data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Each of the five numbers 1, 4, 7, 10, and 13 is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is\n[asy] draw(... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 6000,
"solution": "\nLet's denote the five squares as follows, where $a$, $b$, $c$, $d$, and $e$ represent the numbers 1, 4, 7, 10, and 13 placed in the squares:\n\n\\[\n\\begin{array}{|c|c|}\n\\hline\na & b \\\\\n\\hline\nc & d \\\\\n\\hline\n\\end{array}\n\\quad\n\\begin{array}{|c|}... |
deepscaler | [
{
"role": "user",
"content": "If $k \\in [-2, 2]$, find the probability that for the value of $k$, there can be two tangents drawn from the point A(1, 1) to the circle $x^2 + y^2 + kx - 2y - \\frac{5}{4}k = 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 6001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Al and Barb start their new jobs on the same day. Al's schedule is 3 work-days followed by 1 rest-day. Barb's schedule is 7 work-days followed by 3 rest-days. On how many of their first 1000 days do both have rest-days on the same day? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 6002,
"solution": "1. **Identify the Cycles**: Al's schedule is a cycle of 4 days (3 work-days followed by 1 rest-day), and Barb's schedule is a cycle of 10 days (7 work-days followed by 3 rest-days).\n\n2. **Calculate the Least Common Multiple (LCM)**: To find when their schedules al... |
deepscaler | [
{
"role": "user",
"content": "The length of rectangle $ABCD$ is 5 inches and its width is 3 inches. Diagonal $AC$ is divided into three equal segments by points $E$ and $F$. The area of triangle $BEF$, expressed in square inches, is: Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 6003,
"solution": "1. **Draw the rectangle and identify key components**: Let rectangle $ABCD$ have $AB = 5$ inches (length) and $AD = 3$ inches (width). Diagonal $AC$ is drawn, which divides the rectangle into two right triangles, $ABC$ and $ADC$.\n\n2. **Calculate the length of diag... |
deepscaler | [
{
"role": "user",
"content": "(1) Given the complex number $z=3+bi$ ($i$ is the imaginary unit, $b$ is a positive real number), and $(z-2)^{2}$ is a pure imaginary number, find the complex number $z$;\n(2) Given that the sum of all binomial coefficients in the expansion of $(3x+ \\frac{1}{ \\sqrt{x}})^{n}$ ... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 6004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a polynomial $P(x)$ with integer coefficients such that\\[P(x)=\\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}\\]holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "220"
} | {
"split": "train",
"index": 6005,
"solution": "Note that $2022 = 210\\cdot 9 +132$. Since the only way to express $132$ in terms of $105$, $70$, $42$, or $30$ is $135 = 30+30+30+42$, we are essentially just counting the number of ways to express $210*9$ in terms of these numbers. Since $210 = 2*105=3*70=5*42=7*3... |
deepscaler | [
{
"role": "user",
"content": "The unknown real numbers $x, y, z$ satisfy the equations $$\\frac{x+y}{1+z}=\\frac{1-z+z^{2}}{x^{2}-x y+y^{2}} ; \\quad \\frac{x-y}{3-z}=\\frac{9+3 z+z^{2}}{x^{2}+x y+y^{2}}$$ Find $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{14}"
} | {
"split": "train",
"index": 6006,
"solution": "$\\sqrt[3]{14}$ Cross-multiplying in both equations, we get, respectively, $x^{3}+y^{3}=$ $1+z^{3}, x^{3}-y^{3}=27-z^{3}$. Now adding gives $2 x^{3}=28$, or $x=\\sqrt[3]{14}$."
