data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "An isosceles triangle is a triangle with two sides of equal length. How many of the five triangles on the square grid below are isosceles?\n[asy] for(int a=0; a<12; ++a) { draw((a,0)--(a,6)); } for(int b=0; b<7; ++b) { draw((0,b)--(11,b)); } draw((0,6)--(2,6)--(1,4)--c... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 31849,
"solution": "To determine how many of the triangles are isosceles, we need to calculate the lengths of the sides of each triangle and check if at least two sides are equal in length.\n\n#### Triangle 1:\nVertices: $(0,6), (2,6), (1,4)$\n- Side 1: Distance between $(0,6)$ and $(... |
deepscaler | [
{
"role": "user",
"content": "A $4\\times 4\\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "2+2\\sqrt 7"
} | {
"split": "train",
"index": 31850,
"solution": "1. **Identify the Geometry and Positioning of Spheres:**\n - The large sphere has a radius of $2$ and is centered at point $A$.\n - Each of the eight smaller spheres has a radius of $1$ and is tangent to three sides of the box. Let $C$ be the center of any one ... |
deepscaler | [
{
"role": "user",
"content": "Simplify $(22a+60b)+(10a+29b)-(9a+50b).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23a+39b"
} | {
"split": "train",
"index": 31851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fifteen square tiles with side 10 units long are arranged as shown. An ant walks along the edges of the tiles, always keeping a black tile on its left. Find the shortest distance that the ant would walk in going from point \\( P \\) to point \\( Q \\). Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 31852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The figure below is a map showing $12$ cities and $17$ roads connecting certain pairs of cities. Paula wishes to travel along exactly $13$ of those roads, starting at city $A$ and ending at city $L$, without traveling along any portion of a road more than once. (Paula is allowe... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 31853,
"solution": "1. **Understanding the Problem**: We need to find the number of different routes Paula can take from city $A$ to city $L$, traversing exactly 13 roads, without repeating any road segment.\n\n2. **Vertex Visitation Constraints**: For any vertex $V$ (except $A$ and $... |
deepscaler | [
{
"role": "user",
"content": "When any set of $k$ consecutive positive integers necessarily includes at least one positive integer whose digit sum is a multiple of 11, we call each of these sets of $k$ consecutive positive integers a \"dragon\" of length $k.\" Find the shortest dragon length. Let's think st... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 31854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Numbers $1,2,3,4,5,6,7,$ and $8$ are placed at the vertices of a cube such that the sum of any three numbers belonging to any face of the cube is not less than 10. Find the minimum possible sum of four numbers belonging to one face. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 31855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the integer \\[N = 9 + 99 + 999 + 9999 + \\cdots + \\underbrace{99\\ldots 99}_\\text{321 digits}.\\]Find the sum of the digits of $N$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "342"
} | {
"split": "train",
"index": 31856,
"solution": "Observe how adding results in the last term but with a $1$ concatenated in front and also a $1$ subtracted ($09$, $108$, $1107$, $11106$). Then for any index of terms, $n$, the sum is $11...10-n$, where the first term is of length $n+1$. Here, that is $\\boxed{342}... |
deepscaler | [
{
"role": "user",
"content": "If the digit represented by $\\triangle$ satisfies the following addition problem $$ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c} & & 3 & 2 & 1 & \\triangle_6\\\\ & & & \\triangle & 4 & 0_6\\\\ &+ & & & \\triangle & 2_6\\\\ \\cline{2-6} & & 4 & 2 & \\triangle & 1_6\\\\ \\end{array}... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 31857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[3f(x) - 2 f \\left( \\frac{1}{x} \\right) = x\\]for all $x \\neq 0.$ Find $f(4).$ 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": 31858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many two-digit prime numbers have the property that both digits are also primes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 31859,
"solution": "When considering the 16 two-digit numbers with 2, 3, 5, and 7 as digits, we find that only $23, 37, 53$, and 73 have this property."
