data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Find $(x+1)\\left(x^{2}+1\\right)\\left(x^{4}+1\\right)\\left(x^{8}+1\\right) \\cdots$, where $|x|<1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{1-x}"
} | {
"split": "train",
"index": 37949,
"solution": "Let $S=(x+1)\\left(x^{2}+1\\right)\\left(x^{4}+1\\right)\\left(x^{8}+1\\right) \\cdots=1+x+x^{2}+x^{3}+\\cdots$. Since $x S=x+x^{2}+x^{3}+x^{4}+\\cdots$, we have $(1-x) S=1$, so $S=\\frac{1}{1-x}$."
} |
deepscaler | [
{
"role": "user",
"content": "A cube has a surface area of 216 square centimeters. What is the volume of the cube, in cubic centimeters? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 37950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 3. What is the area of the triangle? Provide your answer as a decimal rounded to the nearest hundredth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8.64"
} | {
"split": "train",
"index": 37951,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The probability of getting rain on any given day in August in Beach Town is \\(\\frac{1}{5}\\). What is the probability that it rains on at most 3 days in the first week of August? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.813"
} | {
"split": "train",
"index": 37952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first $15$ positive even integers is also the sum of six consecutive even integers. What is the smallest of these six integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 37953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To arrange 5 volunteers and 2 elderly people in a row, where the 2 elderly people are adjacent but not at the ends, calculate the total number of different arrangements. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "960"
} | {
"split": "train",
"index": 37954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_{n}\\}$, the sum of the first n terms is $S_{n}$, $S_{2}=7$ and $S_{6}=91$. Calculate $S_{4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 37955,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of 2020 natural numbers is written in a row. Each of them, starting from the third number, is divisible by the previous one and by the sum of the two preceding ones.\n\nWhat is the smallest possible value for the last number in the sequence? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "2019!"
} | {
"split": "train",
"index": 37956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For every $0 < \\alpha < 1$ , let $R(\\alpha)$ be the region in $\\mathbb{R}^2$ whose boundary is the convex pentagon of vertices $(0,1-\\alpha), (\\alpha, 0), (1, 0), (1,1)$ and $(0, 1)$ . Let $R$ be the set of points that belong simultaneously to each of the regions... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"split": "train",
"index": 37957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The letter T is formed by placing two $3\\:\\text{inch}\\!\\times\\!5\\:\\text{inch}$ rectangles to form a T shape. The vertical rectangle is placed in the middle of the horizontal one, overlapping it by $1.5$ inches on both sides. What is the perimeter of the new T, in inches?... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 37958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $3 \\in \\{a, a^2 - 2a\\}$, then the value of the real number $a$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 37959,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jamie counted the number of edges of a cube, Jimmy counted the numbers of corners, and Judy counted the number of faces. They then added the three numbers. What was the resulting sum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 37960,
"solution": "1. **Count the Edges of a Cube**: A cube has 12 edges. Each of the 6 faces of a cube is a square, and each square has 4 edges. However, each edge is shared between 2 faces, so the total number of unique edges is \\( \\frac{4 \\times 6}{2} = 12 \\).\n\n2. **Count th... |
deepscaler | [
{
"role": "user",
"content": "The left and right foci of a hyperbola are $F_{1}$ and $F_{2}$, respectively. A line passing through $F_{2}$ intersects the right branch of the hyperbola at points $A$ and $B$. If $\\triangle F_{1} A B$ is an equilateral triangle, what is the eccentricity of the hyperbola? Let'... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 37961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$2.46\\times 8.163\\times (5.17+4.829)$ is approximately equal to what value? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 37962,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\mathbf{a} \\times \\mathbf{b} = \\begin{pmatrix} 5 \\\\ 4 \\\\ -7 \\end{pmatrix},$ then compute $\\mathbf{a} \\times (3 \\mathbf{b}).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 15 \\\\ 12 \\\\ -21 \\end{pmatrix}"
} | {
"split": "train",
