data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "What is the side length of the larger square if a small square is drawn inside a larger square, and the area of the shaded region and the area of the unshaded region are each $18 \\mathrm{~cm}^{2}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6 \\mathrm{~cm}"
} | {
"split": "train",
"index": 37649,
"solution": "Since the area of the larger square equals the sum of the areas of the shaded and unshaded regions inside, then the area of the larger square equals $2 \\times 18 \\mathrm{~cm}^{2}=36 \\mathrm{~cm}^{2}$. \n Since the larger square has an area of $36 \\mathrm{~cm}^{... |
deepscaler | [
{
"role": "user",
"content": "(The full score for this question is 8 points) Arrange 3 male students and 2 female students in a row, \n (1) The number of all different arrangements; \n(2) The number of arrangements where exactly two male students are adjacent; \n(3) The number of arrangements where male... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 37650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\left(\\dfrac{9819156364}{-24635745744574}\\right)^0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 37651,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ whe... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 37652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\frac{2468_{10}}{123_{5}} \\times 107_{8} + 4321_{9}$? Express your answer in base 10. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7789"
} | {
"split": "train",
"index": 37653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $Z$ be as in problem 15. Let $X$ be the greatest integer such that $|X Z| \\leq 5$. Find $X$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 37654,
"solution": "Problems 13-15 go together. See below."
} |
deepscaler | [
{
"role": "user",
"content": "Alice needs to replace a light bulb located $10$ centimeters below the ceiling in her kitchen. The ceiling is $2.4$ meters above the floor. Alice is $1.5$ meters tall and can reach $46$ centimeters above the top of her head. Standing on a stool, she can just reach the light bul... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 37655,
"solution": "1. **Convert all measurements to centimeters**: \n - The ceiling height is $2.4$ meters, which is $2.4 \\times 100 = 240$ centimeters.\n - Alice's height is $1.5$ meters, which is $1.5 \\times 100 = 150$ centimeters.\n - Alice can reach $46$ centimeters above... |
deepscaler | [
{
"role": "user",
"content": "What is the positive integer $N$ for which $$ 22^2 \\times 55^2 = 10^2 \\times N^2 \\ ? $$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 37656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive integers \\( n \\), let \\( g(n) \\) return the smallest positive integer \\( k \\) such that \\( \\frac{1}{k} \\) has exactly \\( n \\) digits after the decimal point in base 6 notation. Determine the number of positive integer divisors of \\( g(2023) \\). Let's t... | math | {
"style": "rule",
"ground_truth": "4096576"
} | {
"split": "train",
"index": 37657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\[ f(x) =\n\\begin{cases}\n-x^2 & \\text{if } x \\geq 0,\\\\\nx+8& \\text{if } x <0.\n\\end{cases}\n\\]Compute $f(f(f(f(f(1))))).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-33"
} | {
"split": "train",
"index": 37658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If 25,197,624 hot dogs are packaged in sets of 4, how many will be left over? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 37659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? 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}{8}"
} | {
"split": "train",
"index": 37660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The novel takes 630 minutes to read aloud. The disc can hold 80 minutes of reading with at most 4 minutes of unused space. Calculate the number of minutes of reading each disc will contain. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 37661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( S = \\{1, 2, \\cdots, 2005\\} \\). If any \\( n \\) pairwise coprime numbers in \\( S \\) always include at least one prime number, find the minimum value of \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 37662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the set $\\{0.34,0.304,0.034,0.43\\}$. The sum of the smallest and largest numbers in the set is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.464"
} | {
"split": "train",
"index": 37663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group of cows and horses are randomly divided into two equal rows. Each animal in one row is directly opposite an animal in the other row. If 75 of the animals are horses and the number of cows opposite cows is 10 more than the number of horses opposite horses, determine the ... | math | {
"style": "rule",
"ground_truth": "170"
} | {
"split": "train",
