data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Suppose that $N$ is a three digit number divisible by $7$ such that upon removing its middle digit, the remaining two digit number is also divisible by $7$ . What is the minimum possible value of $N$ ?\n\n*2019 CCA Math Bonanza Lightning Round #3.1* Let's think step by st... | math | {
"style": "rule",
"ground_truth": "154"
} | {
"split": "train",
"index": 6400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sandy likes to eat waffles for breakfast. To make them, she centers a circle of waffle batter of radius 3 cm at the origin of the coordinate plane and her waffle iron imprints non-overlapping unit-square holes centered at each lattice point. How many of these holes are containe... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 6401,
"solution": "First, note that each divet must have its sides parallel to the coordinate axes; if the divet centered at the lattice point $(a, b)$ does not have this orientation, then it contains the point $(a+1 / 2, b)$ in its interior, so it necessarily overlaps with the divet ... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{v}_0$ be a vector. The vector $\\mathbf{v}_0$ is projected onto $\\begin{pmatrix} 3 \\\\ 1 \\end{pmatrix},$ resulting in the vector $\\mathbf{v}_1.$ The vector $\\mathbf{v}_1$ is then projected onto $\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix},$ resulting in the ve... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} \\frac{3}{5} & \\frac{1}{5} \\\\ \\frac{3}{5} & \\frac{1}{5} \\end{pmatrix}"
} | {
"split": "train",
"index": 6402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given six cards with the digits $1, 2, 4, 5, 8$ and a comma. Using each card exactly once, various numbers are formed (the comma cannot be at the beginning or at the end of the number). What is the arithmetic mean of all such numbers?\n\n(M. V. Karlukova) Let's think step by st... | math | {
"style": "rule",
"ground_truth": "1234.4321"
} | {
"split": "train",
"index": 6403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three cones are placed on a table on their bases, touching each other. The radii of their bases are 1, 12, and 12, and the apex angles of the cones are $-4 \\operatorname{arctg} \\frac{1}{3}$, $4 \\operatorname{arctg} \\frac{2}{3}$, and $4 \\operatorname{arctg} \\frac{2}{3}$ re... | math | {
"style": "rule",
"ground_truth": "40/21"
} | {
"split": "train",
"index": 6404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value in simplest form of the following expression?\n\\sqrt{1} + \\sqrt{1+3} + \\sqrt{1+3+5} + \\sqrt{1+3+5+7} Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 6405,
"solution": "1. **Identify the pattern in the expression**: The expression given is \\(\\sqrt{1} + \\sqrt{1+3} + \\sqrt{1+3+5} + \\sqrt{1+3+5+7}\\). We need to simplify each term under the square root.\n\n2. **Simplify each term**:\n - The first term is \\(\\sqrt{1}\\).\n - ... |
deepscaler | [
{
"role": "user",
"content": "Given that circle $\\odot M$ passes through the point $(1,0)$ and is tangent to the line $x=-1$, $S$ is a moving point on the trajectory of the center $M$ of the circle, and $T$ is a moving point on the line $x+y+4=0$. Find the minimum value of $|ST|$. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 6406,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the 2011 Shanghai Spring College Entrance Examination, there were 8 colleges recruiting students. If exactly 3 students were admitted by 2 of these colleges, then the number of ways this could happen is __________. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 6407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a $4 \\times 4$ grid, place candies according to the following requirements: (1) Each cell must contain candies; (2) In adjacent cells, the left cell has 1 fewer candy than the right cell and the upper cell has 2 fewer candies than the lower cell; (3) The bottom-right cell c... | math | {
"style": "rule",
"ground_truth": "248"
} | {
"split": "train",
"index": 6408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(x)$ be a polynomial of degree $3n$ such that\n\\begin{align*} P(0) = P(3) = \\dots = P(3n) &= 2, \\\\ P(1) = P(4) = \\dots = P(3n+1-2) &= 1, \\\\ P(2) = P(5) = \\dots = P(3n+2-2) &= 0. \\end{align*}\nAlso, $P(3n+1) = 730$. Determine $n$. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 6409,