} |
deepscaler | [
{
"role": "user",
"content": "In the rhombus \\(A B C D\\), the angle at vertex \\(A\\) is \\(60^{\\circ}\\). Point \\(N\\) divides side \\(A B\\) in the ratio \\(A N: B N = 2: 1\\). Find the tangent of angle \\(D N C\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{243}{121}}"
} | {
"split": "train",
"index": 6007,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $f$ is a function and $f^{-1}$ is the inverse of $f$. If $f(3)=4$, $f(5)=1$, and $f(2)=5$, evaluate $f^{-1}\\left(f^{-1}(5)+f^{-1}(4)\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 6008,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$ and $b$ are positive real numbers satisfying $9a^{2}+b^{2}=1$, find the maximum value of $\\frac{ab}{3a+b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{12}"
} | {
"split": "train",
"index": 6009,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first term of a sequence is 1. Each subsequent term is 4 times the square root of the sum of all preceding terms plus 4.\n\nWhat is the sum of the first 1995 terms of the sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15912121"
} | {
"split": "train",
"index": 6010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least common multiple of 14 and 21? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 6011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \"$x^2-2x-3>0$\" is a necessary but not sufficient condition for \"$x<a$\", then the maximum value of $a$ is. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 6012,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given non-zero vectors $\\overrightarrow{a}, \\overrightarrow{b}$, if $(\\overrightarrow{a} - 2\\overrightarrow{b}) \\perp \\overrightarrow{a}$ and $(\\overrightarrow{b} - 2\\overrightarrow{a}) \\perp \\overrightarrow{b}$, find the angle between $\\overrightarrow{a}$ and $\\ove... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 6013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be positive real numbers such that $4x + 9y = 60.$ Find the maximum value of $xy.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 6014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 6015,
"solution": "1. **Initial Set and Operation Definition**: We start with a set of tiles numbered from 1 to 100. The operation, denoted as $P(x)$, involves removing all tiles numbered with a perfect square and renumbering the remaining tiles consecutively starting with 1.\n\n2. **... |
deepscaler | [
{
"role": "user",
"content": "At a school trip, there are 8 students and a teacher. They want to take pictures in groups where each group consists of either 4 or 5 students. How many different group combinations can they make? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 6016,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse \\( C: \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 \\) where \\( a > b > 0 \\), with the left focus at \\( F \\). A tangent to the ellipse is drawn at a point \\( A \\) on the ellipse and intersects the \\( y \\)-axis at point \\( Q \\). Let \\( O \\) be the or... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{3}"
} | {
"split": "train",
"index": 6017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose we flip four coins simultaneously: a penny, a nickel, a dime, and a quarter. What is the probability that the penny and nickel both come up heads? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{4}"
} | {
"split": "train",
"index": 6018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive real numbers $\\lambda$ such that for all integers $n\\geq 2$ and all positive real numbers $a_1,a_2,\\cdots,a_n$ with $a_1+a_2+\\cdots+a_n=n$, the following inequality holds:\n$\\sum_{i=1}^n\\frac{1}{a_i}-\\lambda\\prod_{i=1}^{n}\\frac{1}{a_i}\\leq n-\\lambda... | math | {
"style": "rule",
"ground_truth": "\\lambda \\geq e"
} | {
"split": "train",
"index": 6019,
"solution": "\nFind all positive real numbers \\(\\lambda\\) such that for all integers \\(n \\geq 2\\) and all positive real numbers \\(a_1, a_2, \\ldots, a_n\\) with \\(a_1 + a_2 + \\cdots + a_n = n\\), the following inequality holds:\n\\[\n\\sum_{i=1}^n \\frac{1}{a_i} - \\lam... |
deepscaler | [
{
"role": "user",
"content": "Consider two circles defined by the equations $x^2 + y^2 - 6x + 4y - 20 = 0$ and $x^2 + y^2 - 2x - 6y + 10 = 0$. Find the slope of the line that passes through their intersection points. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.4"
} | {
"split": "train",
"index": 6020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle and a regular hexagon have equal perimeters. If the triangle's area is 4, what is the area of the hexagon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 6021,
"solution": "1. **Assign Variables:**\n Let the side length of the equilateral triangle be $s$ and the side length of the regular hexagon be $t$.\n\n2. **Equal Perimeters:**\n Since the triangle and the hexagon have equal perimeters, and the triangle has 3 sides while the he... |