"index": 37963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the numbers, $100,101,\\cdots,999$ have three different digits in increasing order or in decreasing order? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 37964,
"solution": "To solve this problem, we need to count how many numbers between $100$ and $999$ have digits that are either in strictly increasing or strictly decreasing order. Each number in this range has three digits.\n\n#### Step 1: Counting numbers with digits in decreasing ... |
deepscaler | [
{
"role": "user",
"content": "Given a company needs to select 8 engineering and technical personnel from its 6 subsidiaries to form a task force, with each subsidiary contributing at least one person, calculate the total number of ways to allocate these 8 positions. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 37965,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two students, A and B, are doing rope skipping training. It is known that A takes the same amount of time to skip 300 times as B does to skip 270 times. A skips 19 more times per minute than B. Find out how many times each of them skips per minute. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "171"
} | {
"split": "train",
"index": 37966,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All the complex roots of $(z - 2)^6 = 64z^6$ when plotted in the complex plane, lie on a circle. Find the radius of this circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 37967,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nThe edge of the cube \\(E F G H E_1 F_1 G_1 H_1\\) is equal to 2. Points \\(A\\) and \\(B\\) are taken on the edges \\(E H\\) and \\(H H_1\\) such that \\(\\frac{E A}{A H} = 2\\) and \\(\\frac{B H}{B H_1} = \\frac{1}{2}\\). A plane is drawn through the points \\(A\\), \\(B\\)... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{\\frac{2}{11}}"
} | {
"split": "train",
"index": 37968,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{7}-\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 37969,
"solution": "1. **Understanding the Geometry**: The problem describes a round table with radius $4$ and six rectangular place mats each with width $1$ and length $x$. Each place mat has two corners on the edge of the table, and the inner corners of each mat touch the inner corn... |
deepscaler | [
{
"role": "user",
"content": "In isosceles $\\triangle ABC$ with $AB=AC$, let $D$ be the midpoint of $AC$ and $BD=1$. Find the maximum area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 37970,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ilya has a one-liter bottle filled with freshly squeezed orange juice and a 19-liter empty jug. Ilya pours half of the bottle's contents into the jug, then refills the bottle with half a liter of water and mixes everything thoroughly. He repeats this operation a total of 10 tim... | math | {
"style": "rule",
"ground_truth": "0.05"
} | {
"split": "train",
"index": 37971,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon $ABCDEF$ has sides of length three. Find the area of $\\bigtriangleup ACE$. Express your answer in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 37972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different positive values of \\( x \\) will make this statement true: there are exactly 3 three-digit multiples of \\( x \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 37973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of $\\log_{10}{579}$ is between the consecutive integers $a$ and $b$. Find $a+b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 37974,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum number of points $X_{i}$ such that for each $i$, $\\triangle A B X_{i} \\cong \\triangle C D X_{i}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 37975,
"solution": "One of the sides $A X_{i}$ or $B X_{i}$ is equal to $C D$, thus $X_{i}$ is on one of the circles of radius $C D$ and center $A$ or $B$. In the same way $X_{i}$ is on one of circles of radius $A B$ with center $C$ or $D$. The intersection of these four circles has n... |
deepscaler | [
{
"role": "user",
"content": "Graphistan has $2011$ cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the maximum possible value of the integer $k$ such that no matter how these flights are arranged it is possible to travel between any two cities in Graphi... | math | {
"style": "rule",
"ground_truth": "1005"
} | {
"split": "train",
"index": 37976,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 3.$ Find the minimum value of\n\\[\\frac{4}{x} + \\frac{9}{y} + \\frac{16}{z}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 37977,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of triangle $MNP$ given below:\n\n[asy]\nunitsize(1inch);\npair M,N,P;\nM = (0,0);\nN= (sqrt(3),0);\nP = (0,1);\ndraw (M--N--P--M, linewidth(0.9));\ndraw(rightanglemark(N,M,P,3));\nlabel(\"$M$\",M,S);\nlabel(\"$N$\",N,S);\nlabel(\"$P$\",P,N);\nlabel(\"$15$\",(N+P)... | math | {
