data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "For what values of the constant $c$ does the graph of $f(x) = \\frac{x^2-x+c}{x^2+x-20}$ have exactly one vertical asymptote?\n\nEnter all possible values, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-12 \\text{ or } -30"
} | {
"split": "train",
"index": 32649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 120 five-digit numbers formed by the digits 1, 2, 3, 4, 5, arranged in descending order. The 95th number is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21354"
} | {
"split": "train",
"index": 32650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the last 3 digits of \\(1 \\times 3 \\times 5 \\times 7 \\times \\cdots \\times 2005\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "375"
} | {
"split": "train",
"index": 32651,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.3 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (... | math | {
"style": "rule",
"ground_truth": "f(4 - x)"
} | {
"split": "train",
"index": 32652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express the sum as a common fraction: $.1 + .02 + .003 + .0004 + .00005.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{2469}{20,\\!000}"
} | {
"split": "train",
"index": 32653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An ant starts at the point $(0,0)$ in the Cartesian plane. In the first minute, the ant faces towards $(1,0)$ and walks one unit. Each subsequent minute, the ant chooses an angle $\\theta$ uniformly at random in the interval $\\left[-90^{\\circ}, 90^{\\circ}\\right]$, and then ... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 32654,
"solution": "Let $\\alpha_{k}$ be a random variable that represents the turn made after step $k$, choosing $\\alpha_{k}$ uniformly at random on the complex plane among the arc of the unit circle containing 1 from $-i$ to $i$. It is well known that $\\mathbb{E}\\left[\\alpha_{k}... |
deepscaler | [
{
"role": "user",
"content": "a) In how many ways can 9 people arrange themselves I) on a bench II) around a circular table? \nb) In how many ways can 5 men and 4 women arrange themselves on a bench such that I) no two people of the same gender sit next to each other? II) the men and women sit in separate ... | math | {
"style": "rule",
"ground_truth": "5760"
} | {
"split": "train",
"index": 32655,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nThe hour and minute hands of a clock move continuously and at constant speeds. A moment of time $X$ is called interesting if there exists such a moment $Y$ (the moments $X$ and $Y$ do not necessarily have to be different), so that the hour hand at moment $Y$ will be where the... | math | {
"style": "rule",
"ground_truth": "143"
} | {
"split": "train",
"index": 32656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mia is researching a yeast population. There are 50 yeast cells present at 10:00 a.m. and the population triples every 5 minutes. Assuming none of the yeast cells die, how many yeast cells are present at 10:18 a.m. the same day? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "1350"
} | {
"split": "train",
"index": 32657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that player A needs to win 2 more games and player B needs to win 3 more games, and the probability of winning each game for both players is $\\dfrac{1}{2}$, calculate the probability of player A ultimately winning. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\dfrac{11}{16}"
} | {
"split": "train",
"index": 32658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five balls are arranged around a circle. Chris chooses two adjacent balls at random and interchanges them. Then Silva does the same, with her choice of adjacent balls to interchange being independent of Chris's. What is the expected number of balls that occupy their original po... | math | {
"style": "rule",
"ground_truth": "2.2"
} | {
"split": "train",
"index": 32659,
"solution": "To solve this problem, we need to calculate the expected number of balls that return to their original positions after two successive transpositions by Chris and Silva. We will use the linearity of expectation to find the expected number of balls in their original ... |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\lfloor\\sqrt{80}\\rfloor$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 32660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x)=4^x$ then $f(x+1)-f(x)$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3f(x)"
} | {
"split": "train",
"index": 32661,