deepscaler | [
{
"role": "user",
"content": "Twenty cubical blocks are arranged as shown. First, 10 are arranged in a triangular pattern; then a layer of 6, arranged in a triangular pattern, is centered on the 10; then a layer of 3, arranged in a triangular pattern, is centered on the 6; and finally one block is centered ... | math | {
"style": "rule",
"ground_truth": "114"
} | {
"split": "train",
"index": 6022,
"solution": "To solve this problem, we need to understand how the numbers assigned to each block in the pyramid structure contribute to the number on the top block. The structure is built in layers with each block in the upper layers resting on three blocks from the layer below.... |
deepscaler | [
{
"role": "user",
"content": "Timur and Alexander are counting the trees growing around the house. They move in the same direction but start counting from different trees. How many trees are growing around the house if the tree that Timur counted as the 12th, Alexander counted as the 33rd, and the tree that... | math | {
"style": "rule",
"ground_truth": "118"
} | {
"split": "train",
"index": 6023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the parliament of the island nation Promenade-and-Tornado, only native inhabitants, who are divided into knights and liars, can be elected. Knights always tell the truth, and liars always lie. In the latest term, 2020 deputies were elected. At the first plenary session of th... | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 6024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\left\\{a_{n}\\right\\}$ is defined such that $a_{n}$ is the last digit of the sum $1 + 2 + \\cdots + n$. Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\\left\\{a_{n}\\right\\}$. Calculate $S_{2016}$. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "7066"
} | {
"split": "train",
"index": 6025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $\\sec x+\\tan x=\\frac{22}7$ and that $\\csc x+\\cot x=\\frac mn,$ where $\\frac mn$ is in lowest terms. Find $m+n^{}_{}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 6026,
"solution": "We are given that $\\frac{1+\\sin x}{\\cos x}=\\frac{22}7\\implies\\frac{1+\\sin x}{\\cos x}\\cdot\\frac{1-\\sin x}{1-\\sin x}=\\frac{1-\\sin^2x}{\\cos x(1-\\sin x)}=\\frac{\\cos^2x}{\\cos x(1-\\sin x)}$ $=\\frac{\\cos x}{1-\\sin x}$, or equivalently, $\\cos x=\\fra... |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x$ and $y$ are positive real numbers such that $x^{2}-xy+2y^{2}=8$. Find the maximum possible value of $x^{2}+xy+2y^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{72+32 \\sqrt{2}}{7}"
} | {
"split": "train",
"index": 6027,
"solution": "Let $u=x^{2}+2y^{2}$. By AM-GM, $u \\geq \\sqrt{8}xy$, so $xy \\leq \\frac{u}{\\sqrt{8}}$. If we let $xy=ku$ where $k \\leq \\frac{1}{\\sqrt{8}}$, then we have $u(1-k)=8$ and $u(1+k)=x^{2}+xy+2y^{2}$, that is, $u(1+k)=8 \\cdot \\frac{1+k}{1-k}$. It is not hard to se... |
deepscaler | [
{
"role": "user",
"content": "Given vectors $a$ and $b$ that satisfy $(a+2b)\\cdot(5a-4b)=0$, and $|a|=|b|=1$, find the angle $\\theta$ between $a$ and $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\pi}{3}"
} | {
"split": "train",
"index": 6028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the four real roots of the quartic polynomial $f(x)$ form an arithmetic sequence with a common difference of $2$, calculate the difference between the maximum root and the minimum root of $f'(x)$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 6029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first $n$ terms of the arithmetic sequences ${a_n}$ and ${b_n}$ are $S_n$ and $T_n$ respectively. If $$\\frac {S_{n}}{T_{n}}= \\frac {2n+1}{3n+2}$$, find the value of $$\\frac {a_{3}+a_{11}+a_{19}}{b_{7}+b_{15}}$$. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{129}{130}"
} | {
"split": "train",
"index": 6030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $k$ so that the line $3x + 5y + k = 0$ is tangent to the parabola $y^2 = 24x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 6031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\frac{x^2}{1+x^2}$.\n\n$(1)$ Calculate the values of $f(2) + f\\left(\\frac{1}{2}\\right)$, $f(3) + f\\left(\\frac{1}{3}\\right)$, $f(4) + f\\left(\\frac{1}{4}\\right)$, and infer a general conclusion (proof not required);\n\n$(2)$ Calculate the valu... | math | {
"style": "rule",
"ground_truth": "4032"
} | {
"split": "train",
"index": 6032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F$ is the right focus of the hyperbola $C$: ${{x}^{2}}-\\dfrac{{{y}^{2}}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,4)$. When the perimeter of $\\Delta APF$ is minimized, the area of this triangle is \\_\\_\\_. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "\\dfrac{36}{7}"
} | {
"split": "train",