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{S}$ be the set $\\lbrace1,2,3,\\ldots,10\\rbrace$ Let $n$ be the number of sets of two non-empty disjoint subsets of $\\mathcal{S}$. (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when $n$ is divided by $1000$.\n Let... | math | {
"style": "rule",
"ground_truth": "501"
} | {
"split": "train",
"index": 31860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sides of rhombus \\( EFGH \\) are the hypotenuses of the isosceles right triangles \\( EAF, FDG, GCH, \\) and \\( HBE \\), and all these triangles have common interior points with the rhombus \\( EFGH \\). The sum of the areas of quadrilateral \\( ABCD \\) and rhombus \\( E... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 31861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A quadrilateral $ABCD$ has a right angle at $\\angle ABC$ and satisfies $AB = 12$ , $BC = 9$ , $CD = 20$ , and $DA = 25$ . Determine $BD^2$ .\n.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "769"
} | {
"split": "train",
"index": 31862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $(128)^{\\frac{1}{3}}(729)^{\\frac{1}{2}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108 \\cdot 2^{\\frac{1}{3}}"
} | {
"split": "train",
"index": 31863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle \\(ABC\\) with sides \\(AB = 13\\), \\(BC = 14\\), and \\(AC = 15\\). Point \\(K\\) is marked on side \\(AB\\), point \\(L\\) is marked on side \\(BC\\), and point \\(N\\) is marked on side \\(AC\\). It is known that \\(BK = \\frac{14}{13}\\), \\(AN = 10\\), an... | math | {
"style": "rule",
"ground_truth": "\\frac{36503}{1183}"
} | {
"split": "train",
"index": 31864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum of the expression \\(6 t^{2} + 3 s^{2} - 4 s t - 8 t + 6 s + 5\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{7}"
} | {
"split": "train",
"index": 31865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles trapezoid \\(ABCD\\), \\(AB\\) is parallel to \\(CD\\), \\(AB = 6\\), \\(CD = 14\\), \\(\\angle AEC\\) is a right angle, and \\(CE = CB\\). What is \\(AE^2\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 31866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many four-digit whole numbers does the sum of the digits equal $30$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 31867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "According to Moor's Law, the number of shoes in Moor's room doubles every year. In 2013, Moor's room starts out having exactly one pair of shoes. If shoes always come in unique, matching pairs, what is the earliest year when Moor has the ability to wear at least 500 mismatche... | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"split": "train",
"index": 31868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $b\\sin C + c\\sin B = 4a\\sin B\\sin C$ and $b^2 + c^2 - a^2 = 8$. Find the area of $\\triangle ABC$. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 31869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of solutions in natural numbers to the equation \\(\\left\\lfloor \\frac{x}{10} \\right\\rfloor = \\left\\lfloor \\frac{x}{11} \\right\\rfloor + 1\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 31870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Reimu has a wooden cube. In each step, she creates a new polyhedron from the previous one by cutting off a pyramid from each vertex of the polyhedron along a plane through the trisection point on each adjacent edge that is closer to the vertex. For example, the polyhedron after... | math | {
"style": "rule",
"ground_truth": "974"
} | {
"split": "train",
"index": 31871,
"solution": "Notice that the number of vertices and edges triple with each step. We always have 3 edges meeting at one vertex, and slicing off a pyramid doesn't change this (we make new vertices from which one edge from the previous step and two of the pyramid edges emanate). S... |
deepscaler | [
{
"role": "user",
"content": "\nMonsieur Dupont remembered that today is their wedding anniversary and invited his wife to dine at a fine restaurant. Upon leaving the restaurant, he noticed that he had only one fifth of the money he initially took with him. He found that the centimes he had left were equal ... | math | {
"style": "rule",
"ground_truth": "7996"
} | {
"split": "train",
"index": 31872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $2-(-2)^{-2}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{4}"
} | {
"split": "train",
"index": 31873,