"style": "rule",
"ground_truth": "28.125\\sqrt{3}"
} | {
"split": "train",
"index": 37978,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point \\( P \\) is located on the side \\( AB \\) of the square \\( ABCD \\) such that \\( AP: PB = 2:3 \\). Point \\( Q \\) lies on the side \\( BC \\) of the square and divides it in the ratio \\( BQ: QC = 3 \\). Lines \\( DP \\) and \\( AQ \\) intersect at point \\( E \\). F... | math | {
"style": "rule",
"ground_truth": "4:9"
} | {
"split": "train",
"index": 37979,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On hypotenuse $AB$ of a right triangle $ABC$ a second right triangle $ABD$ is constructed with hypotenuse $AB$. If $BC=1$, $AC=b$, and $AD=2$, then $BD$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{b^2-3}"
} | {
"split": "train",
"index": 37980,
"solution": "1. **Identify the triangles and their properties**: We have two right triangles, $ABC$ and $ABD$, sharing the hypotenuse $AB$. Triangle $ABC$ has legs $\\overline{BC} = 1$ and $\\overline{AC} = b$. Triangle $ABD$ has one leg $\\overline{AD} = 2$.\n\n2. **Apply the ... |
deepscaler | [
{
"role": "user",
"content": "Express -2013 in base -4. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200203_{-4}"
} | {
"split": "train",
"index": 37981,
"solution": "-2013 \\equiv 3(\\bmod 4)$, so the last digit is $3 ;$ now $\\frac{-2013-3}{-4}=504 \\equiv 0$, so the next digit (to the left) is 0 ; then $\\frac{504-0}{-4}=-126 \\equiv 2 ; \\frac{-126-2}{-4}=32 \\equiv 0 ; \\frac{32-0}{-4}=-8 \\equiv 0 ; \\frac{-8-0}{-4}=2$. Th... |
deepscaler | [
{
"role": "user",
"content": "What is the maximum possible product of three different numbers from the set $\\{-9, -7, -2, 0, 4, 6, 8\\}$, where the product contains exactly one negative number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-96"
} | {
"split": "train",
"index": 37982,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular polygon has interior angles of 120 degrees. How many sides does the polygon have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 37983,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the coordinate plane, let $A = (-8, 0)$ , $B = (8, 0)$ , and $C = (t, 6)$ . What is the maximum value of $\\sin m\\angle CAB \\cdot \\sin m\\angle CBA$ , over all real numbers $t$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3/8"
} | {
"split": "train",
"index": 37984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The square of $a$ and the square root of $b$ vary inversely. If $a=2$ when $b=81$, then find $b$ when $ab=48$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 37985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the ages of three people A, B, and C, denoted as \\(x, y, z\\), is 120, with \\(x, y, z \\in (20, 60)\\). How many ordered triples \\((x, y, z)\\) satisfy this condition? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1198"
} | {
"split": "train",
"index": 37986,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_{1}$ is $\\begin{cases}x=a\\cos \\varphi \\\\ y=b\\sin \\varphi\\end{cases} (a > b > 0,$ $\\varphi$ is the parameter), and the point $M(2, \\sqrt{3})$ on curve $C1$ corresponds to the parameter $\\var... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"split": "train",
"index": 37987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two spheres touch the plane of triangle \\(ABC\\) at points \\(A\\) and \\(B\\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 9, and the distance between their centers is \\(\\sqrt{305}\\). The center of a third sphere with a radius o... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{14}"
} | {
"split": "train",
"index": 37988,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function $y=f(x)$ have the domain $D$. If for any $x_{1}, x_{2} \\in D$, when $x_{1}+x_{2}=2a$, it always holds that $f(x_{1})+f(x_{2})=2b$, then the point $(a,b)$ is called the symmetry center of the graph of the function $y=f(x)$. Study a symmetry point of the graph o... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 37989,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a simulation experiment, 20 groups of random numbers were generated: 6830, 3013, 7055, 7430, 7740, 4422, 7884, 2604, 3346, 0952, 6807, 9706, 5774, 5725, 6576, 5929, 9768, 6071, 9138, 6754. If the numbers 1, 2, 3, 4, 5, 6 each appear exactly three times among these, it repres... | math | {
"style": "rule",
"ground_truth": "25\\%"
} | {
"split": "train",
"index": 37990,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_{0} = 2$, $a_{1} = 5$, and $a_{2} = 8$, and for $n > 2$ define $a_{n}$ recursively to be the remainder when $4$($a_{n-1}$ $+$ $a_{n-2}$ $+$ $a_{n-3}$) is divided by $11$. Find $a_{2018} \\cdot a_{2020} \\cdot a_{2022}$. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 37991,