"index": 6033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f : \\mathbb{N}\\rightarrow \\mathbb{N}$ satisfying following condition:\n\\[f(n+1)>f(f(n)), \\quad \\forall n \\in \\mathbb{N}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "{f(n)=n}"
} | {
"split": "train",
"index": 6034,
"solution": "\nWe are tasked with finding all functions \\( f: \\mathbb{N} \\rightarrow \\mathbb{N} \\) that satisfy the condition:\n\\[\nf(n+1) > f(f(n)), \\quad \\forall n \\in \\mathbb{N}.\n\\]\n\nTo solve this problem, let us first analyze the condition given:\n\\[\nf(n+1) >... |
deepscaler | [
{
"role": "user",
"content": "John has 15 marbles of different colors, including two reds, two greens, and two blues. In how many ways can he choose 5 marbles, if exactly one of the chosen marbles must be red and one must be green? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "660"
} | {
"split": "train",
"index": 6035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside triangle $ABC$, there are 1997 points. Using the vertices $A, B, C$ and these 1997 points, into how many smaller triangles can the original triangle be divided? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3995"
} | {
"split": "train",
"index": 6036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the consumer, a single discount of $n\\%$ is more advantageous than any of the following discounts:\n(1) two successive $15\\%$ discounts\n(2) three successive $10\\%$ discounts\n(3) a $25\\%$ discount followed by a $5\\%$ discount\nWhat is the smallest possible positive i... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 6037,
"solution": "Let the original price be $x$. We need to find the smallest integer $n$ such that a single discount of $n\\%$ is more advantageous than any of the given series of discounts.\n\n1. **Calculating the effective discount for option (1):**\n Two successive $15\\%$ disc... |
deepscaler | [
{
"role": "user",
"content": "A function $g$ from the integers to the integers is defined as follows:\n\\[g(n) = \\left\\{\n\\begin{array}{cl}\nn + 5 & \\text{if $n$ is odd}, \\\\\nn/2 & \\text{if $n$ is even}.\n\\end{array}\n\\right.\\]\nSuppose $m$ is odd and $g(g(g(m))) = 39.$ Find $m.$ Let's think step... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 6038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the integers \\(1, 2, 3, \\ldots, 9\\) is assigned to a vertex of a regular 9-sided polygon (every vertex receives exactly one integer from \\(\\{1, 2, \\ldots, 9\\}\\), and no two vertices receive the same integer) so that the sum of the integers assigned to any three ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 6039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A radio program has a quiz consisting of $3$ multiple-choice questions, each with $3$ choices. A contestant wins if he or she gets $2$ or more of the questions right. The contestant answers randomly to each question. What is the probability of winning? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{27}"
} | {
"split": "train",
"index": 6040,
"solution": "To solve this problem, we need to calculate the probability that the contestant guesses at least 2 out of 3 questions correctly. We will consider the following cases:\n\n1. **Case 1: The contestant guesses all three questions correctly.**\n2. **Case 2: The contestan... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the equation of circle $M$ is given by $x^2+y^2-4x\\cos \\alpha-2y\\sin \\alpha+3\\cos^2\\alpha=0$ (where $\\alpha$ is a parameter), and the parametric equation of line $l$ is $\\begin{cases}x=\\tan \\theta \\\\ y=1+t\\sin \\theta\\end{... | math | {
"style": "rule",
"ground_truth": "\\dfrac {4 \\sqrt {3}}{3}"
} | {
"split": "train",
"index": 6041,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( AEF \\) be a triangle with \\( EF = 20 \\) and \\( AE = AF = 21 \\). Let \\( B \\) and \\( D \\) be points chosen on segments \\( AE \\) and \\( AF \\), respectively, such that \\( BD \\) is parallel to \\( EF \\). Point \\( C \\) is chosen in the interior of triangle \... | math | {
"style": "rule",
"ground_truth": "5300"
} | {
"split": "train",
"index": 6042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "My grandpa has 10 pieces of art, including 3 prints by Escher. If he hangs the pieces of art in a row in a random order, what is the probability that all three pieces by Escher will be placed consecutively? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{15}"
} | {
"split": "train",