"solution": "To solve for $n$, we start by analyzing the polynomial $P(x)$ given its values at specific points and its degree. We use Lagrange Interpolation Formula to express $P(x)$, and then evaluate it at $x = 3n+1$ to find $n$.\n\n1. **Constructing the Polynomial Using Lagra... |
deepscaler | [
{
"role": "user",
"content": "Calculate: $(128)^{\\frac{7}{3}}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65536 \\cdot \\sqrt[3]{2}"
} | {
"split": "train",
"index": 6410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of two positive integers plus their sum is 103. The integers are relatively prime, and each is less than 20. What is the sum of the two integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 6411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A boat has a speed of $15$ mph in still water. In a stream that has a current of $5$ mph it travels a certain distance downstream and returns. The ratio of the average speed for the round trip to the speed in still water is: Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"split": "train",
"index": 6412,
"solution": "1. **Identify speeds and distances:**\n - Speed of the boat in still water: $15$ mph.\n - Speed of the current: $5$ mph.\n - Downstream speed (boat speed + current speed): $15 + 5 = 20$ mph.\n - Upstream speed (boat speed - current speed): $15 - 5 = 10$ mph.... |
deepscaler | [
{
"role": "user",
"content": "Determine the value of $a^3 + b^3$ given that $a+b=12$ and $ab=20$, and also return the result for $(a+b-c)(a^3+b^3)$, where $c=a-b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4032"
} | {
"split": "train",
"index": 6413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the complex number $z$ satisfying the equation $2z-3i\\bar{z}=-7+3i$. Note that $\\bar{z}$ denotes the conjugate of $z$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1+3i"
} | {
"split": "train",
"index": 6414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCDEF$ be a regular hexagon with each side length $s$. Points $G$, $H$, $I$, $J$, $K$, and $L$ are the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$, respectively. The segments $\\overline{AH}$, $\\overline{BI}$, $\\overline{CJ}$, $\\overline{DK}$, $\\overline... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 6415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct digits can appear as the units digit of an integral perfect-square number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 6416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For an arithmetic sequence $a_1, a_2, a_3, \\dots,$ let\n\\[ S_n = a_1 + a_2 + a_3 + \\dots + a_n, \\]\nand let\n\\[ T_n = S_1 + S_2 + S_3 + \\dots + S_n. \\]\nGiven the value of $S_{2023}$, determine the smallest integer $n$ for which you can uniquely determine the value of $T... | math | {
"style": "rule",
"ground_truth": "3034"
} | {
"split": "train",
"index": 6417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x_1, x_2, \\ldots, x_7$ be natural numbers, and $x_1 < x_2 < \\ldots < x_6 < x_7$, also $x_1 + x_2 + \\ldots + x_7 = 159$, then the maximum value of $x_1 + x_2 + x_3$ is. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 6418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify and then evaluate the expression: $$(1- \\frac {1}{a-2})÷ \\frac {a^{2}-6a+9}{2a-4}$$, where $$a=2 \\sqrt {3}+3$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 6419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $f(x+3)=3x^2 + 7x + 4$ and $f(x)=ax^2 + bx + c$. What is $a+b+c$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 6420,
"solution": "1. **Given Equations**: We are given that $f(x+3) = 3x^2 + 7x + 4$ and $f(x) = ax^2 + bx + c$.\n\n2. **Expression for $f(x+3)$**: Using the expression for $f(x)$, we can write:\n \\[\n f(x+3) = a(x+3)^2 + b(x+3) + c\n \\]\n Expanding $(x+3)^2$ and $(x+3)$, w... |
deepscaler | [
{
"role": "user",
"content": "For an arithmetic sequence \\(a_1, a_2, a_3, \\dots\\), let\n\\[ S_n = a_1 + a_2 + a_3 + \\dots + a_n, \\]\nand let\n\\[ T_n = S_1 + S_2 + S_3 + \\dots + S_n. \\]\nIf you know the value of \\( S_{2023}, \\) then you can uniquely determine the value of \\( T_n \\) for some integ... | math | {
"style": "rule",
"ground_truth": "3034"
} | {
"split": "train",
"index": 6421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x \\) and \\( y \\) be positive integers such that\n\n\\[ x^2 + y^2 - 2017xy > 0 \\]\n\nand it is not a perfect square. Find the minimum value of \\( x^2 + y^2 - 2017xy \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2019"
} | {
"split": "train",