"solution": "When given a sequence problem, one good thing to do is to check if the sequence repeats itself or if there is a pattern.\n After computing more values of the sequence, it can be observed that the sequence repeats itself every 10 terms starting at $a_{0}$.\n$a_{0} =... |
deepscaler | [
{
"role": "user",
"content": "In a certain high school, there are 300 freshmen students, including 180 boys and 120 girls. In order to understand the height information of the freshmen students, a stratified random sampling method is used to select samples according to the proportion of the sample size. It ... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 37992,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of the first three terms of a geometric sequence is 2, the product of the last three terms is 4, and the product of all terms is 64. Find the number of terms in the sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 37993,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A trapezoid inscribed in a circle with a radius of $13 \\mathrm{~cm}$ has its diagonals located $5 \\mathrm{~cm}$ away from the center of the circle. What is the maximum possible area of the trapezoid? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 37994,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\tan \\alpha=-2$, the focus of the parabola $y^{2}=2px (p > 0)$ is $F(-\\sin \\alpha\\cos \\alpha,0)$, and line $l$ passes through point $F$ and intersects the parabola at points $A$ and $B$ with $|AB|=4$, find the distance from the midpoint of segment $AB$ to the ... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{10}"
} | {
"split": "train",
"index": 37995,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Factor the expression $3x(x+1) + 7(x+1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(3x+7)(x+1)"
} | {
"split": "train",
"index": 37996,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$, $b$, $c$, and $d$ are the solutions of the equation $x^4 - bx - 3 = 0$, then an equation whose solutions are $\\frac{a + b + c}{d^2}$, $\\frac{a + b + d}{c^2}$, $\\frac{a + c + d}{b^2}$, $\\frac{b + c + d}{a^2}$ is Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "3x^4 - bx^3 - 1 = 0"
} | {
"split": "train",
"index": 37997,
"solution": "1. **Identify the roots of the given polynomial**: Given the polynomial $x^4 - bx - 3 = 0$, we know that its roots are $a$, $b$, $c$, and $d$.\n\n2. **Use Vieta's formulas**: Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots... |
deepscaler | [
{
"role": "user",
"content": "Let \\( A, B, C \\) be points on the same plane with \\( \\angle ACB = 120^\\circ \\). There is a sequence of circles \\( \\omega_0, \\omega_1, \\omega_2, \\ldots \\) on the same plane (with corresponding radii \\( r_0, r_1, r_2, \\ldots \\) where \\( r_0 > r_1 > r_2 > \\cdots ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2} + \\sqrt{3}"
} | {
"split": "train",
"index": 37998,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lines $x = \\frac{1}{4}y + a$ and $y = \\frac{1}{4}x + b$ intersect at the point $(1,2)$. What is $a + b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"split": "train",
"index": 37999,
"solution": "1. **Given Equations and Intersection Point**: We are given the equations of two lines:\n - $x = \\frac{1}{4}y + a$\n - $y = \\frac{1}{4}x + b$\n These lines intersect at the point $(1,2)$.\n\n2. **Substitute the Intersection Point into Each Equation**:\n -... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $k$ if the side lengths of four squares are shown, and the area of the fifth square is $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 38000,
"solution": "Let $s$ be the side length of the square with area $k$. \n The sum of the heights of the squares on the right side is $3+8=11$. \n The sum of the heights of the squares on the left side is $1+s+4=s+5$. \n Since the two sums are equal, then $s+5=11$, and so $s=6$. \... |
deepscaler | [
{
"role": "user",
"content": "A chessboard’s squares are labeled with numbers as follows:\n\n[asy]\nunitsize(0.8 cm);\n\nint i, j;\n\nfor (i = 0; i <= 8; ++i) {\n draw((i,0)--(i,8));\n draw((0,i)--(8,i));\n}\n\nfor (i = 0; i <= 7; ++i) {\nfor (j = 0; j <= 7; ++j) {\n label(\"$\\frac{1}{\" + string(9 - i ... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"split": "train",
"index": 38001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In writing the integers from 100 through 199 inclusive, how many times is the digit 7 written? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 38002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider $7$ points on a circle. Compute the number of ways there are to draw chords between pairs of points such that two chords never intersect and one point can only belong to one chord. It is acceptable to draw no chords. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 38003,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Of the natural numbers greater than 1000 that are composed of the digits $0, 1, 2$ (where each digit can be used any number of times or not at all), in ascending order, what is the position of 2010? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 38004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the rectangular region is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": ".088 m^2"
