
"style": "rule",
"ground_truth": "22.21"
} | {
"split": "train",
"index": 28599,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The length of a circular race track is $1 \\mathrm{~km}$. Two motorcyclists start simultaneously from a given point $A$ on the track, traveling in opposite directions. One of them travels at a constant speed, while the other accelerates uniformly. They first meet at point $B$ o... | math | {
"style": "rule",
"ground_truth": "\\frac{-1 + \\sqrt{5}}{2}"
} | {
"split": "train",
"index": 28600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Juan, Carlos and Manu take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Manu will win? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"split": "train",
"index": 28601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $PQRS$ with $PQ$ parallel to $RS$, the diagonals $PR$ and $QS$ intersect at $T$. If the area of triangle $PQT$ is 75 square units, and the area of triangle $PST$ is 45 square units, what is the area of trapezoid $PQRS$? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 28602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $n$ is an integer and $0 < 4n <30$, what is the sum of all possible integer values of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 28603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I have three 30-sided dice that each have 6 purple sides, 8 green sides, 10 blue sides, and 6 silver sides. If I roll all three dice, what is the probability that they all show the same color? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{25}"
} | {
"split": "train",
"index": 28604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On January 15 in the stormy town of Stormville, there is a $50\\%$ chance of rain. Every day, the probability of it raining has a $50\\%$ chance of being $\\frac{2017}{2016}$ times that of the previous day (or $100\\%$ if this new quantity is over $100\\%$ ) and a $50... | math | {
"style": "rule",
"ground_truth": "243/2048"
} | {
"split": "train",
"index": 28605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the distance between (-2,4) and (3,-8)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 28606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the area of $\\triangle ABC$ is $2 \\sqrt {3}$, $BC=2$, $C=120^{\\circ}$, find the length of side $AB$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {7}"
} | {
"split": "train",
"index": 28607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $XYZ$ with $\\angle Z = 90^\\circ$, the lengths of sides $XY$ and $XZ$ are 8 and 3 respectively. Find $\\cos Y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 28608,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{3}{5}=\\frac{M}{45}=\\frac{60}{N}$, what is $M+N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 28609,
"solution": "Given the equation $\\frac{3}{5}=\\frac{M}{45}=\\frac{60}{N}$, we can separate this into two equations and solve for $M$ and $N$ respectively.\n\n1. **Solving for $M$:**\n \\[\n \\frac{3}{5} = \\frac{M}{45}\n \\]\n Cross-multiplying gives:\n \\[\n 3 \\t... |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $I_{A}, I_{B}, I_{C}$ be the $A, B, C$ excenters of this triangle, and let $O$ be the circumcenter of the triangle. Let $\\gamma_{A}, \\gamma_{B}, \\gamma_{C}$ be the corresponding excircles and $\\omega$ be the circu... | math | {
"style": "rule",
"ground_truth": "-\\frac{49}{65}"
} | {
"split": "train",
"index": 28610,
"solution": "Let $r_{A}, r_{B}, r_{C}$ be the exradii. Using $O X=R, X I_{A}=r_{A}, O I_{A}=\\sqrt{R\\left(R+2 r_{A}\\right)}$ (Euler's theorem for excircles), and the Law of Cosines, we obtain $$\\cos \\angle O X I_{A}=\\frac{R^{2}+r_{A}^{2}-R\\left(R+2 r_{A}\\right)}{2 R r_{A... |
deepscaler | [
{
"role": "user",
"content": "If $y=x^2+px+q$, then if the least possible value of $y$ is zero $q$ is equal to: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{p^2}{4}"
} | {
"split": "train",
"index": 28611,
"solution": "1. **Identify the vertex of the quadratic function**: The quadratic function given is $y = x^2 + px + q$. The vertex form of a quadratic function $y = ax^2 + bx + c$ is given by the vertex $(h, k)$ where $h = -\\frac{b}{2a}$ and $k$ is the value of the function at ... |
deepscaler | [
{
"role": "user",
"content": "Let $p, q, r,$ and $s$ be the roots of the polynomial $3x^4 - 8x^3 - 15x^2 + 10x - 2 = 0$. Find $pqrs$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 28612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive multiples of 3 that are less than 150 have a units digit of 3 or 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 28613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $$ \\sum_{k=1}^{40} \\left( \\sqrt{1 + \\frac{1}{k^{2}} + \\frac{1}{(k + 1)^{2}}}\\right) = a + \\frac {b}{c} $$ where $a, b, c \\in \\mathbb{N}, b < c, gcd(b,c) =1 ,$ then what is the value of $a+ b ?$ Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 28614,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 28615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all the positive divisors of 91? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 28616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 7 light bulbs arranged in a row. It is required to light up at least 3 of the bulbs, and adjacent bulbs cannot be lit at the same time. Determine the total number of different ways to light up the bulbs. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 28617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cara is sitting at a round table with her eight friends. How many different pairs of friends could Cara be potentially sitting between? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 28618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all pairs $(n, p)$ of positive integers such that $p$ is prime and\n\\[ 1 + 2 + \\cdots + n = 3 \\cdot (1^2 + 2^2 + \\cdot + p^2). \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(5, 2)"