"index": 37664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a given isosceles right triangle, a square is inscribed such that its one vertex touches the right angle vertex of the triangle and its two other vertices touch the legs of the triangle. If the area of this square is found to be $784 \\text{cm}^2$, determine the area of anot... | math | {
"style": "rule",
"ground_truth": "784"
} | {
"split": "train",
"index": 37665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cube of side length $8$ and balls of clay of radius $1.5$, determine the maximum number of balls that can completely fit inside the cube when the balls are reshaped but not compressed. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 37666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\n\\left( 1 + \\sin \\frac {\\pi}{12} \\right) \\left( 1 + \\sin \\frac {5\\pi}{12} \\right) \\left( 1 + \\sin \\frac {7\\pi}{12} \\right) \\left( 1 + \\sin \\frac {11\\pi}{12} \\right).\n\\] Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 37667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\frac{1}{(-7^{3})^{3}}\\cdot(-7)^{10}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 37668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a regular octagon. How many triangles can be formed whose vertices are the vertices of the octagon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 37669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \\ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \\le 50$ is $f_{50}(n) = 12?$ Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 37670,
"solution": "To solve this problem, we need to understand the function $f_j(n)$ and its behavior as $j$ increases. We start by analyzing the function $f_1(n)$, which is defined as twice the number of positive integer divisors of $n$. We then recursively apply $f_1$ to its own o... |
deepscaler | [
{
"role": "user",
"content": "Trapezoid \\(ABCD\\) is inscribed in the parabola \\(y = x^2\\) such that \\(A = (a, a^2)\\), \\(B = (b, b^2)\\), \\(C = (-b, b^2)\\), and \\(D = (-a, a^2)\\) for some positive reals \\(a, b\\) with \\(a > b\\). If \\(AD + BC = AB + CD\\), and \\(AB = \\frac{3}{4}\\), what is \... | math | {
"style": "rule",
"ground_truth": "\\frac{27}{40}"
} | {
"split": "train",
"index": 37671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rachelle uses 3 pounds of meat to make 8 hamburgers for her family. How many pounds of meat does she need to make 24 hamburgers for a neighbourhood picnic? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 37672,
"solution": "1. **Determine the amount of meat per hamburger**: Rachelle uses 3 pounds of meat to make 8 hamburgers. Therefore, the amount of meat required for one hamburger is calculated by dividing the total pounds of meat by the number of hamburgers:\n \\[\n \\text{Meat ... |
deepscaler | [
{
"role": "user",
"content": "If \\( a \\) and \\( b \\) are given real numbers, and \\( 1 < a < b \\), then the absolute value of the difference between the average and the median of the four numbers \\( 1, a+1, 2a+b, a+b+1 \\) is ______. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 37673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The symphony orchestra has more than 200 members but fewer than 300 members. When they line up in rows of 6, there are two extra members; when they line up in rows of 8, there are three extra members; and when they line up in rows of 9, there are four extra members. How many me... | math | {
"style": "rule",
"ground_truth": "260"
} | {
"split": "train",
"index": 37674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $512^{x}=64^{240}$, what is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 37675,
"solution": "We note that $64=2^{6}$ and $512=2^{9}$. Therefore, the equation $512^{x}=64^{240}$ can be rewritten as $(2^{9})^{x}=(2^{6})^{240}$ or $2^{9x}=2^{6(240)}$. Since the bases in this last equation are equal, then the exponents are equal, so $9x=6(240)$ or $x=\\frac{14... |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively in $\\triangle ABC$, with $a=4$ and $(4+b)(\\sin A-\\sin B)=(c-b)\\sin C$, find the maximum value of the area of $\\triangle ABC$. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 37676,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle XYZ$, the medians $\\overline{XU}$ and $\\overline{YV}$ intersect at right angles. If $XU = 18$ and $YV = 24$, find the area of $\\triangle XYZ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 37677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square flag features a symmetric red cross with uniform width and two identical blue squares in the center on a white background. The entire cross, including the red arms and the two blue centers, occupies 40% of the flag's area. Determine what percent of the flag's area is o... | math | {
"style": "rule",
"ground_truth": "20\\%"
} | {
"split": "train",
"index": 37678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\(\\triangle DEF\\), where \\(DE=28\\), \\(EF=30\\), and \\(FD=16\\), calculate the area of \\(\\triangle DEF\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "221.25"