"solution": "1. We start with the function $f(x) = 4^x$. We need to find $f(x+1) - f(x)$.\n\n2. Substitute $x+1$ into the function:\n \\[\n f(x+1) = 4^{x+1}\n \\]\n\n3. Now, calculate $f(x+1) - f(x)$:\n \\[\n f(x+1) - f(x) = 4^{x+1} - 4^x\n \\]\n\n4. Using the prope... |
deepscaler | [
{
"role": "user",
"content": "What fraction of the area of a regular hexagon of side length 1 is within distance $\\frac{1}{2}$ of at least one of the vertices? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi \\sqrt{3} / 9"
} | {
"split": "train",
"index": 32662,
"solution": "The hexagon has area $6(\\sqrt{3} / 4)(1)^{2}=3 \\sqrt{3} / 2$. The region we want consists of six $120^{\\circ}$ arcs of circles of radius $1 / 2$, which can be reassembled into two circles of radius $1 / 2$. So its area is $\\pi / 2$, and the ratio of areas is $\... |
deepscaler | [
{
"role": "user",
"content": "Solve \\[\\frac{2x+4}{x^2+4x-5}=\\frac{2-x}{x-1}\\]for $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 32663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle \\( ABC \\) is isosceles, and \\( \\angle ABC = x^\\circ \\). If the sum of the possible measures of \\( \\angle BAC \\) is \\( 240^\\circ \\), find \\( x \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 32664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three congruent isosceles triangles $DAO$, $AOB$, and $OBC$ have $AD=AO=OB=BC=13$ and $AB=DO=OC=15$. These triangles are arranged to form trapezoid $ABCD$. Point $P$ is on side $AB$ such that $OP$ is perpendicular to $AB$.\n\nPoint $X$ is the midpoint of $AD$ and point $Y$ is t... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 32665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 7 volunteers, among which 3 people only speak Russian, and 4 people speak both Russian and English. From these, 4 people are to be selected to serve as translators for the opening ceremony of the \"Belt and Road\" summit, with 2 people serving as English translators a... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 32666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The dilation, centered at $-1 + 4i,$ with scale factor $-2,$ takes $2i$ to which complex number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3 + 8i"
} | {
"split": "train",
"index": 32667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the ancient Chinese mathematical masterpiece \"Nine Chapters on the Mathematical Art,\" there is a problem called \"Division of Grains in a Granary\": When a granary is opened to receive grain, a person brings 1534 stones of rice. Upon inspection, it is found that there are ... | math | {
"style": "rule",
"ground_truth": "169.1"
} | {
"split": "train",
"index": 32668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider two solid spherical balls, one centered at $\\left( 0, 0, \\frac{21}{2} \\right),$ with radius 6, and the other centered at $(0,0,1)$ with radius $\\frac{9}{2}.$ How many points $(x,y,z)$ with only integer coefficients are there in the intersection of the balls? Let's... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 32669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four football teams participate in a round-robin tournament, where each team plays a match against every other team. In each match, the winning team earns 3 points, the losing team earns 0 points, and in the case of a draw, both teams earn 1 point each. After all matches are co... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 32670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(x) \\) be a function with the property that \\( f(x) + f\\left(\\frac{x-1}{3x-2}\\right) = x \\) for all real numbers \\( x \\) other than \\( \\frac{2}{3} \\). What is the sum \\( f(0) + f(1) + f(2) \\)? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac{87}{40}"
} | {
"split": "train",
"index": 32671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that a positive integer $N$ can be expressed as the sum of $k$ consecutive positive integers \\[ N = a + (a+1) +(a+2) + \\cdots + (a+k-1) \\] for $k=2017$ but for no other values of $k>1$. Considering all positive integers $N$ with this property, what is the smallest po... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 32672,
"solution": "We prove that the smallest value of $a$ is 16. Note that the expression for $N$ can be rewritten as $k(2a+k-1)/2$, so that $2N = k(2a+k-1)$. In this expression, $k>1$ by requirement; $k < 2a+k-1$ because $a>1$; and obviously $k$ and $2a+k-1$ have opposite parity. C... |
deepscaler | [
{
"role": "user",
