data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "You are trapped in ancient Japan, and a giant enemy crab is approaching! You must defeat it by cutting off its two claws and six legs and attacking its weak point for massive damage. You cannot cut off any of its claws until you cut off at least three of its legs, and you canno... | math | {
"style": "rule",
"ground_truth": "14400"
} | {
"split": "train",
"index": 21349,
"solution": "The answer is given by $6!2!\\binom{5}{2}$, because we can cut off the claws and legs in any order and there are $\\binom{5}{2}$ ways to decide when to cut off the two claws (since we can do it at any time among the last 5 cuts)."
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can 7 people sit around a round table, considering that two seatings are the same if one is a rotation of the other, and additionally, one specific person must sit between two particular individuals? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 21350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle \\( EFGH \\) is 10 cm by 6 cm. \\( P \\) is the midpoint of \\( \\overline{EF} \\), and \\( Q \\) is the midpoint of \\( \\overline{GH} \\). Calculate the area of region \\( EPGQ \\).\n\n** Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 21351,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways can a schedule of 4 mathematics courses - algebra, geometry, number theory, and calculus - be created in an 8-period day if exactly one pair of these courses can be taken in consecutive periods, and the other courses must not be consecutive? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"split": "train",
"index": 21352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A company needs to renovate its new office building. If the renovation is done solely by Team A, it would take 18 weeks, and if done solely by Team B, it would take 12 weeks. The result of the bidding is that Team A will work alone for the first 3 weeks, and then both Team A an... | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 21353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.\n ~ pi_is_3.14 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "307"
} | {
"split": "train",
"index": 21354,
"solution": "If $16p+1 =k$, we have $k \\equiv 1 \\mod 16$, so $k^3 \\equiv 1 \\mod 16$. If $k=1$ we have $p=0$, which is not prime. If $k=17$ we have $16p+1=4913$, or $p=\\boxed{307}$"
} |
deepscaler | [
{
"role": "user",
"content": "A store received a large container of milk. The salesperson has a balance scale that lacks weights (milk bottles can be placed on the scale), and there are 3 identical milk bottles, two of which are empty, and one has 1 liter of milk. How can exactly 85 liters of milk be measur... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 21355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 15x^2 + 50x - 60$. It is given that there exist real numbers $A$, $B$, and $C$ such that \\[\\dfrac{1}{s^3 - 15s^2 + 50s - 60} = \\dfrac{A}{s-p} + \\dfrac{B}{s-q} + \\frac{C}{s-r}\\]for all $s\\not\\in\\{p,q,r... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 21356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are both unit vectors, and their angle is $120^{\\circ}$, calculate the magnitude of the vector $|\\overrightarrow{a}-2\\overrightarrow{b}|$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"split": "train",
"index": 21357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. Wha... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 21358,
"solution": "1. **Define Variables:**\n Let the length of the track be $x$ meters.\n\n2. **Analyze First Meeting:**\n Since Brenda and Sally start at diametrically opposite points, they start $x/2$ meters apart. When they first meet, Brenda has run 100 meters. Therefore, Sa... |
deepscaler | [
{
"role": "user",
"content": "Given $\\overrightarrow{a}=(-3,4)$, $\\overrightarrow{b}=(5,2)$, find $|\\overrightarrow{a}|$, $|\\overrightarrow{b}|$, and $\\overrightarrow{a}\\cdot \\overrightarrow{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 21359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two positive integers $p,q \\in \\mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$. Determine the smallest $n$ so that such operati... | math | {
"style": "rule",
"ground_truth": "\\frac{p+q}{\\gcd(p,q)}"
} | {
"split": "train",
"index": 21360,
"solution": "\nGiven two positive integers \\( p \\) and \\( q \\), we are to determine the smallest number \\( n \\) such that the operation of choosing two identical numbers \\( a, a \\) on the blackboard and replacing them with \\( a+p \\) and \\( a+q \\) can go on infinitel... |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of $2023 \\cdot n$. What is the minimum value of $k(n)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 21361,