"solution": "1. **Evaluate the exponentiation and inversion:**\n The expression given is $2 - (-2)^{-2}$. We start by evaluating $(-2)^{-2}$. By the properties of exponents, $a^{-n} = \\frac{1}{a^n}$, so:\n \\[\n (-2)^{-2} = \\frac{1}{(-2)^2}\n \\]\n\n2. **Calculate $(-... |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)=\\frac{1}{x}$, calculate the limit of $\\frac{f(2+3\\Delta x)-f(2)}{\\Delta x}$ as $\\Delta x$ approaches infinity. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"split": "train",
"index": 31874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the $2010$ roots of $(x-1)^{2010} + 2(x-2)^{2009} + 3(x-3)^{2008} + \\cdots + 2009(x-2009)^2 + 2010(x-2010)^1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2008"
} | {
"split": "train",
"index": 31875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an angle measuring $54^{\\circ}$, use only a compass to divide it into three equal parts (that is, find such points that rays passing through the vertex of the given angle and these points divide the angle into three equal parts). Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 31876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ are vectors such that $\\|\\mathbf{a}\\| = \\|\\mathbf{b}\\| = 1,$ $\\|\\mathbf{a} + \\mathbf{b}\\| = \\sqrt{3},$ and\n\\[\\mathbf{c} - \\mathbf{a} - 2 \\mathbf{b} = 3 (\\mathbf{a} \\times \\mathbf{b}),\\]then find $\\mathbf{b}... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 31877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempte... | math | {
"style": "rule",
"ground_truth": "849"
} | {
"split": "train",
"index": 31878,
"solution": "Let $q$ be the number of questions Beta takes on day 1 and $a$ be the number he gets right. Let $b$ be the number he gets right on day 2.\n These inequalities follow: \\[\\frac{a}{q} < \\frac{160}{300} = \\frac{8}{15}\\] \\[\\frac{b}{500-q} < \\frac{140}{200} = \\f... |
deepscaler | [
{
"role": "user",
"content": "In the parallelepiped $ABCD-A_{1}B_{1}C_{1}D_{1}$, three edges with vertex $A$ as an endpoint are all of length $2$, and their angles with each other are all $60^{\\circ}$. Determine the cosine value of the angle between the line $BD_{1}$ and the line $AC$. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{6}"
} | {
"split": "train",
"index": 31879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit positive integers are multiples of 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1286"
} | {
"split": "train",
"index": 31880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular square pyramid \\( P-ABCD \\) with a base side length \\( AB=2 \\) and height \\( PO=3 \\). \\( O' \\) is a point on the segment \\( PO \\). Through \\( O' \\), a plane parallel to the base of the pyramid is drawn, intersecting the edges \\( PA, PB, PC, \\) and ... | math | {
"style": "rule",
"ground_truth": "16/27"
} | {
"split": "train",
"index": 31881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357$, $89$, and $5$ are all uphill integers, but $32$, $1240$, and $466$ are not. How many uphill integers are divisible by $15$? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 31882,
"solution": "1. **Divisibility by 15**: A number is divisible by 15 if and only if it is divisible by both 3 and 5. \n\n2. **Divisibility by 5**: A number is divisible by 5 if it ends in 0 or 5. Since we are considering positive integers and uphill integers cannot end in 0 (as ... |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers such that $a + b = 4.$ Find the maximum value of\n\\[a^4 b + a^3 b + a^2 b + ab + ab^2 + ab^3 + ab^4.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7225}{56}"
} | {
"split": "train",
"index": 31883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a regular tetrahedron and Let $E$ be a point inside the face $ABC.$ Denote by $s$ the sum of the distances from $E$ to the faces $DAB, DBC, DCA,$ and by $S$ the sum of the distances from $E$ to the edges $AB, BC, CA.$ Then $\\frac{s}{S}$ equals Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 31884,
"solution": "The problem statement and the solution provided seem to be mismatched. The problem statement is about geometry involving a tetrahedron, while the solution provided discusses logarithms and is unrelated to the problem. Let's solve the original problem about the tetr... |
deepscaler | [
{
"role": "user",
"content": "How many factors are there in the product $1 \\cdot 2 \\cdot 3 \\cdot \\ldots \\cdot n$ if we know that it ends with 1981 zeros? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7935"
} | {
"split": "train",