"content": "A convex 2019-gon \\(A_{1}A_{2}\\ldots A_{2019}\\) is cut into smaller pieces along its 2019 diagonals of the form \\(A_{i}A_{i+3}\\) for \\(1 \\leq i \\leq 2019\\), where \\(A_{2020}=A_{1}, A_{2021}=A_{2}\\), and \\(A_{2022}=A_{3}\\). What is the least possible number of resul... | math | {
"style": "rule",
"ground_truth": "5049"
} | {
"split": "train",
"index": 32673,
"solution": "Each time we draw in a diagonal, we create one new region, plus one new region for each intersection on that diagonal. So, the number of regions will be \\(1+\\text{ (number of diagonals) }+ \\text{ (number of intersections) }\\) where (number of intersections) cou... |
deepscaler | [
{
"role": "user",
"content": "In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?\n$\\begin{array}{cccccc}&A&B&B&C&B\\ +&B&C&A&D&A\\ \\hline &D&B&D&D&D\\end{array}$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 32674,
"solution": "1. **Analyze the first column**: The sum $A + B$ must yield a single digit, $D$, and no carry-over since the sum of the digits in the first column is a single digit. This implies $A + B < 10$.\n\n2. **Analyze the fourth column**: The sum $C + D$ must yield the digi... |
deepscaler | [
{
"role": "user",
"content": "Five friends — Sarah, Lily, Emma, Nora, and Kate — performed in a theater as quartets, with one friend sitting out each time. Nora performed in 10 performances, which was the most among all, and Sarah performed in 6 performances, which was the fewest among all. Calculate the to... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 32675,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular triangular pyramid \\(P-ABC\\), where points \\(P\\), \\(A\\), \\(B\\), and \\(C\\) all lie on the surface of a sphere with radius \\(\\sqrt{3}\\), and \\(PA\\), \\(PB\\), and \\(PC\\) are mutually perpendicular, find the distance from the center of the sphere t... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 32676,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many solutions does the equation $\\tan x = \\tan (\\tan x + \\frac{\\pi}{4})$ have in the interval $0 \\leq x \\leq \\tan^{-1} 1884$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 32677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Acute-angled $\\triangle ABC$ is inscribed in a circle with center at $O$; $\\stackrel \\frown {AB} = 100^\\circ$ and $\\stackrel \\frown {BC} = 80^\\circ$.\nA point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. The task is to determine the ratio of th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 32678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest positive integer $n$ such that\n\\[\\sin^n x + \\cos^n x \\ge \\frac{1}{n}\\]for all real numbers $x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 32679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$24$ lemons are required to make $32$ gallons of lemonade. How many lemons are needed to make $4$ gallons of lemonade? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 32680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is $6$. How many two-digit numbers have this property? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 32681,
"solution": "1. **Represent the two-digit number**: Let the two-digit number be $10a + b$, where $a$ is the tens digit and $b$ is the units digit. Here, $a$ and $b$ are integers such that $1 \\leq a \\leq 9$ and $0 \\leq b \\leq 9$.\n\n2. **Formulate the given condition**: Acco... |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), the angle at vertex \\(B\\) is \\(\\frac{\\pi}{3}\\), and the line segments connecting the incenter to vertices \\(A\\) and \\(C\\) are 4 and 6, respectively. Find the radius of the circle inscribed in triangle \\(ABC\\). Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "\\frac{6 \\sqrt{3}}{\\sqrt{19}}"
} | {
"split": "train",
"index": 32682,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When arranging the performance order of 6 singers, requiring that both singers B and C are either before or after singer A, 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": "480"
} | {
"split": "train",
"index": 32683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the tens digit of $2015^{2016}-2017?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 32684,
"solution": "1. **Understanding the Problem**: We need to find the tens digit of $2015^{2016} - 2017$. This can be simplified by considering the expression modulo 100, as the tens digit of a number is the second last digit in its decimal representation.\n\n2. **Simplifying $201... |