"solution": "The minimum is $3$. \\n\\n\\textbf{First solution.} We record the factorization $2023 = 7\\cdot 17^2$. We first rule out $k(n)=1$ and $k(n)=2$. If $k(n)=1$, then $2023n = 2^a$ for some $a$, which clearly cannot happen. If $k(n)=2$, then $2023n=2^a+2^b=2^b(1+2^{a-b}... |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle and a square both have perimeters of 48 inches. What is the ratio of the length of the side of the triangle to the length of the side of the square? Express your answer as a common fraction. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac43"
} | {
"split": "train",
"index": 21362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the integral $\\int_{0}^{\\frac{\\pi}{2}} \\sin^{2} \\frac{x}{2} dx =$ \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4} - \\frac{1}{2}"
} | {
"split": "train",
"index": 21363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five million times eight million equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40,000,000,000,000"
} | {
"split": "train",
"index": 21364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[f(x) = \\sqrt{x(50 - x)} + \\sqrt{x(2 - x)}\\]for $0 \\le x \\le 2.$ Let $M$ be the maximum value of $f(x),$ and let $x = x_0$ be the point where the maximum value is attained. Enter the ordered pair $(x_0,M).$ Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{25}{13}, 10 \\right)"
} | {
"split": "train",
"index": 21365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two angles of a triangle measure 30 and 45 degrees. If the side of the triangle opposite the 30-degree angle measures $6\\sqrt2$ units, what is the sum of the lengths of the two remaining sides? Express your answer as a decimal to the nearest tenth. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "28.4"
} | {
"split": "train",
"index": 21366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A chess piece called \"the four-liner\" attacks two vertical and two horizontal squares adjacent to the square on which it stands. What is the maximum number of non-attacking four-liners that can be placed on a $10 \\times 10$ board? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 21367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of the sum $\\frac{3}{4}+\\frac{3^2}{4^2}+\\frac{3^3}{4^3}+ \\ldots +\\frac{3^{15}}{4^{15}}$? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3177884751}{1073741824}"
} | {
"split": "train",
"index": 21368,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$100_{10}$ in base $b$ has exactly $5$ digits. What is the value of $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 21369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation wi... | math | {
"style": "rule",
"ground_truth": "294"
} | {
"split": "train",
"index": 21370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The total number of matches played in the 2006 World Cup competition can be calculated by summing the number of matches determined at each stage of the competition. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 21371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of the geometric body formed by points whose distance to line segment AB is no greater than three units is $216 \\pi$. Calculate the length of the line segment AB. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 21372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle with integral sides and an isosceles perimeter of 11, calculate the area of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{2.75}}{2}"
} | {
"split": "train",
"index": 21373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x)$ is a function defined on $\\mathbb{R}$, and for any $x \\in \\mathbb{R}$, it holds that $f(x+2) = f(2-x) + 4f(2)$, if the graph of the function $y=f(x+1)$ is symmetric about the point $(-1,0)$ and $f(1)=3$, then find $f(2015)$. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 21374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the real numbers \\(a\\) and \\(b\\) satisfying \\(\\left(a - \\frac{b}{2}\\right)^2 = 1 - \\frac{7}{4} b^2\\), let \\(t_{\\max}\\) and \\(t_{\\min}\\) denote the maximum and minimum values of \\(t = a^2 + 2b^2\\), respectively. Find the value of \\(t_{\\text{max}} + t_{\... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{7}"
} | {
"split": "train",
"index": 21375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ is an integer, find the largest integer that always divides the expression \\[(12x + 2)(8x + 14)(10x + 10)\\] when $x$ is odd. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 21376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We say that a positive real number $d$ is good if there exists an infinite sequence $a_{1}, a_{2}, a_{3}, \\ldots \\in(0, d)$ such that for each $n$, the points $a_{1}, \\ldots, a_{n}$ partition the interval $[0, d]$ into segments of length at most $1 / n$ each. Find $\\sup \\{... | math | {
"style": "rule",
"ground_truth": "\\ln 2"
} | {
"split": "train",
"index": 21377,