"index": 31885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a finite sequence $B = (b_1, b_2, \\dots, b_m)$ of numbers, the Cesaro sum of $B$ is defined to be\n\\[\\frac{T_1 + T_2 + \\dots + T_m}{m},\\]\nwhere $T_k = b_1 + b_2 + \\dots + b_k$ for $1 \\leq k \\leq m$.\n\nIf the Cesaro sum of the 50-term sequence $(b_1, b_2, \\dots, b... | math | {
"style": "rule",
"ground_truth": "492"
} | {
"split": "train",
"index": 31886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a $50$-question multiple choice math contest, students receive $4$ points for a correct answer, $0$ points for an answer left blank, and $-1$ point for an incorrect answer. Jesse’s total score on the contest was $99$. What is the maximum number of questions that Jesse could ... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 31887,
"solution": "Let's define the variables:\n- Let $a$ be the number of questions Jesse answered correctly.\n- Let $b$ be the number of questions Jesse left blank.\n- Let $c$ be the number of questions Jesse answered incorrectly.\n\nGiven the total number of questions is $50$, we ... |
deepscaler | [
{
"role": "user",
"content": "Calculate: $5 \\times 13 \\times 31 \\times 73 \\times 137$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20152015"
} | {
"split": "train",
"index": 31888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a skewed six-sided die is structured so that rolling an odd number is twice as likely as rolling an even number, calculate the probability that, after rolling the die twice, the sum of the numbers rolled is odd. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 31889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $16^{7/4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 31890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nine tiles are numbered $1, 2, 3, \\cdots, 9,$ respectively. Each of three players randomly selects and keeps three of the tiles, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive inte... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 31891,
"solution": "Let $O$ stand for an odd number and $E$ an even. Therefore, one person must pick $OOO$, the other person must pick $OEE$ and the last person must pick $OEE$. Since any permutation of the order of who is picking or change in the order of the even numbers (e.g. $EOE$... |
deepscaler | [
{
"role": "user",
"content": "The operation $\\star$ is defined as $a \\star b = a^2 \\div b$. For how many integer values of $x$ will the value of $10 \\star x$ be a positive integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 31892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the random variable $\\eta\\sim B(n,p)$, and $E(2\\eta)=8$, $D(4\\eta)=32$, find the respective values of $n$ and $p$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 31893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two lines \\( l \\) and \\( m \\) have equations \\( y = -2x + 8 \\), and \\( y = -3x + 9 \\), what is the probability that a point randomly selected in the 1st quadrant and below \\( l \\) will fall between \\( l \\) and \\( m \\)? Express your answer as a decimal to the ne... | math | {
"style": "rule",
"ground_truth": "0.16"
} | {
"split": "train",
"index": 31894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation\n\\[(x - \\sqrt[3]{20})(x - \\sqrt[3]{70})(x - \\sqrt[3]{120}) = \\frac{1}{2}\\] has three distinct solutions $u,$ $v,$ and $w.$ Calculate the value of $u^3 + v^3 + w^3.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "211.5"
} | {
"split": "train",
"index": 31895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The fare in Moscow with the \"Troika\" card in 2016 is 32 rubles for one trip on the metro and 31 rubles for one trip on ground transportation. What is the minimum total number of trips that can be made at these rates, spending exactly 5000 rubles? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "157"
} | {
"split": "train",
"index": 31896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The vertices of $\\triangle ABC$ are $A = (0,0)\\,$, $B = (0,420)\\,$, and $C = (560,0)\\,$. The six faces of a die are labeled with two $A\\,$'s, two $B\\,$'s, and two $C\\,$'s. Point $P_1 = (k,m)\\,$ is chosen in the interior of $\\triangle ABC$, and points $P_2\\,$, $P_3\\,$... | math | {
"style": "rule",
"ground_truth": "344"
} | {
"split": "train",
"index": 31897,
"solution": "If we have points $(p,q)$ and $(r,s)$ and we want to find $(u,v)$ so $(r,s)$ is the midpoint of $(u,v)$ and $(p,q)$, then $u=2r-p$ and $v=2s-q$. So we start with the point they gave us and work backwards. We make sure all the coordinates stay within the triangle. W... |
deepscaler | [
{
"role": "user",