} | {
"split": "train",
"index": 37679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When measuring a part, random errors occur that follow a normal distribution with a parameter $\\sigma=10$ mm. Find the probability that the measurement is made with an error not exceeding $15$ mm. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.8664"
} | {
"split": "train",
"index": 37680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\vec{a}\\cdot \\vec{b}$, where $\\vec{a}=(2\\cos x,\\sqrt{3}\\sin 2x)$, $\\vec{b}=(\\cos x,1)$, and $x\\in \\mathbb{R}$.\n\n(Ⅰ) Find the period and the intervals of monotonic increase for the function $y=f(x)$;\n\n(Ⅱ) In $\\triangle ABC$, the sides opp... | math | {
"style": "rule",
"ground_truth": "\\frac{7\\sqrt{3}}{6}"
} | {
"split": "train",
"index": 37681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a line, 6 points are given, and on a parallel line, 8 points are given. How many triangles can be formed with the vertices at the given points? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 37682,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\\in P$ with integer coordinates is it true that $|4x+3y| \\leq 1000$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 37683,
"solution": "1. **Identify the axis of symmetry**: Given the focus of the parabola $P$ at $(0,0)$ and points $(4,3)$ and $(-4,-3)$ on $P$, we observe that the line connecting these points has a slope of $\\frac{3 - (-3)}{4 - (-4)} = \\frac{6}{8} = \\frac{3}{4}$. This suggests t... |
deepscaler | [
{
"role": "user",
"content": "In the adjoining figure $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, then the area of $CMN$ in square inches is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}-3"
} | {
"split": "train",
"index": 37684,
"solution": "1. **Identify the variables and apply the Pythagorean Theorem:**\n Let $BN = x$, which implies $AN = 1 - x$ because $AB = 1$ (since the area of square $ABCD$ is 1 square inch, each side of the square is 1 inch). \n \n In $\\triangle NBC$, which is a right tri... |
deepscaler | [
{
"role": "user",
"content": "A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{729}"
} | {
"split": "train",
"index": 37685,
"solution": "1. **Understanding the Problem**: A bug starts at one vertex of a cube and moves along the edges. At each vertex, it can choose any of the three connected edges with equal probability. We need to find the probability that after seven moves, the bug visits every ver... |
deepscaler | [
{
"role": "user",
"content": "For $\\{1, 2, 3, ..., n\\}$ and each of its non-empty subsets, a unique alternating sum is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. Find the sum of all such a... | math | {
"style": "rule",
"ground_truth": "5120"
} | {
"split": "train",
"index": 37686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three lines are drawn parallel to each of the three sides of $\\triangle ABC$ so that the three lines intersect in the interior of $ABC$ . The resulting three smaller triangles have areas $1$ , $4$ , and $9$ . Find the area of $\\triangle ABC$ .\n\n[asy]\ndefaultpen(line... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 37687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given complex numbers \\( z, z_{1}, z_{2} \\left( z_{1} \\neq z_{2} \\right) \\) such that \\( z_{1}^{2}=z_{2}^{2}=-2-2 \\sqrt{3} \\mathrm{i} \\), and \\(\\left|z-z_{1}\\right|=\\left|z-z_{2}\\right|=4\\), find \\(|z|=\\ \\ \\ \\ \\ .\\) Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 37688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a parallelogram with $\\angle ABC=135^\\circ$, $AB=14$ and $BC=8$. Extend $\\overline{CD}$ through $D$ to $E$ so that $DE=3$. If $\\overline{BE}$ intersects $\\overline{AD}$ at $F$, then find the length of segment $FD$.\nA) $\\frac{6}{17}$ \nB) $\\frac{18}{17}$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{24}{17}"
} | {
"split": "train",
"index": 37689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two adjacent faces of a tetrahedron, each being equilateral triangles with side length 1, form a dihedral angle of 45 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane co... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 37690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $q_1(x)$ and $r_1$ are the quotient and remainder, respectively, when the polynomial $x^8$ is divided by $x + \\frac{1}{2}$, and if $q_2(x)$ and $r_2$ are the quotient and remainder, respectively, when $q_1(x)$ is divided by $x + \\frac{1}{2}$, then $r_2$ equals Let's think ... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{16}"
} | {
"split": "train",
"index": 37691,
"solution": "1. **Divide \\(x^8\\) by \\(x + \\frac{1}{2}\\):**\n We use synthetic division to divide \\(x^8\\) by \\(x + \\frac{1}{2}\\). Setting \\(x + \\frac{1}{2} = 0\\) gives \\(x = -\\frac{1}{2}\\). We substitute \\(x = -\\frac{1}{2}\\) into \\(x^8\\) to find the remain... |
deepscaler | [
{