deepscaler | [
{
"role": "user",
"content": "In tetrahedron \\( SABC \\), the circumradius of triangles \\( \\triangle SAB \\), \\( \\triangle SBC \\), and \\( \\triangle SCA \\) are all 108. The center of the inscribed sphere is \\( I \\) and \\( SI = 125 \\). Let \\( R \\) be the circumradius of \\( \\triangle ABC \\). ... | math | {
"style": "rule",
"ground_truth": "11665"
} | {
"split": "train",
"index": 32685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $PQR$ has vertices $P(0, 6)$, $Q(3, 0)$, $R(9, 0)$. A line through $Q$ cuts the area of $\\triangle PQR$ in half. Find the sum of the slope and $y$-intercept of this line. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 32686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right rectangular prism has 6 faces, 12 edges, and 8 vertices. A new pyramid is to be constructed using one of the rectangular faces as the base. Calculate the maximum possible sum of the number of exterior faces, vertices, and edges of the combined solid (prism and pyramid).... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 32687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of $10$ hours per week helping around the house for $6$ weeks. For the first $5$ weeks she helps around the house for $8$, $11$, $7$, $12$ and $10$ hours. How many hours must she ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 32688,
"solution": "1. **Define the variable**: Let $x$ be the number of hours Theresa needs to work in the sixth week to achieve an average of 10 hours per week over 6 weeks.\n\n2. **Set up the equation for the average**: The average number of hours worked per week is given by the to... |
deepscaler | [
{
"role": "user",
"content": "Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\\mathbf{v}=(v_1,\\ldots,v_{2022})$ and $\\mathbf{w}=(w_1,\\ldots,w_{2022})$ that she has already written, and apply one of the fol... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 32689,
"solution": "\nTo solve the problem, we need to determine the minimum number \\( s \\) of initial integer-valued \\( 2022 \\)-tuples that Lucy has to write on the blackboard such that any other integer-valued \\( 2022 \\)-tuple can be formed using the operations defined.\n\n###... |
deepscaler | [
{
"role": "user",
"content": "On a 4x4 grid (where each unit distance is 1), calculate how many rectangles can be formed where each of the rectangle's vertices is a point on this grid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 32690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a real number $y$, find the maximum value of\n\\[\n\\frac{y^6}{y^{12} + 3y^9 - 9y^6 + 27y^3 + 81}.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{27}"
} | {
"split": "train",
"index": 32691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all positive integer solutions less than or equal to $30$ to the congruence $15(3x-4) \\equiv 30 \\pmod{10}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 32692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The center of the circle with equation $x^2+y^2=4x+12y-39$ is the point $(h,k)$. What is $h+k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 32693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a positive real number \\(\\alpha\\), determine the greatest real number \\(C\\) such that the inequality\n\n$$\n\\left(1+\\frac{\\alpha}{x^{2}}\\right)\\left(1+\\frac{\\alpha}{y^{2}}\\right)\\left(1+\\frac{\\alpha}{z^{2}}\\right) \\geq C \\cdot\\left(\\frac{x}{z}+\\frac{... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 32694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that all three vertices of \\(\\triangle ABC\\) lie on the parabola defined by \\(y = 4x^2\\), with \\(A\\) at the origin and \\(\\overline{BC}\\) parallel to the \\(x\\)-axis, calculate the length of \\(BC\\), given that the area of the triangle is 128. Let's think step ... | math | {
"style": "rule",
"ground_truth": "4\\sqrt[3]{4}"
} | {
"split": "train",
"index": 32695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sets \\( P=\\left\\{m^{2}-4, m+1,-3\\right\\} \\) and \\( Q=\\{m-3, 2m-1, 3m+1\\} \\), if \\( P \\cap Q = \\{-3\\} \\), find the value of the real number \\( m \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}"
} | {
"split": "train",
"index": 32696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two individuals, A and B, start traveling towards each other from points A and B, respectively, at the same time. They meet at point C, after which A continues to point B and B rests for 14 minutes before continuing to point A. Both A and B, upon reaching points B and A, immedi... | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"split": "train",