"index": 6422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different lines pass through at least two points in this 4-by-4 grid of lattice points? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 6423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vehicle A and Vehicle B start from points A and B, respectively, at the same time and travel towards each other. They meet after 3 hours, at which point Vehicle A turns back towards point A, and Vehicle B continues forward. After Vehicle A reaches point A, it turns around and h... | math | {
"style": "rule",
"ground_truth": "7.2"
} | {
"split": "train",
"index": 6424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The region shown is bounded by the arcs of circles having radius 5 units, each with a central angle measure of 45 degrees, intersecting at points of tangency. The area of the region can be expressed in the form $a\\sqrt{b} + c\\pi$ square units, where $\\sqrt{b}$ is a radical i... | math | {
"style": "rule",
"ground_truth": "-12.625"
} | {
"split": "train",
"index": 6425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of $999 + 999$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"split": "train",
"index": 6426,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first term of a geometric sequence is 250. If the sum of the first 50 terms is 625 and the sum of the first 100 terms is 1225, find the sum of the first 150 terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1801"
} | {
"split": "train",
"index": 6427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all integers \\( n \\) such that \\( n^{4} + 6n^{3} + 11n^{2} + 3n + 31 \\) is a perfect square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 6428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Riquinho distributed 1000,00 reais among his friends Antônio, Bernardo, and Carlos in the following manner: he gave, successively, 1 real to Antônio, 2 reais to Bernardo, 3 reais to Carlos, 4 reais to Antônio, 5 reais to Bernardo, etc. How much money did Bernardo receive? Let's... | math | {
"style": "rule",
"ground_truth": "345"
} | {
"split": "train",
"index": 6429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Increasing the radius of a cylinder by $6$ units increased the volume by $y$ cubic units. Increasing the height of the cylinder by $6$ units also increases the volume by $y$ cubic units. If the original height is $2$, then the original radius is:\n$\\text{(A) } 2 \\qquad \\text... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 6430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = ax+b$, where $a$ and $b$ are real constants, and $g(x) = 2x - 5$. Suppose that for all $x$, it is true that $g(f(x)) = 3x + 4$. What is $a+b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 6431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\sqrt{2}\\sin \\left( 2x- \\frac{\\pi}{4} \\right)$, where $x\\in\\mathbb{R}$, if the maximum and minimum values of $f(x)$ in the interval $\\left[ \\frac{\\pi}{8}, \\frac{3\\pi}{4} \\right]$ are $a$ and $b$ respectively, then the value of $a+b$ is __... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}-1"
} | {
"split": "train",
"index": 6432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two natural ways to inscribe a square in a given isosceles right triangle. If it is done as in Figure 1 below, then one finds that the area of the square is $441 \\text{cm}^2$. What is the area (in $\\text{cm}^2$) of the square inscribed in the same $\\triangle ABC$ a... | math | {
"style": "rule",
"ground_truth": "392"
} | {
"split": "train",
"index": 6433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The region consisting of all points in three-dimensional space within 4 units of line segment $\\overline{CD}$, plus a cone with the same height as $\\overline{CD}$ and a base radius of 4 units, has a total volume of $448\\pi$. Find the length of $\\textit{CD}$. Let's think ste... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 6434,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\(P Q R S T\\) is a pentagon with \\(P Q=8\\), \\(Q R=2\\), \\(R S=13\\), \\(S T=13\\), and \\(T P=8\\). Also, \\(\\angle T P Q=\\angle P Q R=90^\\circ\\). What is the area of pentagon \\(P Q R S T\\) ? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 6435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define\n\\[ A' = \\frac{1}{1^2} + \\frac{1}{7^2} - \\frac{1}{11^2} - \\frac{1}{13^2} + \\frac{1}{19^2} + \\frac{1}{23^2} - \\dotsb, \\]\nwhich omits all terms of the form $\\frac{1}{n^2}$ where $n$ is an odd multiple of 5, and\n\\[ B' = \\frac{1}{5^2} - \\frac{1}{25^2} + \\frac... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 6436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the line passing through the point $P(2,1)$ intersects the graph of the function $f(x)= \\frac{2x+3}{2x-4}$ at points $A$ and $B$, and $O$ is the origin, calculate the value of $( \\overrightarrow{OA}+ \\overrightarrow{OB})\\cdot \\overrightarrow{OP}$. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 6437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lines $L_1, L_2, \\dots, L_{100}$ are distinct. All lines $L_{4n}$, where $n$ is a positive integer, are parallel to each other. All lines $L_{4n-3}$, where $n$ is a positive integer, pass through a given point $A$. The maximum number of points of intersection of pairs of lines... | math | {
"style": "rule",
"ground_truth": "4351"
} | {
"split": "train",
"index": 6438,
"solution": "We begin by partitioning the set of lines $\\{L_1, L_2, \\dots, L_{100}\\}$ into three subsets based on their properties:\n\n1. **Set $X$**: Contains lines $L_n$ where $n \\equiv 0 \\pmod{4}$. These lines are all parallel to each other.\n2. **Set $Y$**: Contains lin... |
deepscaler | [
{
"role": "user",
"content": "Cyclic quadrilateral \\(ABCD\\) has side lengths \\(AB = 1\\), \\(BC = 2\\), \\(CD = 3\\), and \\(AD = 4\\). Determine \\(\\frac{AC}{BD}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5/7"
} | {
"split": "train",
"index": 6439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube of mass $m$ slides down the felt end of a ramp semicircular of radius $h$ , reaching a height $h/2$ at the opposite extreme.\nFind the numerical coefficient of friction $\\mu_k$ bretween the cube and the surface.\n\n*Proposed by Danilo Tejeda, Atlantida* Let's thi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{1 - \\left(\\frac{1}{2\\pi}\\right)^2}}"
} | {
"split": "train",
"index": 6440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the roots of the equation $\\tan^2x - 8\\tan x + 2 = 0$ that are between $x = 0$ and $x = 2\\pi$ radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\pi"
} | {
"split": "train",
"index": 6441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Agricultural researchers randomly selected $50$ plants of a certain crop in an experimental field for research. The frequency distribution of the mass (in grams) of a single plant falling into each group is shown in the table below:\n\n| Mass of a Single Plant (in grams) | Freq... | math | {
"style": "rule",
"ground_truth": "0.8413"
} | {
"split": "train",
"index": 6442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many cubic centimeters are in the volume of a cone having a diameter of 10cm and a height of 6cm? Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50 \\pi"
} | {
"split": "train",
"index": 6443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers between $100$ and $150$ have three different digits in increasing order? One such integer is $129$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 6444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the terminal side of the angle $α+ \\frac {π}{6}$ passes through point P($-1$, $-2\\sqrt {2}$), find the value of $\\sinα$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1-2\\sqrt {6}}{6}"
} | {
"split": "train",
"index": 6445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given three rays $AB$, $BC$, $BB_{1}$ are not coplanar, and the diagonals of quadrilaterals $BB_{1}A_{1}A$ and $BB_{1}C_{1}C$ bisect each other, and $\\overrightarrow{AC_{1}}=x\\overrightarrow{AB}+2y\\overrightarrow{BC}+3z\\overrightarrow{CC_{1}}$, find the value of $x+y+z$. Le... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{6}"
} | {
"split": "train",
"index": 6446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 10 people who want to choose a committee of 5 people among them. They do this by first electing a set of $1,2,3$, or 4 committee leaders, who then choose among the remaining people to complete the 5-person committee. In how many ways can the committee be formed, assum... | math | {
"style": "rule",
"ground_truth": "7560"
} | {
"split": "train",
"index": 6447,
"solution": "There are $\\binom{10}{5}$ ways to choose the 5-person committee. After choosing the committee, there are $2^{5}-2=30$ ways to choose the leaders. So the answer is $30 \\cdot\\binom{10}{5}=7560$."