"index": 6043,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If Alex gives Bob a penny, Bob will have three times as many pennies as Alex has. If Bob gives Alex a penny, Bob will have twice as many pennies as Alex has. How many pennies does Bob currently have? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 6044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integer triples $(x,y,z)$ are there such that $\\begin{array}{rcl} x - yz^2&\\equiv & 1 \\pmod {13} \\\\ xz+y&\\equiv& 4 \\pmod {13} \\end{array}$ where $0\\leq x < 13$ , $0\\leq y <13$ , and $0\\leq z< 13$ . Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 6045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can the number 1024 be factored into three natural factors such that the first factor is a multiple of the second, and the second is a multiple of the third? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 6046,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular 2015-gon \\( A_{1} A_{2} \\cdots A_{2015} \\) is inscribed in a unit circle \\( O \\). What is the probability that for any two distinct vertices \\( A_{i}, A_{j} \\), the magnitude \\( \\left|\\overrightarrow{O A_{i}}+\\overrightarrow{O A_{j}}\\right| \\geqslant 1 \\... | math | {
"style": "rule",
"ground_truth": "671/1007"
} | {
"split": "train",
"index": 6047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A parabola is given by the equation $y^{2}=4x$. Let $F$ be its focus, and let $A$ and $B$ be the points where a line passing through $F$ intersects the parabola. Let $O$ be the origin of the coordinate system. If $|AF|=3$, find the area of $\\triangle AOB$. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 6048,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the random variable $\\xi$ follows the normal distribution $N(1, 4)$, if $P(\\xi > 4) = 0.1$, then $P(-2 \\leq \\xi \\leq 4)$ equals _______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.8"
} | {
"split": "train",
"index": 6049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$, where $a_{n+1} + (-1)^n a_n = 2n - 1$, calculate the sum of the first 12 terms of $\\{a_n\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 6050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=4\\sin({8x-\\frac{π}{9}})$, $x\\in \\left[0,+\\infty \\right)$, determine the initial phase of this harmonic motion. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\pi}{9}"
} | {
"split": "train",
"index": 6051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can the sequence $1,2,3,4,5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 6052,
"solution": "To solve this problem, we need to count the number of permutations of the sequence $1, 2, 3, 4, 5$ such that no three consecutive terms are either strictly increasing or strictly decreasing. We analyze the problem by considering the possible patterns of increases an... |
deepscaler | [
{
"role": "user",
"content": "Let $P(x) = (x-1)(x-2)(x-3)$. For how many polynomials $Q(x)$ does there exist a polynomial $R(x)$ of degree 3 such that $P\\left(Q(x)\\right) = P(x)\\cdot R(x)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 6053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $d_1 = a^2 + 2^a + a \\cdot 2^{(a+1)/2} + a^3$ and $d_2 = a^2 + 2^a - a \\cdot 2^{(a+1)/2} + a^3$. If $1 \\le a \\le 300$, how many integral values of $a$ are there such that $d_1 \\cdot d_2$ is a multiple of $3$? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 6054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three players play tic-tac-toe together. In other words, the three players take turns placing an \"A\", \"B\", and \"C\", respectively, in one of the free spots of a $3 \\times 3$ grid, and the first player to have three of their label in a row, column, or diagonal wins. How ma... | math | {
"style": "rule",
"ground_truth": "148"
} | {
"split": "train",
"index": 6055,
"solution": "In all winning cases for the third player, every spot in the grid must be filled. There are two ways that player C wins along a diagonal, and six ways that player C wins along a row or column. In the former case, any arrangement of the As and Bs is a valid board, si... |
deepscaler | [
{
"role": "user",
"content": "The $52$ cards in a deck are numbered $1, 2, \\cdots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with hig... | math | {
"style": "rule",
"ground_truth": "263"
} | {
"split": "train",
"index": 6056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On an island, \\( \\frac{2}{3} \\) of all men are married and \\( \\frac{3}{5} \\) of all women are married. What fraction of the island's population is married? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12/19"
} | {
"split": "train",
"index": 6057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Lakers are playing the Celtics in the NBA finals. To win the NBA finals, a team must win 4 games before the other team does. If the Celtics win each game with probability $\\dfrac{2}{3}$ and there are no ties, what is the probability that the Lakers will win the NBA finals ... | math | {
"style": "rule",
"ground_truth": "\\frac{160}{2187}"
} | {
"split": "train",
"index": 6058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If real numbers \\( x \\) and \\( y \\) satisfy the relation \\( xy - x - y = 1 \\), calculate the minimum value of \\( x^{2} + y^{2} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6 - 4\\sqrt{2}"