"index": 32697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Chemistry class, Samantha finds that she can make a certain solution by mixing $.04$ liters of chemical A with $.02$ liters of water (this gives her $.06$ liters of the solution). She wants to make a total of $.48$ liters of this new solution. To do so, how many liters of w... | math | {
"style": "rule",
"ground_truth": "0.16"
} | {
"split": "train",
"index": 32698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of different complex numbers $z$ such that $|z|=1$ and $z^{7!}-z^{6!}$ is a real number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 32699,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $PQRS$ is a square. A circle with center $S$ has arc $PXC$. A circle with center $R$ has arc $PYC$. If $PQ = 3$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined? Express your answer as a decimal to the n... | math | {
"style": "rule",
"ground_truth": "5.1"
} | {
"split": "train",
"index": 32700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A ball thrown vertically upwards has its height above the ground expressed as a quadratic function with respect to its time of motion. Xiaohong throws two balls vertically upwards one after the other, with a 1-second interval between them. Assume the initial height above the gr... | math | {
"style": "rule",
"ground_truth": "1.6"
} | {
"split": "train",
"index": 32701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two teams, Team A and Team B, participate in a table tennis group match. The ratio of their strength is $3:2$. Assuming both teams play to their normal levels, find the probability that Team A wins after playing 4 games in a best-of-5 series (3 wins out of 5 games). Let's think... | math | {
"style": "rule",
"ground_truth": "\\frac{162}{625}"
} | {
"split": "train",
"index": 32702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a cubic polynomial with $P(0) = k$, $P(1) = 2k$, and $P(-1) = 3k$. What is $P(2) + P(-2)$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14k"
} | {
"split": "train",
"index": 32703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nLet the function \\( f(x) = x^2 - x + 1 \\). Define \\( f^{(n)}(x) \\) as follows:\n$$\nf^{(1)}(x) = f(x), \\quad f^{(n)}(x) = f\\left(f^{(n-1)}(x)\\right).\n$$\n\nLet \\( r_{n} \\) be the arithmetic mean of all the roots of \\( f^{(n)}(x) = 0 \\). Find \\( r_{2015} \\). Let'... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 32704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( OP \\) be the diameter of the circle \\( \\Omega \\), and let \\( \\omega \\) be a circle with its center at point \\( P \\) and a radius smaller than that of \\( \\Omega \\). The circles \\( \\Omega \\) and \\( \\omega \\) intersect at points \\( C \\) and \\( D \\). T... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 32705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the integer $n,$ $-180 \\le n \\le 180,$ such that $\\cos n^\\circ = \\cos 430^\\circ.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-70"
} | {
"split": "train",
"index": 32706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "My frisbee group often calls \"best of five\" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, what i... | math | {
"style": "rule",
"ground_truth": "3/2"
} | {
"split": "train",
"index": 32707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest quotient that can be formed using two numbers chosen from the set $\\{-30, -6, -1, 3, 5, 20\\}$, where one of the numbers must be negative? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-0.05"
} | {
"split": "train",
"index": 32708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The minimum value of the function $y = \\sin 2 \\cos 2x$ is ______. 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": 32709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coefficients of the polynomial\n\\[x^4 + bx^3 + cx^2 + dx + e = 0\\]are all integers. Let $n$ be the exact number of integer roots of the polynomial, counting multiplicity. For example, the polynomial $(x + 3)^2 (x^2 + 4x + 11) = 0$ has two integer roots counting multipli... | math | {
"style": "rule",
"ground_truth": "0, 1, 2, 4"
} | {
"split": "train",
"index": 32710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $9 \\times 99 \\times 999 \\times \\cdots \\times \\underbrace{99\\cdots9}_{\\text{999 9's}}$ is divided by $1000$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "109"