"solution": "Let $d^{\\star}=\\sup \\{d \\mid d$ is good $\\}$. We will show that $d^{\\star}=\\ln (2) \\doteq 0.693$. 1. $d^{\\star} \\leq \\ln 2:$ Assume that some $d$ is good and let $a_{1}, a_{2}, \\ldots$ be the witness sequence. Fix an integer $n$. By assumption, the pref... |
deepscaler | [
{
"role": "user",
"content": "Let P be a moving point on the line $3x+4y+3=0$, and through point P, two tangents are drawn to the circle $C: x^2+y^2-2x-2y+1=0$, with the points of tangency being A and B, respectively. Find the minimum value of the area of quadrilateral PACB. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 21378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the subsets of the set $\\{1,2, \\cdots, 100\\}$, calculate the maximum number of elements a subset can have if it does not contain any pair of numbers where one number is exactly three times the other. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 21379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of sets ${a,b,c}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,61$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "728"
} | {
"split": "train",
"index": 21380,
"solution": "Note that the prime factorization of the product is $3^{2}\\cdot 7 \\cdot 11 \\cdot 17 \\cdot 31 \\cdot 41 \\cdot 61$. Ignoring overcounting, by stars and bars there are $6$ ways to choose how to distribute the factors of $3$, and $3$ ways to distribute the factors... |
deepscaler | [
{
"role": "user",
"content": "I have 5 marbles numbered 1 through 5 in a bag. Suppose I take out two different marbles at random. What is the expected value of the sum of the numbers on the marbles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 21381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five points, no three of which are collinear, are given. What is the least possible value of the numbers of convex polygons whose some corners are from these five points? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 21382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sarah buys $3$ gumballs from a gumball machine that contains $10$ orange, $6$ green, and $9$ yellow gumballs. What is the probability that the first gumball is orange, the second is green or yellow, and the third is also orange? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "9/92"
} | {
"split": "train",
"index": 21383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( ABCD \\) and \\( WXYZ \\) be two squares that share the same center such that \\( WX \\parallel AB \\) and \\( WX < AB \\). Lines \\( CX \\) and \\( AB \\) intersect at \\( P \\), and lines \\( CZ \\) and \\( AD \\) intersect at \\( Q \\). If points \\( P, W \\), and \\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"split": "train",
"index": 21384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "1. How many four-digit numbers with no repeated digits can be formed using the digits 1, 2, 3, 4, 5, 6, 7, and the four-digit number must be even?\n2. How many five-digit numbers with no repeated digits can be formed using the digits 0, 1, 2, 3, 4, 5, and the five-digit number ... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 21385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a, b \\in \\mathbb{R}$, and $a^2 + 2ab - 3b^2 = 1$, find the minimum value of $a^2 + b^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} + 1}{4}"
} | {
"split": "train",
"index": 21386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A shop specializing in small refrigerators sells an average of 50 refrigerators per month. The cost of each refrigerator is $1200. Since shipping is included in the sale, the shop needs to pay a $20 shipping fee for each refrigerator. In addition, the shop has to pay a \"storef... | math | {
"style": "rule",
"ground_truth": "1824"
} | {
"split": "train",
"index": 21387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest common factor of 120, 180, and 300? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 21388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "While entertaining his younger sister Alexis, Michael drew two different cards from an ordinary deck of playing cards. Let $a$ be the probability that the cards are of different ranks. Compute $\\lfloor 1000a\\rfloor$ . Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "941"
} | {
"split": "train",
"index": 21389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers, and let $r, s$, and $t$ be the roots of $f(x)=x^{3}+a x^{2}+b x-1$. Also, $g(x)=x^{3}+m x^{2}+n x+p$ has roots $r^{2}, s^{2}$, and $t^{2}$. If $g(-1)=-5$, find the maximum possible value of $b$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "1+\\sqrt{5}"
} | {
"split": "train",
"index": 21390,
"solution": "By Vieta's Formulae, $m=-\\left(r^{2}+s^{2}+t^{2}\\right)=-a^{2}+2 b, n=r^{2} s^{2}+s^{2} t^{2}+t^{2} r^{2}= b^{2}+2 a$, and $p=-1$. Therefore, $g(-1)=-1-a^{2}+2 b-b^{2}-2 a-1=-5 \\Leftrightarrow(a+1)^{2}+(b-1)^{2}=5$. This is an equation of a circle, so $b$ reache... |
deepscaler | [
{
"role": "user",