"role": "user",
"content": "Given that $x = -1$ is a solution to the equation $7x^3 - 3x^2 + kx + 5 = 0$, find the value of $k^3 + 2k^2 - 11k - 85$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-105"
} | {
"split": "train",
"index": 37692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is isosceles with $AC = BC$ and $\\angle ACB = 106^\\circ.$ Point $M$ is in the interior of the triangle so that $\\angle MAC = 7^\\circ$ and $\\angle MCA = 23^\\circ.$ Find the number of degrees in $\\angle CMB.$\n[asy] pointpen = black; pathpen = black+linewidt... | math | {
"style": "rule",
"ground_truth": "83"
} | {
"split": "train",
"index": 37693,
"solution": "Without loss of generality, let $AC = BC = 1$. Then, using the Law of Sines in triangle $AMC$, we get $\\frac {1}{\\sin 150} = \\frac {MC}{\\sin 7}$, and using the sine addition formula to evaluate $\\sin 150 = \\sin (90 + 60)$, we get $MC = 2 \\sin 7$.\n Then, usi... |
deepscaler | [
{
"role": "user",
"content": "Consider a rectangle \\( ABCD \\) where the side lengths are \\( \\overline{AB}=4 \\) and \\( \\overline{BC}=8 \\). Points \\( M \\) and \\( N \\) are fixed on sides \\( BC \\) and \\( AD \\), respectively, such that the quadrilateral \\( BMDN \\) is a rhombus. Calculate the ar... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 37694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $\\text{palindrome}$, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 37695,
"solution": "\n#### Step 1: Understand the properties of $x$ and $x+32$\nGiven that $x$ is a three-digit palindrome and $x+32$ is a four-digit palindrome, we need to find the possible values of $x$ and $x+32$ that satisfy these conditions.\n\n#### Step 2: Determine the range of... |
deepscaler | [
{
"role": "user",
"content": "Consider triangle $A B C$ with side lengths $A B=4, B C=7$, and $A C=8$. Let $M$ be the midpoint of segment $A B$, and let $N$ be the point on the interior of segment $A C$ that also lies on the circumcircle of triangle $M B C$. Compute $B N$. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{210}}{4}"
} | {
"split": "train",
"index": 37696,
"solution": "Let $\\angle B A C=\\theta$. Then, $\\cos \\theta=\\frac{4^{2}+8^{2}-7^{2}}{2 \\cdot 4 \\cdot 8}$. Since $A M=\\frac{4}{2}=2$, and power of a point gives $A M \\cdot A B=A N \\cdot A C$, we have $A N=\\frac{2 \\cdot 4}{8}=1$, so $N C=8-1=7$. Law of cosines on trian... |
deepscaler | [
{
"role": "user",
"content": "Given the sets of consecutive integers where each set starts with one more element than the preceding one and the first element of each set is one more than the last element of the preceding set, find the sum of the elements in the 21st set. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "4641"
} | {
"split": "train",
"index": 37697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a common difference of $2$ and an even number of terms, the sum of all odd terms is $15$, and the sum of all even terms is $25$, calculate the number of terms in this sequence. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 37698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "5. Let $S$ denote the set of all positive integers whose prime factors are elements of $\\{2,3,5,7,11\\}$ . (We include 1 in the set $S$ .) If $$ \\sum_{q \\in S} \\frac{\\varphi(q)}{q^{2}} $$ can be written as $\\frac{a}{b}$ for relatively prime positive integers $a$ ... | math | {
"style": "rule",
"ground_truth": "1537"
} | {
"split": "train",
"index": 37699,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$M$ is an $8 \\times 8$ matrix. For $1 \\leq i \\leq 8$, all entries in row $i$ are at least $i$, and all entries on column $i$ are at least $i$. What is the minimum possible sum of the entries of $M$ ? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "372"
} | {
"split": "train",
"index": 37700,
"solution": "Let $s_{n}$ be the minimum possible sum for an $n$ by $n$ matrix. Then, we note that increasing it by adding row $n+1$ and column $n+1$ gives $2 n+1$ additional entries, each of which has minimal size at least $n+1$. Consequently, we obtain $s_{n+1}=s_{n}+(2 n+1)(n... |
deepscaler | [
{
"role": "user",
"content": "The ratio of the areas of two squares is $\\frac{192}{80}$. After rationalizing the denominator, the ratio of their side lengths can be expressed in the simplified form $\\frac{a\\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integers. What is the value of the sum $a+b+c$? Let's thi... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 37701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose we flip five coins simultaneously: a penny (1 cent), a nickel (5 cents), a dime (10 cents), a quarter (25 cents), and a half-dollar (50 cents). What is the probability that at least 40 cents worth of coins come up heads? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{16}"
} | {
"split": "train",