} | {
"split": "train",
"index": 32711,
"solution": "Note that $999\\equiv 9999\\equiv\\dots \\equiv\\underbrace{99\\cdots9}_{\\text{999 9's}}\\equiv -1 \\pmod{1000}$ (see modular arithmetic). That is a total of $999 - 3 + 1 = 997$ integers, so all those integers multiplied out are congruent to $- 1\\pmod{1000}$. Thu... |
deepscaler | [
{
"role": "user",
"content": "There are 20 rooms, some with lights on and some with lights off. The occupants of these rooms prefer to match the majority of the rooms. Starting from room one, if the majority of the remaining 19 rooms have their lights on, the occupant will turn the light on; otherwise, they... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 32712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at $4$ cents per glob and $J$ blobs of jam at $5$ cents per blob. The cost of the peanut butter and jam to make all the sandwiches is $\\$2.53$. Assume that $B$, $J$, and $N$ are ... | math | {
"style": "rule",
"ground_truth": "\\$1.65"
} | {
"split": "train",
"index": 32713,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $2\\times 3$ rectangle and a $3\\times 4$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square? Let's think step b... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 32714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Dima calculated the factorials of all natural numbers from 80 to 99, found the reciprocals of them, and printed the resulting decimal fractions on 20 endless ribbons (for example, the last ribbon had the number \\(\\frac{1}{99!}=0. \\underbrace{00\\ldots00}_{155 \\text{ zeros}}... | math | {
"style": "rule",
"ground_truth": "155"
} | {
"split": "train",
"index": 32715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A motel bills its customers by charging a flat fee for the first night and then adding on a fixed amount for every night thereafter. If it costs George $\\$155$ to stay in the motel for 3 nights and Noah $\\$290$ to stay in the motel for 6 nights, how much is the flat fee for t... | math | {
"style": "rule",
"ground_truth": "\\$65"
} | {
"split": "train",
"index": 32716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $AOB$ is a sector of a circle with $\\angle AOB=60^\\circ.$ $OY$ is drawn perpendicular to $AB$ and intersects $AB$ at $X.$ What is the length of $XY\n?$ [asy]\ndraw((0,0)--(12,0),black+linewidth(1));\ndraw((0,0)--(10.3923,-6)..(12,0)..(10.3923,6)--(0,0),black+... | math | {
"style": "rule",
"ground_truth": "12 - 6\\sqrt{3}"
} | {
"split": "train",
"index": 32717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Angie's class has 2 girls for every 3 boys. If there are 20 students in the class, how many girls are in Angie's class? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 32718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a table, there are five clocks with hands. It is allowed to move any number of them forward. For each clock, the time by which it is moved will be referred to as the translation time. It is required to set all clocks such that they show the same time. What is the minimum tot... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 32719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle RED$, $\\measuredangle DRE=75^{\\circ}$ and $\\measuredangle RED=45^{\\circ}$. $RD=1$. Let $M$ be the midpoint of segment $\\overline{RD}$. Point $C$ lies on side $\\overline{ED}$ such that $\\overline{RC}\\perp\\overline{EM}$. Extend segment $\\overline{DE}$ thr... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 32720,
"solution": "Let $P$ be the foot of the perpendicular from $A$ to $\\overline{CR}$, so $\\overline{AP}\\parallel\\overline{EM}$. Since triangle $ARC$ is isosceles, $P$ is the midpoint of $\\overline{CR}$, and $\\overline{PM}\\parallel\\overline{CD}$. Thus, $APME$ is a parallelo... |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $C\\_1: \\frac{x^2}{8} + \\frac{y^2}{4} = 1$ with left and right foci $F\\_1$ and $F\\_2$, a line $l\\_1$ is drawn through point $F\\_1$ perpendicular to the $x$-axis. Line $l\\_2$ is perpendicular to $l\\_1$ at point $P$, and the perpendicular bisector of seg... | math | {
"style": "rule",
"ground_truth": "\\frac{64}{9}"
} | {
"split": "train",
"index": 32721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x+\\frac{1}{x}=6$, then what is the value of $x^{2}+\\frac{1}{x^{2}}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 32722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are face cards (Jacks, Queens, or Kings) totaling to a numeric value of 20? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{221}"