"content": "Mark writes the expression $\\sqrt{d}$ for each positive divisor $d$ of 8 ! on the board. Seeing that these expressions might not be worth points on HMMT, Rishabh simplifies each expression to the form $a \\sqrt{b}$, where $a$ and $b$ are integers such that $b$ is not divisible... | math | {
"style": "rule",
"ground_truth": "3480"
} | {
"split": "train",
"index": 21391,
"solution": "Let $\\sqrt{n}$ simplify to $a_{n} \\sqrt{b_{n}}$. Notice that both $a_{n}$ and $b_{n}$ are multiplicative. Thus, $\\sum_{d \\mid n} a_{d}$ and $\\sum_{d \\mid n} b_{d}$ are multiplicative. We consider the sum $\\sum_{d \\mid p^{k}} a_{d}$ and $\\sum_{d \\mid p^{k}... |
deepscaler | [
{
"role": "user",
"content": "Let $a\\star b = ab+2b-a$. If $5\\star x = 37$, find the value of $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 21392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of three numbers \\( a \\), \\( b \\), and \\( c \\) is 150. If we increase \\( a \\) by 10, decrease \\( b \\) by 5, and multiply \\( c \\) by 7, the three resulting numbers are equal. What is the value of \\( b \\)? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "77.\\overline{3}"
} | {
"split": "train",
"index": 21393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers $n$ satisfy the inequality $-8\\pi\\le n\\le10\\pi$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 21394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sector with central angle $60^\\circ$ is cut from a circle of radius 10. Determine both the radius of the circle circumscribed about the sector and the inradius of the triangle formed by the sector. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 21395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In acute triangle $ \\Delta ABC $, the sides opposite angles $A, B, C$ are denoted as $a, b, c$, respectively. Let vector $ \\overrightarrow{m}=(2,c) $ and vector $ \\overrightarrow{n} =\\left(\\frac{b}{2}\\cos C-\\sin A, \\cos B\\right) $, given that $b=\\sqrt{3}$ and $ \\over... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 21396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice has $24$ apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "190"
} | {
"split": "train",
"index": 21397,
"solution": "To solve the problem, we need to find the number of ways Alice, Becky, and Chris can each receive at least two apples from a total of 24 apples. Let's denote the number of apples received by Alice, Becky, and Chris as $a$, $b$, and $c$ respectively. The condition g... |
deepscaler | [
{
"role": "user",
"content": "$a,b,c$ are distinct real roots of $x^3-3x+1=0$. $a^8+b^8+c^8$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "186"
} | {
"split": "train",
"index": 21398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $|\\vec{a}|=1$, $|\\vec{b}|=6$, and $\\vec{a}\\cdot(\\vec{b}-\\vec{a})=2$, calculate the angle between $\\vec{a}$ and $\\vec{b}$. 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": 21399,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$ and $b$ are real numbers, and $\\frac{a}{1-i} + \\frac{b}{2-i} = \\frac{1}{3-i}$, find the sum of the first 100 terms of the arithmetic sequence ${an + b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-910"
} | {
"split": "train",
"index": 21400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the area of the triangle formed by the tangent line to the curve $y=x^{-\\frac{1}{2}}$ at the point $(a,a^{-\\frac{1}{2}})$ and the two coordinate axes is $18$, find the real number $a=\\_\\_\\_\\_\\_\\_\\_\\_.$ Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 21401,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways can a student schedule $3$ mathematics courses -- algebra, geometry, and number theory -- in a $6$-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other $3$ periods is of no concern here.) Le... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 21402,
"solution": "To solve this problem, we need to find the number of ways to schedule 3 mathematics courses (algebra, geometry, and number theory) in a 6-period day such that no two mathematics courses are taken in consecutive periods.\n\n#### Step 1: Total ways to schedule withou... |
deepscaler | [
{
"role": "user",
"content": "Circle $\\Gamma$ with radius $1$ is centered at point $A$ on the circumference of circle $\\omega$ with radius $7$ . Suppose that point $P$ lies on $\\omega$ with $AP=4$ . Determine the product of the distances from $P$ to the two intersections of $\\omega$ an... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 21403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $ABCD$ satisfies $\\angle ABC = \\angle ACD = 90^{\\circ}, AC=20,$ and $CD=30.$ Diagonals $\\overline{AC}$ and $\\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$ Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 21404,