"content": "Given the geometric sequence with 8 inserted numbers between 1 and 3, find the product of these 8 inserted numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 31898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If\n\\[\\frac{\\sin^4 \\theta}{a} + \\frac{\\cos^4 \\theta}{b} = \\frac{1}{a + b},\\]then find the value of\n\\[\\frac{\\sin^8 \\theta}{a^3} + \\frac{\\cos^8 \\theta}{b^3}\\]in terms of $a$ and $b.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{(a + b)^3}"
} | {
"split": "train",
"index": 31899,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an isosceles triangle DEF with DE = DF = 5√3, a circle with radius 6 is tangent to DE at E and to DF at F. If the altitude from D to EF intersects the circle at its center, find the area of the circle that passes through vertices D, E, and F. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "36\\pi"
} | {
"split": "train",
"index": 31900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\\sin \\frac{B}{2}-\\cos \\frac{B}{2}= \\frac{1}{5}$.\n\n(I) Find the value of $\\cos B$;\n\n(II) If $b^{2}-a^{2}=ac$, find the value of $\\frac{\\sin C}{\\sin A}$. Let'... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{25}"
} | {
"split": "train",
"index": 31901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Maria is 54 inches tall, and Samuel is 72 inches tall. Using the conversion 1 inch = 2.54 cm, how tall is each person in centimeters? Additionally, what is the difference in their heights in centimeters? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "45.72"
} | {
"split": "train",
"index": 31902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the range of the function \\[f(x) = \\frac{x}{x^2-x+1},\\]where $x$ can be any real number. (Give your answer in interval notation.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "[-\\tfrac13, 1]"
} | {
"split": "train",
"index": 31903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of arrangements of the letters a, b, c, d, e in a sequence such that neither a nor b is adjacent to c. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 31904,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $F$ be the set of functions from $\\mathbb{R}^{+}$ to $\\mathbb{R}^{+}$ such that $f(3x) \\geq f(f(2x)) + x$. Maximize $\\alpha$ such that $\\forall x \\geq 0, \\forall f \\in F, f(x) \\geq \\alpha x$. 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": 31905,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB = 12$, $BC = 25$, and $CA = 17$. Rectangle $PQRS$ has vertex $P$ on $\\overline{AB}$, vertex $Q$ on $\\overline{AC}$, and vertices $R$ and $S$ on $\\overline{BC}$. In terms of the side length $PQ = \\omega$, the area of $PQRS$ can be expresse... | math | {
"style": "rule",
"ground_truth": "161"
} | {
"split": "train",
"index": 31906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The length of the escalator is 200 steps. When Petya walks down the escalator, he counts 50 steps. How many steps will he count if he runs twice as fast? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 31907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( p, q, r, s, \\) and \\( t \\) be the roots of the polynomial \n\\[ x^5 + 10x^4 + 20x^3 + 15x^2 + 6x + 3 = 0. \\]\nFind the value of\n\\[ \\frac{1}{pq} + \\frac{1}{pr} + \\frac{1}{ps} + \\frac{1}{pt} + \\frac{1}{qr} + \\frac{1}{qs} + \\frac{1}{qt} + \\frac{1}{rs} + \\fra... | math | {
"style": "rule",
"ground_truth": "\\frac{20}{3}"
} | {
"split": "train",
"index": 31908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to arrange three indistinguishable rooks on a $6 \\times 6$ board such that no two rooks are attacking each other? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2400"
} | {
"split": "train",
"index": 31909,
"solution": "There are $6 \\times 6=36$ possible places to place the first rook. Since it cannot be in the same row or column as the first, the second rook has $5 \\times 5=25$ possible places, and similarly, the third rook has $4 \\times 4=16$ possible places. However, the roo... |
deepscaler | [
{
"role": "user",
"content": "In the sequence $\\{a_n\\}$, if $a_1=-2$ and for any $n\\in\\mathbb{N}^*$, $a_{n+1}=1+2a_n$, then the sum of the first $10$ terms of the sequence $\\{a_n\\}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1033"
} | {
"split": "train",
"index": 31910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation $x^{x^{x^{.^{.^.}}}}=2$ is satisfied when $x$ is equal to: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 31911,
"solution": "1. **Define the Infinite Power Tower**: Let's consider the infinite power tower $x^{x^{x^{.^{.^.}}}}$. We denote this expression by $y$, so we have:\n \\[\n y = x^{x^{x^{.^{.^.}}}}\n \\]\n\n2. **Substitute and Simplify**: Given that $y = 2$, we substitute thi... |