"index": 37702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pentagon is obtained by joining, in order, the points \\((0,0)\\), \\((1,2)\\), \\((3,3)\\), \\((4,1)\\), \\((2,0)\\), and back to \\((0,0)\\). The perimeter of the pentagon can be written in the form \\(a + b\\sqrt{c} + d\\sqrt{e}\\), where \\(a\\), \\(b\\), \\(c\\), \\(d\\)... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 37703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find how many positive integers $k\\in\\{1,2,\\ldots,2022\\}$ have the following property: if $2022$ real numbers are written on a circle so that the sum of any $k$ consecutive numbers is equal to $2022$ then all of the $2022$ numbers are equal. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "672"
} | {
"split": "train",
"index": 37704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ whose right focus is $F$, and a circle with center $F$: $x^2 + y^2 - 4x - 32 = 0$ that is tangent to the right directrix of the ellipse, find the eccentricity of the ellipse. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 37705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least common multiple of 105 and 360? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2520"
} | {
"split": "train",
"index": 37706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Automobile license plates for a state consist of four letters followed by a dash and two single digits. How many different license plate combinations are possible if exactly two letters are repeated once each (meaning two pairs of identical letters), and digits can be repeated?... | math | {
"style": "rule",
"ground_truth": "390000"
} | {
"split": "train",
"index": 37707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $y=f(x)$ is shown below, with $1$ unit between grid lines. Assume $f(x)$ is defined only on the domain shown.\n\nWhat is the sum of all integers $c$ for which the equation $f(x)=c$ has exactly $6$ solutions?\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\... | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 37708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A yogurt shop sells four flavors of yogurt and has six different toppings. How many combinations of one flavor and two different toppings are available? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 37709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the beginning of my bike ride I feel good, so I can travel 20 miles per hour. Later, I get tired and travel only 12 miles per hour. If I travel a total of 122 miles in a total time of 8 hours, for how many hours did I feel good? Express your answer as a common fract... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{4}"
} | {
"split": "train",
"index": 37710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "19) A puck is kicked up a ramp, which makes an angle of $30^{\\circ}$ with the horizontal. The graph below depicts the speed of the puck versus time. What is the coefficient of friction between the puck and the ramp?\n\nA) 0.07\nB) 0.15\nC) 0.22\nD) 0.29\nE) 0.37 Let's think ... | math | {
"style": "rule",
"ground_truth": "0.29"
} | {
"split": "train",
"index": 37711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ be the function defined on $-1\\le x\\le 1$ by the formula $$f(x)=1-\\sqrt{1-x^2}.$$This is a graph of $y=f(x)$: [asy]\nimport graph; size(4cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.5,xmax=1.5,ymin=-1.5,ymax=1... | math | {
"style": "rule",
"ground_truth": "0.57"
} | {
"split": "train",
"index": 37712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( n! \\), in decimal notation, has exactly 57 ending zeros, find the sum of all possible values of \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1185"
} | {
"split": "train",
"index": 37713,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\left\\lceil\\sqrt{2}\\,\\right\\rceil+\\left\\lceil\\sqrt{22}\\,\\right\\rceil+\\left\\lceil\\sqrt{222}\\,\\right\\rceil$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 37714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\(\\sum^{100}_{i=1} \\sum^{100}_{j=1} (i+j)\\), find the value of the expression. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1010000"
} | {
"split": "train",
"index": 37715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the mass of the plate $D$ with surface density $\\mu = \\frac{x^2}{x^2 + y^2}$, bounded by the curves\n\n$$\ny^2 - 4y + x^2 = 0, \\quad y^2 - 8y + x^2 = 0, \\quad y = \\frac{x}{\\sqrt{3}}, \\quad x = 0.\n$$ Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\pi + \\frac{3\\sqrt{3}}{8}"
} | {
"split": "train",
"index": 37716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $PQR$, angle $R$ is a right angle and the altitude from $R$ meets $\\overline{PQ}$ at $S$. The lengths of the sides of $\\triangle PQR$ are integers, $PS=17^3$, and $\\cos Q = a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$. Let's think st... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 37717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate using your preferred method!\n\n100 - 54 - 46 \n234 - (134 + 45) \n125 × 7 × 8 \n15 × (61 - 45) \n318 ÷ 6 + 165. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "218"