"solution": "1. **Assign Variables and Use Pythagorean Theorem in $\\triangle ABC$:**\n Let $AB = x$ and $BC = y$. Since $\\angle ABC = 90^\\circ$, by the Pythagorean theorem, we have:\n \\[\n x^2 + y^2 = AC^2 = 20^2 = 400.\n \\]\n\n2. **Calculate Area of $\\triangle AC... |
deepscaler | [
{
"role": "user",
"content": "A student travels from his university to his hometown, a distance of 150 miles, in a sedan that averages 25 miles per gallon. For the return trip, he drives his friend's truck, which averages only 15 miles per gallon. Additionally, they make a detour of 50 miles at an average o... | math | {
"style": "rule",
"ground_truth": "16.67"
} | {
"split": "train",
"index": 21405,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5$. The solution set for $x$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\{x \\mid 2 < x < 3\\}"
} | {
"split": "train",
"index": 21406,
"solution": "1. **Understanding the Problem:**\n The problem asks us to find the set of values for $x$ such that the sum of the greatest integer less than or equal to $x$ (denoted $\\lfloor x \\rfloor$) and the least integer greater than or equal to $x$ (denoted $\\lceil x \\... |
deepscaler | [
{
"role": "user",
"content": "There is a target on the wall consisting of five zones: a central circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the number of points for hitting each zone is inversely proportional to the probability of... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 21407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive odd integers greater than 1 and less than $200$ are square-free? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "79"
} | {
"split": "train",
"index": 21408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let point $P$ lie on the curve $y= \\frac {1}{2}e^{x}$, and point $Q$ lie on the curve $y=\\ln (2x)$. Find the minimum value of $|PQ|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}(1-\\ln 2)"
} | {
"split": "train",
"index": 21409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Out of 8 shots, 3 hit the target. The total number of ways in which exactly 2 hits are consecutive is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 21410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute: $\\sin 187^{\\circ}\\cos 52^{\\circ}+\\cos 7^{\\circ}\\sin 52^{\\circ}=\\_\\_\\_\\_\\_\\_ \\cdot$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 21411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of men. For each of the three factors, the probability that a randomly selected man in the population has only this risk factor (and none of the others) is 0.1. For any t... | math | {
"style": "rule",
"ground_truth": "76"
} | {
"split": "train",
"index": 21412,
"solution": "We first assume a population of $100$ to facilitate solving. Then we simply organize the statistics given into a Venn diagram.\n[asy] pair A,B,C,D,E,F,G; A=(0,55); B=(60,55); C=(60,0); D=(0,0); draw(A--B--C--D--A); E=(30,35); F=(20,20); G=(40,20); draw(circle(E,15)... |
deepscaler | [
{
"role": "user",
"content": "A ferry boat shuttles tourists to an island every half-hour from 10 AM to 3 PM, with 100 tourists on the first trip and 2 fewer tourists on each successive trip. Calculate the total number of tourists taken to the island that day. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "990"
} | {
"split": "train",
"index": 21413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\\dfrac{1}{2}$ is $\\dfrac{a-b\\pi}{c}$, where $a$, $b$, and $c$ are positive integers with... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 21414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x_{1}, x_{2}, x_{3}, x_{4} \\) be non-negative real numbers satisfying the equation:\n\\[ \nx_{1} + x_{2} + x_{3} + x_{4} = 1 \n\\]\nFind the maximum value of \\( \\sum_{1 \\leq i < j \\leq 4}(x_{i} + x_{j}) \\sqrt{x_{i} x_{j}} \\), and determine the values of \\( x_{1}... | math | {
"style": "rule",
"ground_truth": "3/4"
} | {
"split": "train",
"index": 21415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Elmer’s new car provides a 70% better fuel efficiency and uses a type of fuel that is 35% more expensive per liter, calculate the percentage by which Elmer will save money on fuel costs if he uses his new car for his journey. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "20.6\\%"
} | {
"split": "train",
"index": 21416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jeff has a 50 point quiz at 11 am . He wakes up at a random time between 10 am and noon, then arrives at class 15 minutes later. If he arrives on time, he will get a perfect score, but if he arrives more than 30 minutes after the quiz starts, he will get a 0 , but otherwise, he... | math | {
"style": "rule",
"ground_truth": "\\frac{55}{2}"
} | {
"split": "train",
"index": 21417,
"solution": "If he wakes up between 10:00 and 10:45, he will arrive on time and get a perfect score of 50. If he wakes up between 10:45 and 11:15, he will arrive late and lose points. If he wakes up $k$ minutes after $10: 45$, then he gets $50-k$ points. Finally, if he wakes up... |