deepscaler | [
{
"role": "user",
"content": "If the product of 6 consecutive odd numbers is 135135, what is the sum of these 6 numbers? $\\qquad$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 31912,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Angry reviews about the work of an online store are left by $80\\%$ of dissatisfied customers (those who were poorly served in the store). Of the satisfied customers, only $15\\%$ leave a positive review. A certain online store earned 60 angry and 20 positive reviews. Using thi... | math | {
"style": "rule",
"ground_truth": "0.64"
} | {
"split": "train",
"index": 31913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle \\(ABC\\) has a right angle at \\(B\\), with \\(AB = 3\\) and \\(BC = 4\\). If \\(D\\) and \\(E\\) are points on \\(AC\\) and \\(BC\\), respectively, such that \\(CD = DE = \\frac{5}{3}\\), find the perimeter of quadrilateral \\(ABED\\). Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "28/3"
} | {
"split": "train",
"index": 31914,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression $(-\\frac{1}{343})^{-2/3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 31915,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\sin x\\cos x+1-2\\sin^2x$.\n\n(Ⅰ) Find the smallest positive period of $f(x)$;\n\n(Ⅱ) Find the maximum and minimum values of $f(x)$ in the interval $\\left[-\\frac{\\pi}{3}, \\frac{\\pi}{4}\\right]$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}+1}{2}"
} | {
"split": "train",
"index": 31916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The school organized a picnic with several participants. The school prepared many empty plates. Each attendee counts the empty plates and takes one empty plate to get food (each person can only take one plate, no more). The first attendee counts all the empty plates, the second... | math | {
"style": "rule",
"ground_truth": "1006"
} | {
"split": "train",
"index": 31917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two complex numbers $z_1$ and $z_2$ whose corresponding points in the complex plane are symmetrical about the imaginary axis, and $z_1 = 1 + 2i$ where $i$ is the imaginary unit, calculate the product of $z_1$ and $z_2$. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 31918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the value of $k$ such that the equation\n\\[\\frac{x + 2}{kx - 1} = x\\]has exactly one solution. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 31919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a positive integer $n\\,$, let $p(n)\\,$ be the product of the non-zero digits of $n\\,$. (If $n\\,$ has only one digits, then $p(n)\\,$ is equal to that digit.) Let\n$S=p(1)+p(2)+p(3)+\\cdots+p(999)$\n .\n What is the largest prime factor of $S\\,$? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "103"
} | {
"split": "train",
"index": 31920,
"solution": "Note that $p(1)=p(11), p(2)=p(12), p(3)=p(13), \\cdots p(19)=p(9)$, and $p(37)=3p(7)$. So $p(10)+p(11)+p(12)+\\cdots +p(19)=46$, $p(10)+p(11)+\\cdots +p(99)=46*45=2070$. We add $p(1)+p(2)+p(3)+\\cdots +p(10)=45$ to get 2115. When we add a digit we multiply the sum ... |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be an odd integer with exactly 12 positive divisors. Find the number of positive divisors of $27n^3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 31921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four teams, including Quixajuba, are competing in a volleyball tournament where:\n\n- Each team plays against every other team exactly once;\n- Any match ends with one team winning;\n- In any match, the teams have an equal probability of winning;\n- At the end of the tournament... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 31922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coefficient of $\\frac{1}{x}$ in the expansion of $(1-x^2)^4\\left(\\frac{x+1}{x}\\right)^5$ needs to be determined. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-29"
} | {
"split": "train",
"index": 31923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola \\(\\Gamma: y^{2}=8 x\\) with focus \\(F\\), a line \\(l\\) passing through \\(F\\) intersects parabola \\(\\Gamma\\) at points \\(A\\) and \\(B\\). Tangents to parabola \\(\\Gamma\\) at \\(A\\) and \\(B\\) intersect the \\(y\\)-axis at points \\(P\\) and \\(... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 31924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You are trying to cross a 400 foot wide river. You can jump at most 4 feet, but you have many stones you can throw into the river. You will stop throwing stones and cross the river once you have placed enough stones to be able to do so. You can throw straight, but you can't jud... | math | {