} | {
"split": "train",
"index": 37718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four distinct natural numbers, one of which is an even prime number, have the following properties:\n- The sum of any two numbers is a multiple of 2.\n- The sum of any three numbers is a multiple of 3.\n- The sum of all four numbers is a multiple of 4.\n\nFind the smallest poss... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 37719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $$\\log_2 \\frac{3x+9}{5x-3} +\\log_2\\frac{5x-3}{x-2}=2$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 37720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that $abc = 72$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{72}"
} | {
"split": "train",
"index": 37721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a chorus performance, there are 6 female singers (including 1 lead singer) and 2 male singers arranged in two rows.\n(1) If there are 4 people per row, how many different arrangements are possible?\n(2) If the lead singer stands in the front row and the male singers stand in... | math | {
"style": "rule",
"ground_truth": "5760"
} | {
"split": "train",
"index": 37722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\(0 \\leqslant x \\leqslant 2\\), the function \\(y=4^{x-\\frac{1}{2}}-3 \\cdot 2^{x}+5\\) reaches its minimum value at? 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": 37723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when 2007 is divided by 25? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 37724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when $2^{87} +3$ is divided by $7$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 37725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The same current passes through a copper voltameter and a $10 \\mathrm{ohm}$ resistance wire. In the voltameter, 1.1081 grams of copper are deposited on the negative electrode in 30 minutes. Calculate how many gram calories of heat are generated in the wire. The electrochemical... | math | {
"style": "rule",
"ground_truth": "8.48"
} | {
"split": "train",
"index": 37726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You have 2 six-sided dice. One is a normal fair die, while the other has 2 ones, 2 threes, and 2 fives. You pick a die and roll it. Because of some secret magnetic attraction of the unfair die, you have a 75% chance of picking the unfair die and a 25% chance of picking the fair ... | math | {
"style": "rule",
"ground_truth": "1/7"
} | {
"split": "train",
"index": 37727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a map, a rhombus-shaped park is represented where the scale is given as 1 inch equals 100 miles. The long diagonal of the park on the map measures 10 inches, and the angle between the diagonals of the rhombus is 60 degrees. Calculate the actual area of the park in square mil... | math | {
"style": "rule",
"ground_truth": "200000\\sqrt{3}"
} | {
"split": "train",
"index": 37728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vovochka approached an arcade machine which displayed the number 0 on the screen. The rules of the game stated: \"The screen shows the number of points. If you insert a 1 ruble coin, the number of points increases by 1. If you insert a 2 ruble coin, the number of points doubles... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 37729,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x) = \\frac{1 + x}{1 - 3x}, f_1(x) = f(f(x)), f_2(x) = f(f_1(x)),$ and in general $f_n(x) = f(f_{n-1}(x)),$ then $f_{2023}(2)=$ 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": 37730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\(\\alpha, \\beta \\in \\left[0, \\frac{\\pi}{4}\\right]\\), find the maximum value of \\(\\sin(\\alpha - \\beta) + 2 \\sin(\\alpha + \\beta)\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 37731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube with edge length 2 cm has a dot marked at the center of the top face. The cube is on a flat table and rolls without slipping, making a full rotation back to its initial orientation, with the dot back on top. Calculate the length of the path followed by the dot in terms o... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}\\pi"
} | {
"split": "train",
"index": 37732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many squares are shown in the drawing? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 37733,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a quadratic function in terms of \\\\(x\\\\), \\\\(f(x)=ax^{2}-4bx+1\\\\).\n\\\\((1)\\\\) Let set \\\\(P=\\\\{1,2,3\\\\}\\\\) and \\\\(Q=\\\\{-1,1,2,3,4\\\\}\\\\), randomly pick a number from set \\\\(P\\\\) as \\\\(a\\\\) and from set \\\\(Q\\\\) as \\\\(b\\\\), calculat... | math | {
"style": "rule",
"ground_truth": "\\dfrac{961}{1280}"
} | {
"split": "train",