} | {
"split": "train",
"index": 28619,
"solution": "\nTo find all pairs \\((n, p)\\) of positive integers such that \\( p \\) is prime and:\n\n\\[\n1 + 2 + \\cdots + n = 3 \\cdot (1^2 + 2^2 + \\cdots + p^2)\n\\]\n\nwe first express these sums using known formulas:\n\n1. The sum of the first \\( n \\) positive intege... |
deepscaler | [
{
"role": "user",
"content": "Two fair, eight-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 12? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{49}{64}"
} | {
"split": "train",
"index": 28620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle $H$ is inscribed in a regular hexagon $S$ such that one side of $H$ is parallel to one side of $S$. What is the maximum possible ratio of the area of $H$ to the area of $S$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3/8"
} | {
"split": "train",
"index": 28621,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points \\( A \\), \\( B \\), \\( C \\), \\( D \\), and \\( E \\) are located in 3-dimensional space with \\( AB = BC = CD = DE = 3 \\) and \\( \\angle ABC = \\angle CDE = 60^\\circ \\). Additionally, the plane of triangle \\( ABC \\) is perpendicular to line \\( \\overline{DE} ... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 28622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the difference $(2001 + 2002 + 2003 + \\cdots + 2100) - (51 + 53 + 55 + \\cdots + 149)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200050"
} | {
"split": "train",
"index": 28623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence \\( a_{1}, a_{2}, \\cdots, a_{n}, \\cdots \\) that satisfies \\( a_{1}=a_{2}=1, a_{3}=2 \\), and for any natural number \\( n \\), \\( a_{n} a_{n+1} a_{n+2} \\neq 1 \\). Furthermore, it is given that \\( a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{1} + a_{n+1} + a_{n+... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 28624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what values of $x$ is $\\frac{\\log{(3-x)}}{\\sqrt{x-1}}$ defined? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1,3)"
} | {
"split": "train",
"index": 28625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any $n\\in\\mathbb N$ , denote by $a_n$ the sum $2+22+222+\\cdots+22\\ldots2$ , where the last summand consists of $n$ digits of $2$ . Determine the greatest $n$ for which $a_n$ contains exactly $222$ digits of $2$ . Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "222"
} | {
"split": "train",
"index": 28626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( w \\) and \\( z \\) are complex numbers such that \\( |w+z|=2 \\) and \\( |w^2+z^2|=8 \\), find the smallest possible value of \\( |w^3+z^3| \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 28627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle passes through the point $(0,1),$ and is tangent to the parabola $y = x^2$ at $(2,4).$ Find the center of the circle.\n\n[asy]\nunitsize(0.4 cm);\n\nreal parab (real x) {\n return(x^2);\n}\n\ndraw(graph(parab,-3.5,3.5));\ndraw(Circle((-16/5,53/10),13*sqrt(17)/10));\n... | math | {
"style": "rule",
"ground_truth": "\\left( -\\frac{16}{5}, \\frac{53}{10} \\right)"
} | {
"split": "train",
"index": 28628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The set $S$ consists of 9 distinct positive integers. The average of the two smallest integers in $S$ is 5. The average of the two largest integers in $S$ is 22. What is the greatest possible average of all of the integers of $S$? Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 28629,
"solution": "Since the average of the two smallest integers in $S$ is 5, their sum is $2 \\cdot 5=10$. Since the average of the two largest integers in $S$ is 22, their sum is $2 \\cdot 22=44$. Suppose that the other five integers in the set $S$ are $p<q<r<t<u$. (Note that the ... |
deepscaler | [
{
"role": "user",
"content": "Katrine has a bag containing 4 buttons with distinct letters M, P, F, G on them (one letter per button). She picks buttons randomly, one at a time, without replacement, until she picks the button with letter G. What is the probability that she has at least three picks and her... | math | {
"style": "rule",
"ground_truth": "1/12"
} | {
"split": "train",
"index": 28630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the shaded region $\\text{BEDC}$ in parallelogram $\\text{ABCD}$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 28631,
"solution": "1. **Identify the areas to be calculated**: We need to find the area of the shaded region $\\text{BEDC}$ in parallelogram $\\text{ABCD}$. The area of $\\text{BEDC}$ can be found by subtracting the area of triangle $\\text{ABE}$ from the area of parallelogram $\\tex... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest real number $p$ such that the inequality $\\sqrt{1^2+1}+\\sqrt{2^2+1}+...+\\sqrt{n^2+1} \\le \\frac{1}{2}n(n+p)$ holds for all natural numbers $n$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} - 1"
} | {
"split": "train",