} | {
"split": "train",
"index": 32723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The triangle shown is an equilateral triangle with side length 12 cm. A side of the triangle is the diameter of the circle. If the sum of the areas of the two small shaded regions in square centimeters in simplest radical form is $a\\pi - b\\sqrt{c}$, what is $a+b+c$? [asy]\nim... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 32724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( M = \\{1, 2, \\cdots, 2005\\} \\), and \\( A \\) be a subset of \\( M \\). If for any \\( a_i, a_j \\in A \\) with \\( a_i \\neq a_j \\), an isosceles triangle can be uniquely determined with \\( a_i \\) and \\( a_j \\) as side lengths, find the maximum value of \\( |A|... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 32725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the integer values of $m$, $n$, and $p$ such that the roots of the equation $4x(2x - 5) = -4$ can be expressed in the forms $\\frac{m+\\sqrt{n}}{p}$ and $\\frac{m-\\sqrt{n}}{p}$, and find $m+n+p$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 32726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest constant $m,$ so that for any positive real numbers $a,$ $b,$ $c,$ and $d,$\n\\[\\sqrt{\\frac{a}{b + c + d}} + \\sqrt{\\frac{b}{a + c + d}} + \\sqrt{\\frac{c}{a + b + d}} + \\sqrt{\\frac{d}{a + b + c}} > m.\\] Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 32727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z_1$, $z_2$, $z_3$, $\\dots$, $z_{8}$ be the 8 zeros of the polynomial $z^{8} - 16^8$. For each $j$, let $w_j$ be either $z_j$, $-z_j$, or $iz_j$. Find the maximum possible value of the real part of\n\\[\\sum_{j = 1}^{8} w_j.\\] Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "32 + 32 \\sqrt{2}"
} | {
"split": "train",
"index": 32728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is an arithmetic sequence, $a_1 > 0$, $a_{23} + a_{24} > 0$, and $a_{23} \\cdot a_{24} < 0$, determine the maximum positive integer $n$ for which the sum of the first $n$ terms $S_n > 0$. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 32729,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\cos C$ in the right triangle shown below.\n\n\n[asy]\npair A,B,C;\nA = (0,0);\nB = (6,0);\nC = (0,8);\ndraw(A--B--C--A);\ndraw(rightanglemark(B,A,C,10));\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,SE);\nlabel(\"$C$\",C,N);\nlabel(\"$10$\",(B+C)/2,NE);\nlabel(\"$6$\",B/2,S);... | math | {
"style": "rule",
"ground_truth": "\\frac45"
} | {
"split": "train",
"index": 32730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average age of 6 people in a room is 25 years. A 20-year-old person leaves the room and a new person aged 30 years enters the room. Find the new average age of the people in the room. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{80}{3}"
} | {
"split": "train",
"index": 32731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Peter has three times as many sisters as brothers. His sister Louise has twice as many sisters as brothers. How many children are there in the family? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 32732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $(1-\\frac{m}{{m-3}})\\div \\frac{{{m^2}-3m}}{{{m^2}-6m+9}}$, where $m=4\\sqrt{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 32733,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle PQR$ be a right triangle with angle $Q$ as the right angle. A circle with diameter $QR$ intersects side $PR$ at point $S$. If the area of $\\triangle PQR$ is $192$ and $PR = 32$, what is the length of $QS$? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 32734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A lemming sits at a corner of a square with side length $10$ meters. The lemming runs $6.2$ meters along a diagonal toward the opposite corner. It stops, makes a $90^{\\circ}$ right turn and runs $2$ more meters. A scientist measures the shortest distance between the lemming an... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 32735,