deepscaler | [
{
"role": "user",
"content": "In the interval \\([-6, 6]\\), an element \\(x_0\\) is arbitrarily chosen. If the slope of the tangent line to the parabola \\(y = x^2\\) at \\(x = x_0\\) has an angle of inclination \\(\\alpha\\), find the probability that \\(\\alpha \\in \\left[ \\frac{\\pi}{4}, \\frac{3\\pi}... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{12}"
} | {
"split": "train",
"index": 21418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are four even integers in the top five rows of Pascal's Triangle. How many even integers are in the top 10 rows of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 21419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Burattino got on a train. After travelling half of the total distance, he fell asleep and slept until there was only half of the distance he slept left to travel. What fraction of the total journey did Burattino travel awake? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 21420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of a triangle, given that the radius of the inscribed circle is 1, and the lengths of all three altitudes are integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"split": "train",
"index": 21421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $(-1)^{45} + 2^{(3^2+5^2-4^2)}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "262143"
} | {
"split": "train",
"index": 21422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The vectors $\\mathbf{a} = \\begin{pmatrix} 1 \\\\ 1 \\\\ 1 \\end{pmatrix},$ $\\mathbf{b} = \\begin{pmatrix} 2 \\\\ -3 \\\\ 1 \\end{pmatrix},$ and $\\mathbf{c} = \\begin{pmatrix} 4 \\\\ 1 \\\\ -5 \\end{pmatrix}$ are mutually orthogonal. There exist scalars $p,$ $q,$ and $r$ su... | math | {
"style": "rule",
"ground_truth": "\\left( 2, -\\frac{13}{7}, -\\frac{4}{7} \\right)"
} | {
"split": "train",
"index": 21423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When Cheenu was a boy, he could run $15$ miles in $3$ hours and $30$ minutes. As an old man, he can now walk $10$ miles in $4$ hours. How many minutes longer does it take for him to walk a mile now compared to when he was a boy? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 21424,
"solution": "1. **Calculate the time Cheenu took per mile as a boy:**\n - Cheenu ran 15 miles in 3 hours and 30 minutes.\n - Convert hours to minutes: \\(3 \\text{ hours} = 3 \\times 60 = 180 \\text{ minutes}\\).\n - Add the extra 30 minutes: \\(180 \\text{ minutes} + 30 ... |
deepscaler | [
{
"role": "user",
"content": "Circle $T$ has its center at point $T(-2,6)$. Circle $T$ is reflected across the $y$-axis and then translated 8 units down. What are the coordinates of the image of the center of circle $T$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2, -2)"
} | {
"split": "train",
"index": 21425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the random variable $x$ follows a normal distribution $N(3, \\sigma^2)$, and $P(x \\leq 4) = 0.84$, find $P(2 < x < 4)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.68"
} | {
"split": "train",
"index": 21426,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A person can commute by train or car. If he takes the train to work in the morning, he takes the car in the afternoon; if he takes the train home in the afternoon, he takes the car in the morning. Over $x$ days, this person took the train 9 times, took the car in the morning 8 ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 21427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $103^4 - 4 \\cdot 103^3 + 6 \\cdot 103^2 - 4 \\cdot 103 + 1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108243216"
} | {
"split": "train",
"index": 21428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number 210 is the product of two consecutive positive integers and is also the product of three consecutive integers. What is the sum of those five integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 21429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the le... | math | {
"style": "rule",
"ground_truth": "367"
} | {
"split": "train",
"index": 21430,
"solution": "To solve this problem, we need to understand the pattern in which numbers are placed in the grid. The numbers are placed starting from $1$ in the center of the grid and spiraling outwards in a clockwise manner. The grid is $15 \\times 15$, so it contains numbers fr... |
deepscaler | [
{
"role": "user",
"content": "What is the greatest number of Sundays that can occur in the first $49$ days of a year? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 21431,
"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": 21432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the arithmetic sequence $\\{a_n\\}$, $a_2=4$, $a_4+a_7=15$. \n(Ⅰ) Find the general term formula of sequence $\\{a_n\\}$. \n(Ⅱ) Let $b_{n}= \\frac {1}{a_{n}a_{n+1}}$, find the value of $b_1+b_2+b_3+\\ldots+b_{10}$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{39}"