"style": "rule",
"ground_truth": "712.811"
} | {
"split": "train",
"index": 31925,
"solution": "If we divide the river into 1004-foot sections, then to be able to cross we need to get at least one stone into each section. On average, this takes $$\\frac{100}{100}+\\frac{100}{99}+\\cdots+\\frac{100}{1} \\approx 100 \\ln 100$$ stone throws (it takes $\\frac{100... |
deepscaler | [
{
"role": "user",
"content": "The sequence \\(a_{n}\\) is defined as follows:\n\n\\[ a_{1} = 1, \\quad a_{n+1} = a_{n} + \\frac{2a_{n}}{n} \\text{ for } n \\geq 1 \\]\n\nFind \\(a_{100}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5151"
} | {
"split": "train",
"index": 31926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A, B, C$ in the plane satisfy $\\overline{A B}=2002, \\overline{A C}=9999$. The circles with diameters $A B$ and $A C$ intersect at $A$ and $D$. If $\\overline{A D}=37$, what is the shortest distance from point $A$ to line $B C$? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 31927,
"solution": "$\\angle A D B=\\angle A D C=\\pi / 2$ since $D$ lies on the circles with $A B$ and $A C$ as diameters, so $D$ is the foot of the perpendicular from $A$ to line $B C$, and the answer is the given 37."
} |
deepscaler | [
{
"role": "user",
"content": "In isosceles triangle $\\triangle ABC$ we have $AB=AC=4$. The altitude from $B$ meets $\\overline{AC}$ at $H$. If $AH=3(HC)$ then determine $BC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 31928,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $n \\ge 0$ is an integer and all the roots of $x^3 + \n\\alpha x + 4 - ( 2 \\times 2016^n) = 0$ are integers. Find all possible values of $\\alpha$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 31929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many values of $x$ with $0^\\circ \\le x < 360^\\circ$ satisfy $\\sin x = -0.73$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 31930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using 1 digit of '1', 2 digits of '2', and 2 digits of '3', how many different four-digit numbers can be formed? Fill in the blank with the total number of different four-digit numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 31931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be positive integers such that $7x^5 = 11y^{13}.$ The minimum possible value of $x$ has a prime factorization $a^cb^d.$ What is $a + b + c + d?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 31932,
"solution": "1. **Equation Setup**: We start with the equation given in the problem:\n \\[\n 7x^5 = 11y^{13}\n \\]\n We aim to find the minimum possible value of $x$ in terms of its prime factors.\n\n2. **Prime Factorization of $x$**: Assume $x$ has a prime factorizatio... |
deepscaler | [
{
"role": "user",
"content": "An ant starts at the origin of a coordinate plane. Each minute, it either walks one unit to the right or one unit up, but it will never move in the same direction more than twice in the row. In how many different ways can it get to the point $(5,5)$ ? Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 31933,
"solution": "We can change the ant's sequence of moves to a sequence $a_{1}, a_{2}, \\ldots, a_{10}$, with $a_{i}=0$ if the $i$-th step is up, and $a_{i}=1$ if the $i$-th step is right. We define a subsequence of moves $a_{i}, a_{i+1}, \\ldots, a_{j}$, ( $i \\leq j$ ) as an up ... |
deepscaler | [
{
"role": "user",
"content": "Eight points are chosen on the circumference of a circle, labelled $P_{1}, P_{2}, \\ldots, P_{8}$ in clockwise order. A route is a sequence of at least two points $P_{a_{1}}, P_{a_{2}}, \\ldots, P_{a_{n}}$ such that if an ant were to visit these points in their given order, sta... | math | {
"style": "rule",
"ground_truth": "8744"
} | {
"split": "train",
"index": 31934,
"solution": "Solution 1: How many routes are there if we are restricted to $n$ available points, and we must use all $n$ of them? The answer is $n 2^{n-2}$ : first choose the starting point, then each move after that must visit one of the two neighbors of your expanding region ... |
deepscaler | [
{
"role": "user",
"content": "The ticket price for a cinema is: 6 yuan per individual ticket, 40 yuan for a group ticket for every 10 people, and students enjoy a 10% discount. A school with 1258 students plans to watch a movie (teachers get in for free). The school should pay the cinema at least ____ yuan.... | math | {
"style": "rule",