} | {
"split": "train",
"index": 6059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The remainders when three positive integers are divided by 5 are 1, 2, and 3. Find the remainder when their product is divided by 5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 6060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product \\( \\left(1-\\frac{1}{3}\\right)\\left(1-\\frac{1}{4}\\right)\\left(1-\\frac{1}{5}\\right) \\) is equal to what? 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": 6061,
"solution": "Simplifying, \\( \\left(1-\\frac{1}{3}\\right)\\left(1-\\frac{1}{4}\\right)\\left(1-\\frac{1}{5}\\right)=\\left(\\frac{2}{3}\\right)\\left(\\frac{3}{4}\\right)\\left(\\frac{4}{5}\\right) \\). We can simplify further by dividing equal numerators and denominators to o... |
deepscaler | [
{
"role": "user",
"content": "The image depicts a top-down view of a three-layered pyramid made of 14 identical cubes. Each cube is assigned a natural number in such a way that the numbers corresponding to the cubes in the bottom layer are all different, and the number on any other cube is the sum of the nu... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 6062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the following pseudocode:\n\n```\nS = 0\ni = 1\nDo\n S = S + i\n i = i + 2\nLoop while S ≤ 200\nn = i - 2\nOutput n\n```\n\nWhat is the value of the positive integer $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 6063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define \\( n! = 1 \\times 2 \\times \\ldots \\times n \\), for example \\( 5! = 1 \\times 2 \\times 3 \\times 4 \\times 5 \\). If \\(\\frac{n! \\times (n+1)!}{2}\\) (where \\( \\mathbf{n} \\) is a positive integer and \\( 1 \\leq n \\leq 100 \\)) is a perfect square, what is th... | math | {
"style": "rule",
"ground_truth": "273"
} | {
"split": "train",
"index": 6064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a parallelogram where each vertex has integer coordinates and is located at $(0,0)$, $(4,5)$, $(11,5)$, and $(7,0)$. Calculate the sum of the perimeter and the area of this parallelogram. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9\\sqrt{41}"
} | {
"split": "train",
"index": 6065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a 10cm×10cm×10cm cube cut into 1cm×1cm×1cm small cubes, determine the maximum number of small cubes that can be left unused when reassembling the small cubes into a larger hollow cube with no surface voids. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "134"
} | {
"split": "train",
"index": 6066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the period of the function $y = \\tan(2x) + \\cot(2x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 6067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=|2x+1|-|x|-2$.\n(1) Solve the inequality $f(x)\\geqslant 0$;\n(2) If there exists a real number $x$ such that $f(x)-a\\leqslant |x|$, find the minimum 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": "-3"
} | {
"split": "train",
"index": 6068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose I have a bag with 12 slips of paper in it. Some of the slips have a 2 on them, and the rest have a 7 on them. If the expected value of the number shown on a slip randomly drawn from the bag is $3.25$, then how many slips have a 2? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 6069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of diagonals of a polygon with 150 sides and determine if 9900 represents half of this diagonal count. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11025"
} | {
"split": "train",
"index": 6070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An exchange point conducts two types of operations:\n\n1) Give 2 euros - receive 3 dollars and a candy as a gift;\n2) Give 5 dollars - receive 3 euros and a candy as a gift.\n\nWhen the wealthy Buratino came to the exchange point, he had only dollars. When he left, he had fewer... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 6071,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sqrt{3}\\sin^{2}(x+\\frac{\\pi}{4})-\\cos^{2}x-\\frac{1+\\sqrt{3}}{2} (x\\in\\mathbb{R})$.\n(1) Find the minimum value and the minimum positive period of the function $f(x)$;\n(2) If $A$ is an acute angle, and vector $\\overrightarrow{m}=(1,5)$ is per... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{3}+3}{10}"
} | {
"split": "train",
"index": 6072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose lines $p$ and $q$ in the first quadrant are defined by the equations $y = -2x + 8$ and $y = -3x + 9$, respectively. What is the probability that a point randomly selected in the 1st quadrant and below $p$ will fall between $p$ and $q$? Express your answer as a decimal t... | math | {
"style": "rule",
"ground_truth": "0.16"
} | {
"split": "train",