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=- \\sqrt {3}\\sin ^{2}x+\\sin x\\cos x$.\n(1) Find the value of $f( \\dfrac {25π}{6})$;\n(2) Let $α∈(0,π)$, $f( \\dfrac {α}{2})= \\dfrac {1}{4}- \\dfrac { \\sqrt {3}}{2}$, find the value of $\\sin α$. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\dfrac {1+3 \\sqrt {5}}{8}"
} | {
"split": "train",
"index": 6448,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos \\alpha = -\\frac{4}{5}$, and $\\alpha$ is an angle in the third quadrant, find the values of $\\sin \\alpha$ and $\\tan \\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 6449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many two-digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits is a perfect square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 6450,
"solution": "1. **Express $N$ in terms of its digits**: Let $N = 10t + u$, where $t$ is the tens digit and $u$ is the units digit. This is a standard way to express a two-digit number in terms of its digits.\n\n2. **Formulate the condition given in the problem**: We need to find... |
deepscaler | [
{
"role": "user",
"content": "Three coplanar squares with sides of lengths two, four and six units, respectively, are arranged side-by-side, as shown so that one side of each square lies on line $AB$ and a segment connects the bottom left corner of the smallest square to the upper right corner of the larges... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 6451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f: \\mathbb{N} \\rightarrow \\mathbb{N} \\) be a function that satisfies\n\n\\[ f(1) = 2, \\]\n\\[ f(2) = 1, \\]\n\\[ f(3n) = 3f(n), \\]\n\\[ f(3n + 1) = 3f(n) + 2, \\]\n\\[ f(3n + 2) = 3f(n) + 1. \\]\n\nFind how many integers \\( n \\leq 2014 \\) satisfy \\( f(n) = 2n ... | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 6452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers at minimum must be selected from the set $\\{1, 2, \\ldots, 20\\}$ to ensure that this selection includes two integers \\(a\\) and \\(b\\) such that \\(a - b = 2\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 6453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the sequence \\(\\{a_n\\}\\) satisfies \\(a_1 = \\frac{2}{3}\\) and \\(a_{n+1} - a_n = \\sqrt{\\frac{2}{3} \\left(a_{n+1} + a_n\\right)}\\), then find the value of \\(a_{2015}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1354080"
} | {
"split": "train",
"index": 6454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\[f(x) =\n\\begin{cases}\nx^2+2 &\\text{if } x<n, \\\\\n2x+5 &\\text{if }x\\ge{n}.\n\\end{cases}\n\\]If the graph $y=f(x)$ is continuous, 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": "2"
} | {
"split": "train",
"index": 6455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In spherical coordinates, the point $\\left( 3, \\frac{2 \\pi}{7}, \\frac{8 \\pi}{5} \\right)$ is equivalent to what other point, in the standard spherical coordinate representation? Enter your answer in the form $(\\rho,\\theta,\\phi),$ where $\\rho > 0,$ $0 \\le \\theta < 2 ... | math | {
"style": "rule",
"ground_truth": "\\left( 3, \\frac{9 \\pi}{7}, \\frac{2 \\pi}{5} \\right)"
} | {
"split": "train",
"index": 6456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression: $(a-\\frac{3a}{a+1}$) $÷\\frac{{a}^{2}-4a+4}{a+1}$, then choose a number you like from $-2$, $-1$, and $2$ to substitute for $a$ and calculate the value. 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": 6457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a_n\\}$, $a_1=1$, $a_{n+1} > a_n$, $(a_n+a_{n+1}-1)^2=4a_na_{n+1}$ $(n\\in\\mathbb{N}^*)$.\n\n$(1)$ Find the general formula for the sequence $\\{a_n\\}$;\n\n$(2)$ Let $b_n=\\frac{1}{\\sqrt[4]{a_n}}$, $T_n=b_1+b_2+\\ldots+b_n$, estimate the integer part o... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 6458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an $8 \\times 8$ chessboard, how many ways are there to select 56 squares so that all the black squares are selected, and each row and each column has exactly seven squares selected? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "576"
} | {
"split": "train",
"index": 6459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function\n$$\nf(x) = \\left|8x^3 - 12x - a\\right| + a\n$$\n\nThe maximum value of this function on the interval \\([0, 1]\\) is 0. 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": "-2\\sqrt{2}"
} | {
"split": "train",