} | {
"split": "train",
"index": 21433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are four cards, each with one of the numbers $2$, $0$, $1$, $5$ written on them. Four people, A, B, C, and D, each take one card.\n\nA says: None of the numbers you three have differ by 1 from the number I have.\nB says: At least one of the numbers you three have differs ... | math | {
"style": "rule",
"ground_truth": "5120"
} | {
"split": "train",
"index": 21434,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Assume every 7-digit whole number is a possible telephone number except those that begin with $0$ or $1$. What fraction of telephone numbers begin with $9$ and end with $0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{80}"
} | {
"split": "train",
"index": 21435,
"solution": "1. **Determine the total number of valid 7-digit phone numbers ($b$):**\n - The first digit can be any digit from 2 to 9 (since 0 and 1 are not allowed), giving us 8 choices.\n - Each of the remaining six digits can be any digit from 0 to 9, giving us 10 choice... |
deepscaler | [
{
"role": "user",
"content": "From the 20 natural numbers 1, 2, 3, ..., 20, if three numbers are randomly selected and their sum is an even number greater than 10, then there are $\\boxed{\\text{answer}}$ such sets of numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "563"
} | {
"split": "train",
"index": 21436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB = AC = 15$ and $BC = 14$. Points $D, E, F$ are on sides $\\overline{AB}, \\overline{BC},$ and $\\overline{AC},$ respectively, such that $\\overline{DE} \\parallel \\overline{AC}$ and $\\overline{EF} \\parallel \\overline{AB}$. What is the perimeter of par... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 21437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(\\sin (\\frac{\\omega}{2}x+\\varphi), 1)$ and $\\overrightarrow{b}=(1, \\cos (\\frac{\\omega}{2}x+\\varphi))$, where $\\omega > 0$ and $0 < \\varphi < \\frac{\\pi}{4}$, define the function $f(x)=(\\overrightarrow{a}+\\overrightarrow{b})\\cdot... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 21438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle JKL, where angle J is the right angle, KL measures 20 units, and JL measures 12 units. Calculate $\\tan K$. 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": 21439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If vectors $\\mathbf{a} = (1,0)$, $\\mathbf{b} = (0,1)$, $\\mathbf{c} = k\\mathbf{a} + \\mathbf{b}$ ($k \\in \\mathbb{R}$), $\\mathbf{d} = \\mathbf{a} - \\mathbf{b}$, and $\\mathbf{c} \\parallel \\mathbf{d}$, calculate the value of $k$ and the direction of vector $\\mathbf{c}$.... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 21440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $\\frac{x^{2}}{16}{+}\\frac{y^{2}}{b^{2}}{=}1$, and the line $l$ passing through $F_1$ intersects the ellipse at points $A$ and $B$. If the maximum value of $|AF_2|+|BF_2|$ is $10$, find the eccentricity of t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 21441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\[Q(x) = (3x^3 - 27x^2 + gx + h)(4x^3 - 36x^2 + ix + j),\\] where \\(g, h, i, j\\) are real numbers. Suppose that the set of all complex roots of \\(Q(x)\\) is \\(\\{1, 2, 6\\}\\). Find \\(Q(7).\\) Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "10800"
} | {
"split": "train",
"index": 21442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destin... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 21443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the coordinates of the three vertices of $\\triangle P_{1}P_{2}P_{3}$ are $P_{1}(1,2)$, $P_{2}(4,3)$, and $P_{3}(3,-1)$, the length of the longest edge is ________, and the length of the shortest edge is ________. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\sqrt {10}"
} | {
"split": "train",
"index": 21444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 3 and 9, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and ... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 21445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, points $A$, $B$, and $C$ are located such that $AB = 35$ units, $BC = 40$ units, and $CA = 45$ units. A point $X$ lies on side $AB$ such that $CX$ bisects $\\angle ACB$. Given that $BX = 21$ units, find the length of segment $AX$. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 21446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many factors of 8000 are perfect squares? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 21447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The natural number \\( a \\) is divisible by 55 and has 117 distinct divisors, including 1 and \\( a \\). Find the smallest such \\( a \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12390400"
} | {
"split": "train",
"index": 21448,
"solution": ""
} |
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