"index": 6073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum of the fourth powers of the roots of the equation \\[ x\\sqrt{x} - 8x + 9\\sqrt{x} - 2 = 0, \\] assuming all roots are real and nonnegative. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"split": "train",
"index": 6074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular prism has 4 green faces, 2 yellow faces, and 6 blue faces. What's the probability that when it is rolled, a blue face will be facing up? 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": 6075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are five positive integers that are divisors of each number in the list $$60, 120, -30, 180, 240$$. Find the sum of these five positive integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 6076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The opposite number of $-1 \\frac{1}{2}$ is ______, its reciprocal is ______, and its absolute value is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 6077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group of n friends wrote a math contest consisting of eight short-answer problem $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$ , and four full-solution problems $F_1, F_2, F_3, F_4$ . Each person in the group correctly solved exactly 11 of the 12 problems. We create an 8 x 4 t... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 6078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square pyramid with base $ABCD$ and vertex $E$ has eight edges of length 4. A plane passes through the midpoints of $\\overline{AE}$, $\\overline{BC}$, and $\\overline{CD}$. The plane's intersection with the pyramid has an area that can be expressed as $\\sqrt{p}$. Find $p$. ... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 6079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S=\\{1,2,4,8,16,32,64,128,256\\}$. A subset $P$ of $S$ is called squarely if it is nonempty and the sum of its elements is a perfect square. A squarely set $Q$ is called super squarely if it is not a proper subset of any squarely set. Find the number of super squarely sets... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 6080,
"solution": "Clearly we may biject squarely sets with binary representations of perfect squares between 1 and $2^{0}+\\cdots+2^{8}=2^{9}-1=511$, so there are 22 squarely sets, corresponding to $n^{2}$ for $n=1,2, \\ldots, 22$. For convenience, we say $N$ is (super) squarely if a... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=3^{x}$, where $f(a+2)=27$, and the function $g(x)=λ \\cdot 2^{ax}-4^{x}$ with a domain of $[0,2]$, find:\n\n(1) The value of $a$;\n(2) The value of the real number $λ$ if the maximum value of function $g(x)$ is $\\frac{1}{3}$. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 6081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The $52$ cards in a deck are numbered $1, 2, \\cdots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards form a team, and the two persons with hig... | math | {
"style": "rule",
"ground_truth": "263"
} | {
"split": "train",
"index": 6082,
"solution": "Once the two cards are drawn, there are $\\dbinom{50}{2} = 1225$ ways for the other two people to draw. Alex and Dylan are the team with higher numbers if Blair and Corey both draw below $a$, which occurs in $\\dbinom{a-1}{2}$ ways. Alex and Dylan are the team with ... |
deepscaler | [
{
"role": "user",
"content": "In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, and every head either always tells the truth or always lies. Each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them sai... | math | {
"style": "rule",
"ground_truth": "176"
} | {
"split": "train",
"index": 6083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The $GEB$ sequence $1,3,7,12, \\ldots$ is defined by the following properties: (i) the GEB sequence is increasing (that is, each term is larger than the previous term), (ii) the sequence formed using the differences between each pair of consecutive terms in the GEB sequence (na... | math | {
"style": "rule",
"ground_truth": "5764"
} | {
"split": "train",
"index": 6084,
"solution": "We refer to the two sequences as the GEB sequence and the difference sequence. Since the GEB sequence is increasing and since each positive integer that does not occur in the GEB sequence must occur in the difference sequence, then each positive integer less than 12... |
deepscaler | [
{
"role": "user",
"content": "Suppose that we have a right triangle $ABC$ with the right angle at $B$ such that $AC = \\sqrt{61}$ and $AB = 5.$ A circle is drawn with its center on $AB$ such that the circle is tangent to $AC$ and $BC.$ If $P$ is the point where the circle and side $AC$ meet, then what is $C... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 6085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Price of some item has decreased by $5\\%$ . Then price increased by $40\\%$ and now it is $1352.06\\$ $ cheaper than doubled original price. How much did the item originally cost? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"split": "train",