"solution": "1. **Understanding the problem**: A lemming starts at a corner of a square with side length $10$ meters. It moves $6.2$ meters along a diagonal towards the opposite corner, then makes a $90^{\\circ}$ right turn and runs $2$ meters. We need to find the average of th... |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is an obtuse, isosceles triangle. Angle $A$ measures 20 degrees. What is number of degrees in the measure of the largest interior angle of triangle $ABC$?\n\n[asy]\ndraw((-20,0)--(0,8)--(20,0)--cycle);\nlabel(\"$20^{\\circ}$\",(-13,-0.7),NE);\nlabel(\"$A$\",(-20,... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 32736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Joel now selects an acute angle $x$ (between 0 and 90 degrees) and writes $\\sin x$, $\\cos x$, and $\\tan x$ on three different cards. Each student, Malvina, Paulina, and Georgina, receives one card, but Joel chooses a value $\\sin x$ that is a commonly known value, $\\sin x =... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 32737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jaclyn, Marcelle, Susanna and Wanda are tutors in their school math lab. Their schedule is as follows: Jaclyn works every third school day, Marcelle works every fourth school day, Susanna works every sixth school day, and Wanda works every seventh school day. Today, they are al... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 32738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle ABC, let the lengths of the sides opposite to angles A, B, and C be a, b, and c respectively, and b = 3, c = 1, A = 2B. Find the value of a. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{19}"
} | {
"split": "train",
"index": 32739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct divisors does the number a) 800; b) 126000 have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 32740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each cell of a $3 \\times 3$ grid is labeled with a digit in the set $\\{1,2,3,4,5\\}$. Then, the maximum entry in each row and each column is recorded. Compute the number of labelings for which every digit from 1 to 5 is recorded at least once. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "2664"
} | {
"split": "train",
"index": 32741,
"solution": "We perform casework by placing the entries from largest to smallest. - The grid must have exactly one 5 since an entry equal to 5 will be the maximum in its row and in its column. We can place this in 9 ways. - An entry equal to 4 must be in the same row or column ... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos α= \\frac{1}{7}, \\cos (α+β)=- \\frac{11}{14}$, and $α,β∈(0, \\frac{π}{2})$, find the value of $\\cos β$. 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": 32742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A seven-digit phone number \\(d_{1} d_{2} d_{3}-d_{4} d_{5} d_{6} d_{7}\\) is called \"memorable\" if the initial three digits \\(d_{1} d_{2} d_{3}\\) match either the middle three digits \\(d_{4} d_{5} d_{6}\\) or the last three digits \\(d_{5} d_{6} d_{7}\\) (it is possible f... | math | {
"style": "rule",
"ground_truth": "19990"
} | {
"split": "train",
"index": 32743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given rectangle $R_1$ with one side $2$ inches and area $12$ square inches. Rectangle $R_2$ with diagonal $15$ inches is similar to $R_1$. Expressed in square inches the area of $R_2$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{135}{2}"
} | {
"split": "train",
"index": 32744,
"solution": "1. **Identify the dimensions of $R_1$:**\n Given that one side of rectangle $R_1$ is $2$ inches and its area is $12$ square inches, we can find the other side by dividing the area by the given side:\n \\[\n \\text{Other side} = \\frac{\\text{Area}}{\\text{Giv... |
deepscaler | [
{
"role": "user",
"content": "Which of the following is equal to \\(\\frac{4}{5}\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.8"
} | {
"split": "train",
"index": 32745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ that satisfies $a_1=33$ and $a_{n+1}-a_n=2n$, find the minimum value of $\\frac {a_{n}}{n}$. 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": 32746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon $ABCDEF$ has sides of length 2. Find the area of $\\bigtriangleup ADF$. 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": "4\\sqrt{3}"
} | {
"split": "train",
"index": 32747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola \\( C_1: 2x^2 - y^2 = 1 \\) and the ellipse \\( C_2: 4x^2 + y^2 = 1 \\). If \\( M \\) and \\( N \\) are moving points on the hyperbola \\( C_1 \\) and ellipse \\( C_2 \\) respectively, such that \\( OM \\perp ON \\) and \\( O \\) is the origin, find the dist... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 32748,
"solution": ""
} |
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