"index": 6460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest common divisor of $8!$ and $(6!)^2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7200"
} | {
"split": "train",
"index": 6461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F\\_1$, $F\\_2$. Let $P$ be a point on the ellipse, and $\\triangle F\\_1 P F\\_2$ have centroid $G$ and incenter $I$. If $\\overrightarrow{IG} = λ(1,0) (λ ≠ 0)$, find the eccentri... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 6462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An $10 \\times 25$ rectangle is divided into two congruent polygons, and these polygons are rearranged to form a rectangle again. Determine the length of the smaller side of the resulting rectangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 6463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the $6$ sides and the $9$ diagonals of a regular hexagon are randomly and independently colored red, blue, or green with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the hexagon such that all of its si... | math | {
"style": "rule",
"ground_truth": "\\frac{872}{1000}"
} | {
"split": "train",
"index": 6464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the integer $m > 3$ for which\n\\[\\log_{10} (m - 3)! + \\log_{10} (m - 1)! + 3 = 2 \\log_{10} m!.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 6465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a > b > 0$, and $a + b = 2$, find the minimum value of $$\\frac {3a-b}{a^{2}+2ab-3b^{2}}$$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3+ \\sqrt {5}}{4}"
} | {
"split": "train",
"index": 6466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the $10$ numbers $0-9$, select $3$ numbers. Find:<br/>\n$(1)$ How many unique three-digit numbers can be formed without repeating any digits?<br/>\n$(2)$ How many unique three-digit odd numbers can be formed without repeating any digits? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "320"
} | {
"split": "train",
"index": 6467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A committee of 5 is to be chosen from a group of 9 people. How many ways can it be chosen, if Bill and Karl must serve together or not at all, and Alice and Jane refuse to serve with each other? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 6468,
"solution": "If Bill and Karl are on the committee, there are $\\binom{7}{3}=35$ ways for the other group members to be chosen. However, if Alice and Jane are on the committee with Bill and Karl, there are $\\binom{5}{1}=5$ ways for the last member to be chosen, yielding 5 unacc... |
deepscaler | [
{
"role": "user",
"content": "Given that $F$ is the right focus of the hyperbola $C$: $x^{2}- \\frac {y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6 \\sqrt {6})$. When the perimeter of $\\triangle APF$ is minimized, the area of this triangle is \\_\\_\\_\\_\\_\\_. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "12 \\sqrt {6}"
} | {
"split": "train",
"index": 6469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The solutions to the equation $x(5x+2) = 6(5x+2)$ are ___. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{5}"
} | {
"split": "train",
"index": 6470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line passes through the point $(-2, \\sqrt{3})$ at an angle of inclination of $\\frac{\\pi}{3}$. Find the parametric equation of the line, and if it intersects the curve $y^2 = -x - 1$ at points A and B, find the length $|AB|$. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"split": "train",
"index": 6471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A leak formed in the hold of a ship. A pump was immediately switched on to remove the water, but it couldn't keep up, and after 10 minutes, the water level rose by 20 cm. Then, a second pump of equal power was turned on, and after 5 minutes, the water level dropped by 10 cm. Th... | math | {
"style": "rule",
"ground_truth": "1.25"
} | {
"split": "train",
"index": 6472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of nonzero real numbers. Let $f : S \\to \\mathbb{R}$ be a function such that\n\n(i) $f(1) = 1,$\n(ii) $f \\left( \\frac{1}{x + y} \\right) = f \\left( \\frac{1}{x} \\right) + f \\left( \\frac{1}{y} \\right)$ for all $x,$ $y \\in S$ such that $x + y \\in S,$... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 6473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two identical cups, A and B. Cup A is half-filled with pure water, and cup B is fully filled with a 50% alcohol solution. First, half of the alcohol solution from cup B is poured into cup A and mixed thoroughly. Then, half of the alcohol solution in cup A is poured ba... | math | {