"index": 6086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a $2\\times 3$ grid where each entry is either $0$ , $1$ , or $2$ . For how many such grids is the sum of the numbers in every row and in every column a multiple of $3$ ? One valid grid is shown below: $$ \\begin{bmatrix} 1 & 2 & 0 2 & 1 & 0 \\end{bmatrix} $$ ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 6087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $25^{64} \\cdot 64^{25}$ is the square of a positive integer $N$. In decimal representation, the sum of the digits of $N$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 6088,
"solution": "1. **Identify the expression for $N$:** \n Given that $N^2 = 25^{64} \\cdot 64^{25}$, we find $N$ by taking the square root:\n \\[\n N = \\sqrt{25^{64} \\cdot 64^{25}}\n \\]\n\n2. **Simplify the expression using properties of exponents:** \n Recognize th... |
deepscaler | [
{
"role": "user",
"content": "Alexa wrote the first $16$ numbers of a sequence:\n\\[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …\\]\nThen she continued following the same pattern, until she had $2015$ numbers in total. \nWhat was the last number she wrote?\n Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "1344"
} | {
"split": "train",
"index": 6089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A merchant had 10 barrels of sugar, which he arranged into a pyramid as shown in the illustration. Each barrel, except one, was numbered. The merchant accidentally arranged the barrels such that the sum of the numbers along each row equaled 16. Could you rearrange the barrels s... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 6090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown? [asy]\n/* AMC8 2003 #15 Problem */\ndraw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)-... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 6091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle with a length of 6 cm and a width of 4 cm is rotated around one of its sides. Find the volume of the resulting geometric solid (express the answer in terms of $\\pi$). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144\\pi"
} | {
"split": "train",
"index": 6092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the product \\[ (n-1) \\cdot n \\cdot (n+1) \\cdot (n+2) \\cdot (n+3), \\] where $n=2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 6093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $P$ is inside triangle $\\triangle ABC$. Line $AC$ intersects line $BP$ at $Q$, and line $AB$ intersects line $CP$ at $R$. Given that $AR = RB = CP$, and $CQ = PQ$, find $\\angle BRC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 6094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The percent that $M$ is greater than $N$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{100(M-N)}{N}"
} | {
"split": "train",
"index": 6095,
"solution": "1. **Identify the Increase**: The difference $M - N$ represents the amount by which $M$ is greater than $N$.\n\n2. **Convert to a Fraction**: To find out by what percent $M$ is greater than $N$, we need to express the increase as a fraction of $N$. This fraction is ... |
deepscaler | [
{
"role": "user",
"content": "Assume that the scores $X$ of 400,000 students in a math mock exam in Yunnan Province approximately follow a normal distribution $N(98,100)$. It is known that a student's score ranks among the top 9100 in the province. Then, the student's math score will not be less than ______... | math | {
"style": "rule",
"ground_truth": "118"
} | {
"split": "train",
"index": 6096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an isosceles triangle \\(ABC\\) with \\(\\angle A = 30^\\circ\\) and \\(AB = AC\\). Point \\(D\\) is the midpoint of \\(BC\\). Point \\(P\\) is chosen on segment \\(AD\\), and point \\(Q\\) is chosen on side \\(AB\\) such that \\(PB = PQ\\). What is the measure of angle \... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 6097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow {a}$ and $\\overrightarrow {b}$ with magnitudes $|\\overrightarrow {a}| = 6\\sqrt {3}$ and $|\\overrightarrow {b}| = \\frac {1}{3}$, and their dot product $\\overrightarrow {a} \\cdot \\overrightarrow {b} = -3$, determine the angle $\\theta$ betw... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{6}"
} | {
"split": "train",
"index": 6098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. What is the probability that $3$ draws are required? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 6099,
"solution": "To solve this problem, we need to determine the probability that exactly three draws are required for the sum of the values drawn to exceed $4$. This means that the sum of the values of the first two chips drawn must be $4$ or less, and the sum of the values of the ... |
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