"index": 28632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set of vectors \\(\\mathbf{v}\\) such that\n\\[\n\\mathbf{v} \\cdot \\mathbf{v} = \\mathbf{v} \\cdot \\begin{pmatrix} 4 \\\\ -16 \\\\ 32 \\end{pmatrix}\n\\]\ndetermine the volume of the solid formed in space. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "7776\\pi"
} | {
"split": "train",
"index": 28633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In what ratio does the point \\( P \\) divide the perpendicular segment dropped from vertex \\( A \\) of a regular tetrahedron \\( ABCD \\) to the face \\( BCD \\), given that the lines \\( PB \\), \\( PC \\), and \\( PD \\) are mutually perpendicular to each other? Let's think... | math | {
"style": "rule",
"ground_truth": "1:1"
} | {
"split": "train",
"index": 28634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " $12$ students need to form five study groups. They will form three study groups with $2$ students each and two study groups with $3$ students each. In how many ways can these groups be formed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "138600"
} | {
"split": "train",
"index": 28635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $ABCD$ is inscribed in a circle, $M$ is the point of intersection of its diagonals, $O_1$ and $O_2$ are the centers of the inscribed circles of triangles $ABM$ and $CMD$ respectively, $K$ is the midpoint of the arc $AD$ that does not contain points $B$ and $C$, $\... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 28636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C D$ be a parallelogram with $A B=8, A D=11$, and $\\angle B A D=60^{\\circ}$. Let $X$ be on segment $C D$ with $C X / X D=1 / 3$ and $Y$ be on segment $A D$ with $A Y / Y D=1 / 2$. Let $Z$ be on segment $A B$ such that $A X, B Y$, and $D Z$ are concurrent. Determine t... | math | {
"style": "rule",
"ground_truth": "\\frac{19 \\sqrt{3}}{2}"
} | {
"split": "train",
"index": 28637,
"solution": "Let $A X$ and $B D$ meet at $P$. We have $D P / P B=D X / A B=3 / 4$. Now, applying Ceva's Theorem in triangle $A B D$, we see that $$\\frac{A Z}{Z B}=\\frac{D P}{P B} \\cdot \\frac{A Y}{Y D}=\\frac{3}{4} \\cdot \\frac{1}{2}=\\frac{3}{8}$$ Now, $$\\frac{[A Y Z]}{[A... |
deepscaler | [
{
"role": "user",
"content": "Seven identical bowling balls weigh the same as four identical canoes. If three of the canoes weigh a total of 84 pounds, how many pounds does one of the bowling balls weigh? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 28638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the natural number $N$ be a perfect square, which has at least three digits, its last two digits are not $00$, and after removing these two digits, the remaining number is still a perfect square. Then, the maximum value of $N$ is ____. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "1681"
} | {
"split": "train",
"index": 28639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $1254_6 - 432_6 + 221_6$? Express your answer in base $6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1043_6"
} | {
"split": "train",
"index": 28640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many $n$ with $1 \\leq n \\leq 100$ can a unit square be divided into $n$ congruent figures? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 28641,
"solution": "We can divide the square into congruent rectangles for all $n$, so the answer is 100."
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\text{1 mile} = \\text{8 furlongs}$ and $\\text{1 furlong} = \\text{40 rods}$, the number of rods in one mile is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "320"
} | {
"split": "train",
"index": 28642,
"solution": "To find the number of rods in one mile, we use the given conversion factors between miles, furlongs, and rods.\n\n1. **Convert miles to furlongs:**\n Given that 1 mile equals 8 furlongs, we can write:\n \\[\n 1\\ \\text{mile} = 8\\ \\text{furlongs}\n \\]\n\... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\begin{pmatrix} -4 \\\\ -1 \\end{pmatrix} \\cdot \\begin{pmatrix} 6 \\\\ 8 \\end{pmatrix}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-32"
} | {
"split": "train",
"index": 28643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers such that\n\\[a^3 - 15a^2 + 20a - 50 = 0 \\quad \\text{and} \\quad 8b^3 - 60b^2 - 290b + 2575 = 0.\\]Compute $a + b.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{2}"
} | {
"split": "train",
"index": 28644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C$ with its center at the origin and its foci on the $x$-axis, and its eccentricity equal to $\\frac{1}{2}$. One of its vertices is exactly the focus of the parabola $x^{2}=8\\sqrt{3}y$.\n\n(Ⅰ) Find the standard equation of the ellipse $C$;\n\n(Ⅱ) If the line ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 28645,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two poles $20''$ and $80''$ high are $100''$ apart, then the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16''"
} | {
"split": "train",
"index": 28646,
"solution": "1. **Identify the equations of the lines**: \n - The line from the top of the first pole (20'') to the foot of the second pole can be described by considering the slope and y-intercept. The slope is calculated as the change in height over the change in horizontal... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\frac{13!-12!+144}{11!}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 28647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1 \\le x \\le 5$ and $1 \\le y \\le 3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "416"
} | {
"split": "train",
"index": 28648,
"solution": ""
} |
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