} | {
"split": "train",
"index": 38005,
"solution": "To find the area of a rectangular region, we use the formula:\n\\[ \\text{Area} = \\text{length} \\times \\text{width} \\]\n\nGiven the dimensions of the rectangle are 0.4 meters and 0.22 meters, we calculate the area as follows:\n1. Convert the decimal numbers to ... |
deepscaler | [
{
"role": "user",
"content": "In a game, Jimmy and Jacob each randomly choose to either roll a fair six-sided die or to automatically roll a $1$ on their die. If the product of the two numbers face up on their dice is even, Jimmy wins the game. Otherwise, Jacob wins. The probability Jimmy wins $3$ games... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 38006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each square in the following hexomino has side length 1. Find the minimum area of any rectangle that contains the entire hexomino. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{21}{2}"
} | {
"split": "train",
"index": 38007,
"solution": "If a rectangle contains the entire hexomino, it must also contain its convex hull, which is an origin-symmetric hexagon. It is fairly clear that the smallest rectangle that contains such a hexagon must share one set of parallel sides with the hexagon. There are thr... |
deepscaler | [
{
"role": "user",
"content": "Consider a $2 \\times 2$ grid of squares. Each of the squares will be colored with one of 10 colors, and two colorings are considered equivalent if one can be rotated to form the other. How many distinct colorings are there? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "2530"
} | {
"split": "train",
"index": 38008,
"solution": "This solution will be presented in the general case with $n$ colors. Our problem asks for $n=10$. We isolate three cases: Case 1: Every unit square has the same color In this case there are clearly $n$ ways to color the square. Case 2: Two non-adjacent squares are ... |
deepscaler | [
{
"role": "user",
"content": "How many four-digit positive integers are multiples of 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3000"
} | {
"split": "train",
"index": 38009,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An isosceles triangle has side lengths 8 cm, 8 cm and 10 cm. The longest side of a similar triangle is 25 cm. What is the perimeter of the larger triangle, in centimeters? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 38010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Below is an arithmetic expression where 9 Chinese characters represent the digits 1 to 9, and different characters represent different digits. What is the maximum possible value for the expression?\n\n草 $\\times$ 绿 + 花儿 $\\times$ 红 + 春光明 $\\times$ 媚 Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "6242"
} | {
"split": "train",
"index": 38011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many numbers are in the list $ -48, -41, -34, \\ldots, 65, 72?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 38012,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alexio has 120 cards numbered from 1 to 120, inclusive, and places them in a box. He then randomly picks a card. What is the probability that the number on the card is a multiple of 2, 4, or 6? Express your answer as a common fraction. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 38013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $P$ is randomly placed in the interior of the right triangle below. What is the probability that the area of triangle $PBC$ is less than half of the area of triangle $ABC$? Express your answer as a common fraction. [asy]\nsize(7cm);\ndefaultpen(linewidth(0.7));\npair ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 38014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $(x^2+1)(2x+1)^9 = a_0 + a_1(x+2) + a_2(x+2)^2 + \\ldots + a_{11}(x+2)^{11}$, then calculate the value of $a_0 + a_1 + a_2 + \\ldots + a_{11}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 38015,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 38016,
"solution": "1. **Understanding the Problem**: A sphere with radius $6$ is tangent to all sides of a triangle with sides $15, 15, 24$. We need to find the distance from the center of the sphere, $O$, to the plane of the triangle.\n\n2. **Triangle's Inradius**: The triangle's si... |
deepscaler | [
{
"role": "user",
"content": "In a regular triangular prism \\(ABC-A_1B_1C_1\\), all 9 edges are equal in length. Point \\(P\\) is the midpoint of \\(CC_1\\). The dihedral angle \\(B-A_1P-B_1 = \\alpha\\). Find \\(\\sin \\alpha\\). Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{4}"