"index": 37734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cylindrical water tank, placed horizontally, has an interior length of 15 feet and an interior diameter of 8 feet. If the surface area of the water exposed is 60 square feet, find the depth of the water in the tank. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "4 - 2\\sqrt{3}"
} | {
"split": "train",
"index": 37735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest positive number \\( c \\) such that for every natural number \\( n \\), the inequality \\( \\{n \\sqrt{2}\\} \\geqslant \\frac{c}{n} \\) holds, where \\( \\{n \\sqrt{2}\\} = n \\sqrt{2} - \\lfloor n \\sqrt{2} \\rfloor \\) and \\( \\lfloor x \\rfloor \\) denotes... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2\\sqrt{2}}"
} | {
"split": "train",
"index": 37736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vasya replaced the same digits in two numbers with the same letters, and different digits with different letters. It turned out that the number ZARAZA is divisible by 4, and ALMAZ is divisible by 28. Find the last two digits of the sum ZARAZA + ALMAZ. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 37737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the hyperbola $x^{2}-y^{2}=8$ with left and right foci denoted as $F_{1}$ and $F_{2}$, respectively. Let $P_{n}(x_{n},y_{n})$ be a sequence of points on its right branch such that $|P_{n+1}F_{2}|=|P_{n}F_{1}|$ and $P_{1}F_{2} \\perp F_{1}F_{2}$. Determine the value of ... | math | {
"style": "rule",
"ground_truth": "8064"
} | {
"split": "train",
"index": 37738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of counter examples to the statement: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 37739,
"solution": "We are tasked with finding the number of counterexamples to the statement that a number $N$ with the sum of its digits equal to $4$ and none of its digits being $0$ is prime. We will analyze each possible set of digits that sum to $4$ and check if the resulting num... |
deepscaler | [
{
"role": "user",
"content": "Determine the least possible value of \\((x+2)(x+3)(x+4)(x+5) + 2024\\) where \\(x\\) is a real number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2023"
} | {
"split": "train",
"index": 37740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Estimate the number of primes among the first thousand primes divide some term of the sequence\n \\[2^0+1,2^1+1,2^2+1,2^3+1,\\ldots.\\]\n An estimate of $E$ earns $2^{1-0.02|A-E|}$ points, where $A$ is the actual answer.\n\n\n*2021 CCA Math Bonanza Lightning Round #... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 37741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten positive integers include the numbers 3, 5, 8, 9, and 11. What is the largest possible value of the median of this list of ten positive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 37742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ (where $a > 0$, $b > 0$), a line with an angle of $60^\\circ$ passes through one of the foci and intersects the y-axis and the right branch of the hyperbola. Find the eccentricity of the hyperbola if the point where ... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{3}"
} | {
"split": "train",
"index": 37743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six people enter two rooms, with the conditions that: ①each room receives three people; ②each room receives at least one person. How many distribution methods are there for each condition? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 37744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ and $y$ are positive integers such that $xy - 2x + 5y = 111$, what is the minimal possible value of $|x - y|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "93"
} | {
"split": "train",
"index": 37745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all two-digit positive integers whose squares end with the digits 36? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "194"
} | {
"split": "train",
"index": 37746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line with slope equal to $-1$ and a line with slope equal to $-2$ intersect at the point $P(2,5)$. What is the area of $\\triangle PQR$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6.25"
} | {
"split": "train",
"index": 37747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has $AC=3$, $BC=4$, and $AB=5$. Point $D$ is on $\\overline{AB}$, and $\\overline{CD}$ bisects the right angle. The inscribed circles of $\\triangle ADC$ and $\\triangle BCD$ have radii $r_a$ and $r_b$, respectively. What is $r_a/r_b$? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{28} \\left(10 - \\sqrt{2}\\right)"
} | {
"split": "train",
"index": 37748,
"solution": "1. **Identify the type of triangle and use the Pythagorean theorem**: Triangle $ABC$ with sides $AC=3$, $BC=4$, and $AB=5$ is a right triangle because $3^2 + 4^2 = 5^2$. Thus, $\\angle ACB = 90^\\circ$.\n\n2. **Use the Angle Bisector Theorem**: Since $\\overline{CD... |
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