"style": "rule",
"ground_truth": "3/7"
} | {
"split": "train",
"index": 6474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A woman labels the squares of a very large chessboard $1$ through $64$. On each square $k$, the woman puts $2^k$ grains of rice. How many more grains of rice are placed on the $10^{th}$ square than on the first $8$ squares combined? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "514"
} | {
"split": "train",
"index": 6475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $(2x^3)^3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8x^9"
} | {
"split": "train",
"index": 6476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in triangle $\\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is known that $\\frac{{c \\sin C}}{a} - \\sin C = \\frac{{b \\sin B}}{a} - \\sin A$, $b = 4$. Find:\n\n$(1)$ The measure of angle $B$;\n\n$(2)$ If ... | math | {
"style": "rule",
"ground_truth": "4 + \\frac{{4\\sqrt{3}}}{3}"
} | {
"split": "train",
"index": 6477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $y=(m^2-m-1)x^{m^2-3m-3}$ is a power function, and it is an increasing function on the interval $(0, +\\infty)$. Find the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 6478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The focus of the parabola $y^{2}=4x$ is $F$, and the equation of the line $l$ is $x=ty+7$. Line $l$ intersects the parabola at points $M$ and $N$, and $\\overrightarrow{MF}⋅\\overrightarrow{NF}=0$. The tangents to the parabola at points $M$ and $N$ intersect at point $P$. Find ... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 6479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the function $f(x)$ is monotonic in its domain $(-\\infty, +\\infty)$, and for any real number $x$, it satisfies $f(f(x)+e^{x})=1-e$, where $e$ is the base of the natural logarithm, determine the value of $f(\\ln 2)$. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 6480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-88"
} | {
"split": "train",
"index": 6481,
"solution": "1. **Identify the Roots**: Given that all roots of the polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers, and their sum (as coefficients of $z^5$ with a sign change) is 10, we consider possible sets of roots that sum to 10. The roots are ... |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle and a circle intersect so that each side of the triangle contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the triangle to the area of the circle? Express your answer as a common fraction in term... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{4\\pi}"
} | {
"split": "train",
"index": 6482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(ABC\\) be a triangle with \\(AB=8, AC=12\\), and \\(BC=5\\). Let \\(M\\) be the second intersection of the internal angle bisector of \\(\\angle BAC\\) with the circumcircle of \\(ABC\\). Let \\(\\omega\\) be the circle centered at \\(M\\) tangent to \\(AB\\) and \\(AC\\... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 6483,
"solution": "Redefine \\(D\\) as the reflection of \\(A\\) across the perpendicular bisector \\(l\\) of \\(BC\\). We prove that \\(DB\\) and \\(DC\\) are both tangent to \\(\\omega\\), and hence the two definitions of \\(D\\) align. Indeed, this follows by symmetry; we have that... |
deepscaler | [
{
"role": "user",
"content": "Find the numbers \\( x \\) between 0 and 30 for which the sine of \\( x \\) degrees equals the sine of \\( x \\) radians. How many such numbers exist between 30 and 90? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 6484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integer points $(x, y)$ in the first quadrant satisfy $x + y > 8$ and $x \\leq y \\leq 8$. Determine the number of such integer points $(x, y)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 6485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $b$ and $c$ be real numbers. If the polynomial $x^3 + bx^2 + cx + d$ has exactly one real root and $d = c + b + 1$, find the value of the product of all possible values of $c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 6486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be a positive real number. Find the maximum possible value of $$\\frac{x^2+2-\\sqrt{x^4+4}}{x}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}-2"