} | {
"split": "train",
"index": 38017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive two-digit integers are there in which each of the two digits is prime? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 38018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the real numbers $x, y, z$ are such that $x^2 + 4y^2 + 16z^2 = 48$ and $xy + 4yz + 2zx = 24$ , what is the value of $x^2 + y^2 + z^2$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 38019,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f(x)$ takes positive real numbers to real numbers, such that\n\\[xf(y) - yf(x) = f \\left( \\frac{x}{y} \\right)\\]for all positive real numbers $x$ and $y.$ Find all possible values of $f(100).$ Enter all possible values, separated by commas. Let's think step b... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 38020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The set of vectors $\\mathbf{u}$ such that\n\\[\\mathbf{u} \\cdot \\mathbf{u} = \\mathbf{u} \\cdot \\begin{pmatrix} 6 \\\\ -28 \\\\ 12 \\end{pmatrix}\\] forms a solid in space. Find the volume of this solid. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3} \\pi \\cdot 241^{3/2}"
} | {
"split": "train",
"index": 38021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a,b,c$ are non-negative integers less than $7$ such that \\begin{align*}\na+2b+3c&\\equiv 0\\pmod 7,\\\\\n2a+3b+c&\\equiv 4\\pmod 7,\\\\\n3a+b+2c&\\equiv 4\\pmod 7,\n\\end{align*}then determine the remainder when $abc$ is divided by $7$. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 38022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Thirty teams play in a league where each team plays every other team exactly once, and every game results in a win or loss with no ties. Each game is independent with a $50\\%$ chance of either team winning. Determine the probability that no two teams end up with the same numbe... | math | {
"style": "rule",
"ground_truth": "409"
} | {
"split": "train",
"index": 38023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "95"
} | {
"split": "train",
"index": 38024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of \\(2^{1+2+3}-(2^1+2^2+2^3)?\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 38025,
"solution": "1. **Simplify the Exponent:** \n The expression given is $2^{1+2+3} - (2^1 + 2^2 + 2^3)$. First, simplify the exponent in the term $2^{1+2+3}$:\n \\[\n 1+2+3 = 6\n \\]\n Thus, the expression becomes:\n \\[\n 2^6 - (2^1 + 2^2 + 2^3)\n \\]\n\n2. **Ev... |
deepscaler | [
{
"role": "user",
"content": "Find, with proof, the number of positive integers whose base- $n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by $\\pm 1$ from some digit further to the left. (Your answer should be an explicit function ... | math | {
"style": "rule",
"ground_truth": "2^{n+1} - 2(n+1)"
} | {
"split": "train",
"index": 38026,
"solution": "Let a $k$ -good sequence be a sequence of distinct integers $\\{ a_i \\}_{i=1}^k$ such that for all integers $2\\le i \\le k$ , $a_i$ differs from some preceding term by $\\pm 1$ .\nLemma. Let $a$ be an integer. Then there are $2^{k-1}$ $k$ -good sequences starti... |
deepscaler | [
{
"role": "user",
"content": "Find the value of $c$ such that all the roots of the polynomial $x^3 - 5x^2 + 2bx - c$ are real and positive, given that one root is twice another and four times the third. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1000}{343}"
} | {
"split": "train",
"index": 38027,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three positive integers are each greater than $1$, have a product of $1728$, and are pairwise relatively prime. What is their sum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 38028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{M}$ be a matrix, and let $\\mathbf{v}$ and $\\mathbf{w}$ be vectors, such that\n\\[\\mathbf{M} \\mathbf{v} = \\begin{pmatrix} 2 \\\\ 3 \\end{pmatrix} \\quad \\text{and} \\quad \\mathbf{M} \\mathbf{w} = \\begin{pmatrix} -2 \\\\ -5 \\end{pmatrix}.\\]Compute $\\mathb... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -4 \\\\ -12 \\end{pmatrix}"
} | {
"split": "train",
"index": 38029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equations: \n① $3(x-1)^3 = 24$; \n② $(x-3)^2 = 64$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 38030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A *substring* of a number $n$ is a number formed by removing some digits from the beginning and end of $n$ (possibly a different number of digits is removed from each side). Find the sum of all prime numbers $p$ that have the property that any substring of $p$ is also p... | math | {
"style": "rule",
"ground_truth": "576"
} | {
"split": "train",