"ground_truth": "4536"
} | {
"split": "train",
"index": 31935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $$\\overrightarrow {m} = (\\sin \\omega x + \\cos \\omega x, \\sqrt {3} \\cos \\omega x)$$, $$\\overrightarrow {n} = (\\cos \\omega x - \\sin \\omega x, 2\\sin \\omega x)$$ ($\\omega > 0$), and the function $f(x) = \\overrightarrow {m} \\cdot \\overrightarrow {n}$, if the... | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"split": "train",
"index": 31936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum value of \\( N \\) such that \\( N! \\) has exactly 2013 trailing zeros? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8069"
} | {
"split": "train",
"index": 31937,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many whole numbers between $200$ and $500$ contain the digit $3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "138"
} | {
"split": "train",
"index": 31938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest positive integer \\( n \\) such that \\( n^{3} + 4n^{2} - 15n - 18 \\) is the cube of an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 31939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a flood control emergency, a large oil tank drifting downstream from upstream needs to be exploded by shooting. It is known that there are only $5$ bullets. The first hit can only cause the oil to flow out, and the second hit can cause an explosion. Each shot is independent,... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{27}"
} | {
"split": "train",
"index": 31940,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle has a radius of six inches. Two parallel chords $CD$ and $EF$ are drawn such that the distance from the center of the circle to chord $CD$ is three inches, and the distance to chord $EF$ is four inches. How long is chord $CD$ and $EF$? Express your answer in the simple... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{5}"
} | {
"split": "train",
"index": 31941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first digit of a string of 2002 digits is a 1. Any two-digit number formed by consecutive digits within this string is divisible by 19 or 31. What is the largest possible last digit in this string? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 31942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a five-digit number that, when divided by each of the 12 natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 13, gives different remainders. What is this five-digit number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "83159"
} | {
"split": "train",
"index": 31943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A straight one-way city street has 8 consecutive traffic lights. Every light remains green for 1.5 minutes, yellow for 3 seconds, and red for 1.5 minutes. The lights are synchronized so that each light turns red 10 seconds after the preceding one turns red. Calculate the longes... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 31944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sara baked 60 pies. Of these, one-third contained berries, half contained cream, three-fifths contained nuts, and one-fifth contained coconut. What is the largest possible number of pies that had none of these ingredients? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 31945,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of ${(6x+\\frac{1}{3\\sqrt{x}})}^{9}$, arrange the fourth term in ascending powers of $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{224}{9}"
} | {
"split": "train",
"index": 31946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average age of 33 fifth-graders is 11. The average age of 55 of their parents is 33. What is the average age of all of these parents and fifth-graders? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24.75"
} | {
"split": "train",
"index": 31947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Daniel and Scott are playing a game where a player wins as soon as he has two points more than his opponent. Both players start at par, and points are earned one at a time. If Daniel has a $60 \\%$ chance of winning each point, what is the probability that he will win the game?... | math | {
"style": "rule",
"ground_truth": "9 / 13"
} | {
"split": "train",
"index": 31948,
"solution": "Consider the situation after two points. Daniel has a $9 / 25$ chance of winning, Scott, $4 / 25$, and there is a $12 / 25$ chance that the players will be tied. In the latter case, we revert to the original situation. In particular, after every two points, either ... |
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