} | {
"split": "train",
"index": 6487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all positive rational numbers $r$ such that when the two numbers $r$ and $55r$ are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction.... | math | {
"style": "rule",
"ground_truth": "719"
} | {
"split": "train",
"index": 6488,
"solution": "Denote $r = \\frac{a}{b}$, where $\\left( a, b \\right) = 1$. We have $55 r = \\frac{55a}{b}$. Suppose $\\left( 55, b \\right) = 1$, then the sum of the numerator and the denominator of $55r$ is $55a + b$. This cannot be equal to the sum of the numerator and the den... |
deepscaler | [
{
"role": "user",
"content": "How many distinct, positive factors does $1100$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 6489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equation \\( x^{[x]} = \\frac{9}{2} \\) for real numbers \\( x \\), where \\( [x] \\) represents the greatest integer less than or equal to \\( x \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 6490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \\(ABCD\\) is a rectangle whose diagonals meet at \\(E\\). The perimeter of triangle \\(ABE\\) is \\(10\\pi\\) and the perimeter of triangle \\(ADE\\) is \\(n\\). Compute the number of possible integer values of \\(n\\). Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 6491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 10 different televisions, including 3 type A, 3 type B, and 4 type C. Now, 3 televisions are randomly selected from them. If at least two different types are included, calculate the total number of different ways to select them. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "114"
} | {
"split": "train",
"index": 6492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The distance from city $A$ to city $B$ is $999$ km. Along the highway leading from $A$ to $B$, there are kilometer markers indicating the distances from the marker to $A$ and $B$ as shown:\n\n\n\nHow many of these markers use only two... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 6493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given complex numbers $z_{1}=m+i$ and $z_{2}=2+mi$, where $i$ is the imaginary unit and $m\\in R$. \n$(1)$ If $z_{1}\\cdot z_{2}$ is a pure imaginary number, find the value of $m$. \n$(2)$ If ${z}_{1}^{2}-2{z}_{1}+2=0$, find the imaginary part of $\\frac{{z}_{2}}{{z}_{1}}$. L... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 6494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A dealer bought $n$ radios for $d$ dollars, $d$ a positive integer. He contributed two radios to a community bazaar at half their cost. The rest he sold at a profit of $8 on each radio sold. If the overall profit was $72, then the least possible value of $n$ for the given in... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 6495,
"solution": "1. **Calculate the cost and selling price of the radios:**\n - The dealer bought $n$ radios for $d$ dollars, so the cost per radio is $\\frac{d}{n}$.\n - He sold two radios at half their cost, which is $\\frac{1}{2} \\times \\frac{d}{n} = \\frac{d}{2n}$ each. Th... |
deepscaler | [
{
"role": "user",
"content": "The height of a triangle, equal to 2, divides the angle of the triangle in the ratio 2:1, and the base of the triangle into parts, the smaller of which is equal to 1. Find the area of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11/3"
} | {
"split": "train",
"index": 6496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function $f$ satisfies $f(4x) = 4f(x)$ for all positive real values of $x$, and $f(x) = 2 - |x - 3|$ for $2 \\leq x \\leq 4$. Find the smallest \\( x \\) for which \\( f(x) = f(2022) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2022"
} | {
"split": "train",
"index": 6497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert $645_{10}$ to base 5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10400_5"
} | {
"split": "train",
"index": 6498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular dodecagon \\( Q_1 Q_2 \\dotsb Q_{12} \\) is drawn in the coordinate plane with \\( Q_1 \\) at \\( (4,0) \\) and \\( Q_7 \\) at \\( (2,0) \\). If \\( Q_n \\) is the point \\( (x_n,y_n) \\), compute the numerical value of the product\n\\[\n(x_1 + y_1 i)(x_2 + y_2 i)(x_3... | math | {
"style": "rule",
"ground_truth": "531440"
} | {
"split": "train",
"index": 6499,
"solution": ""
} |
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