"index": 38031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The school choir originally had 36 members, and 9 more people joined. Calculate the total number of people in the choir now. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 38032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a circle centered at $O$, point $A$ is on the circle, and $\\overline{BA}$ is tangent to the circle at $A$. Triangle $ABC$ is right-angled at $A$ with $\\angle ABC = 45^\\circ$. The circle intersects $\\overline{BO}$ at $D$. Chord $\\overline{BC}$ also extends to meet the ci... | math | {
"style": "rule",
"ground_truth": "\\frac{2 - \\sqrt{2}}{2}"
} | {
"split": "train",
"index": 38033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest number with three different prime factors, none of which can be less than 10? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2431"
} | {
"split": "train",
"index": 38034,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number such that when it is multiplied by 9, the resulting number consists of the same digits in a different order. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1089"
} | {
"split": "train",
"index": 38035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the next two smallest juicy numbers after 6, and show a decomposition of 1 into unit fractions for each of these numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12, 15"
} | {
"split": "train",
"index": 38036,
"solution": "12 and 15: $1=\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{6}+\\frac{1}{12}$, $1=\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{10}+\\frac{1}{15}$."
} |
deepscaler | [
{
"role": "user",
"content": "How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5 \\times 5$ square array of dots? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 38037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that 5 students are to be distributed into two groups, A and B, with at least one person in each group, and student A cannot be in group A, calculate the number of different distribution schemes. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 38038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "12. If $p$ is the smallest positive prime number such that there exists an integer $n$ for which $p$ divides $n^{2}+5n+23$, then $p=$ ______ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 38039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many values of $k$ is $60^{10}$ the least common multiple of the positive integers $10^{10}$, $12^{12}$, and $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 38040,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.... | math | {
"style": "rule",
"ground_truth": "\\frac{29}{120}"
} | {
"split": "train",
"index": 38041,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that line $l_{1}$ passes through points $A(m,1)$ and $B(-3,4)$, and line $l_{2}$ passes through points $C(1,m)$ and $D(-1,m+1)$, for what value of $m$ do the lines $l_{1}$ and $l_{2}$\n\n$(1)$ intersect at a right angle;\n\n$(2)$ are parallel;\n\n$(3)$ the angle of inclin... | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 38042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the quadratic function $f(x) = x^2 + ax + b$ such that\n\\[\\frac{f(f(x) + x)}{f(x)} = x^2 + 1776x + 2010.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^2 + 1774x + 235"
} | {
"split": "train",
"index": 38043,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin \\left(\\alpha+ \\frac {\\pi}{3}\\right)=- \\frac {4}{5}$, and $- \\frac {\\pi}{2} < \\alpha < 0$, find $\\cos \\alpha=$ ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3-4 \\sqrt {3}}{10}"
} | {
"split": "train",
"index": 38044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the chord length intercepted by the circle $x^{2}+y^{2}+2x-4y+1=0$ on the line $ax-by+2=0$ $(a > 0, b > 0)$ is 4, find the minimum value of $\\frac{1}{a} + \\frac{1}{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2} + \\sqrt{2}"
} | {
"split": "train",
"index": 38045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to put 6 balls in 4 boxes if the balls are not distinguishable but the boxes are? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 38046,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse in the Cartesian coordinate system $xOy$, its center is at the origin, the left focus is $F(-\\sqrt{3},0)$, and the right vertex is $D(2,0)$. Let point $A\\left( 1,\\frac{1}{2} \\right)$.\n\n(Ⅰ) Find the standard equation of the ellipse;\n\n(Ⅱ) If a line passin... | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}"
} | {
"split": "train",
"index": 38047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2x^{3}+ax^{2}+bx+1$, it reaches an extreme value of $-6$ at $x=1$.\n\n(1) Find the values of the real numbers $a$ and $b$.\n(2) Find the maximum and minimum values of the function $f(x)$ on the interval $[−2,2]$. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 38048,
"solution": ""
} |
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