data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero di... | math | {
"style": "rule",
"ground_truth": "376"
} | {
"split": "train",
"index": 35949,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $0 < x < \\frac{\\pi}{2}$ and $\\sin(2x - \\frac{\\pi}{4}) = -\\frac{\\sqrt{2}}{10}$, find the value of $\\sin x + \\cos x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{10}}{5}"
} | {
"split": "train",
"index": 35950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct, positive factors does $1320$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 35951,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that F is the right focus of the ellipse $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a>b>0)$, and A is one endpoint of the ellipse's minor axis. If F is the trisection point of the chord of the ellipse that passes through AF, calculate the eccentricity of the ellipse. L... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 35952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A neighbor bought a certain quantity of beef at two shillings a pound, and the same quantity of sausages at eighteenpence a pound. If she had divided the same money equally between beef and sausages, she would have gained two pounds in the total weight. Determine the exact amou... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 35953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The steamboat \"Rarity\" travels for three hours at a constant speed after leaving the city, then drifts with the current for an hour, then travels for three hours at the same speed, and so on. If the steamboat starts its journey in city A and goes to city B, it takes it 10 hou... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 35954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIf \\( x_{1} \\) satisfies \\( 2x + 2^{x} = 5 \\) and \\( x_{2} \\) satisfies \\( 2x + 2 \\log_{2}(x - 1) = 5 \\), then \\( x_{1} + x_{2} = \\) ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"split": "train",
"index": 35955,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the arithmetic sequence $\\{a_n\\}$, $a_3 - a_2 = -2$, and $a_7 = -2$, calculate the value of $a_9$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 35956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is easy to place the complete set of ships for the game \"Battleship\" on a $10 \\times 10$ board (see illustration). What is the smallest square board on which this set can be placed? (Recall that according to the rules, ships must not touch each other, even at the corners.... | math | {
"style": "rule",
"ground_truth": "7 \\times 7"
} | {
"split": "train",
"index": 35957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$, let $S_n$ denote the sum of its first $n$ terms. Define $T_n = \\frac{S_1 + S_2 + \\dots + S_n}{n}$ as the \"ideal number\" of the sequence $a_1, a_2, \\dots, a_n$. If the \"ideal number\" of the sequence $a_1, a_2, \\dots, a_{502}$ is $2012$, calc... | math | {
"style": "rule",
"ground_truth": "2010"
} | {
"split": "train",
"index": 35958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose I have 6 shirts and 5 ties. How many shirt-and-tie outfits can I make? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 35959,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular box has width $12$ inches, length $16$ inches, and height $\\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle wit... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 35960,
"solution": "It isn't hard to see that the triangle connecting the centers of the faces of the rectangular prism is congruent to the triangle connecting the midpoints of three edges that concur. So we can now apply de Gua's theorem (https://en.wikipedia.org/wiki/De_Gua%27s_theo... |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be real numbers such that\n\\[4x^2 + 8xy + 5y^2 = 1.\\]Let $m$ and $M$ be the minimum and maximum values of $2x^2 + 3xy + 2y^2,$ respectively. Find the product $mM.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"split": "train",
"index": 35961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let triangle $ABC$ have incircle $\\omega$, which touches $BC, CA$, and $AB$ at $D, E$, and $F$, respectively. Then, let $\\omega_{1}$ and $\\omega_{2}$ be circles tangent to $AD$ and internally tangent to $\\omega$ at $E$ and $F$, respectively. Let $P$ be the intersection of l... | math | {
"style": "rule",
"ground_truth": "3600"
} | {
"split": "train",
"index": 35962,
"solution": "Let the centers of $\\omega_{1}$ and $\\omega_{2}$ be $O_{1}$ and $O_{2}$. Let $DE$ intersect $\\omega_{1}$ again at $Q$, and let $DF$ intersect $\\omega_{2}$ again at $R$. Note that since $\\omega_{1}$ and $\\omega_{2}$ must be tangent to $AD$ at the same point (b... |
deepscaler | [
{
"role": "user",
"content": "Given the vectors $\\overrightarrow{m}=(2\\sin \\omega x, \\cos ^{2}\\omega x-\\sin ^{2}\\omega x)$ and $\\overrightarrow{n}=( \\sqrt {3}\\cos \\omega x,1)$, where $\\omega > 0$ and $x\\in R$. If the minimum positive period of the function $f(x)= \\overrightarrow{m}\\cdot \\ove... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 35963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A mathematician is working on a geospatial software and comes across a representation of a plot's boundary described by the equation $x^2 + y^2 + 8x - 14y + 15 = 0$. To correctly render it on the map, he needs to determine the diameter of this plot. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "10\\sqrt{2}"
} | {
"split": "train",
"index": 35964,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\\{ A,B,C,D,M,N\\}$ . Find the least number $k... | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"split": "train",
"index": 35965,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all the four-digit positive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49495500"
} | {
"split": "train",
"index": 35966,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $ABC$ with right angle $C$, $CA = 30$ and $CB = 16$. Its legs $CA$ and $CB$ are extended beyond $A$ and $B$. Points $O_1$ and $O_2$ lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center $O_1$ is tangent... | math | {
"style": "rule",
"ground_truth": "737"
} | {
"split": "train",
"index": 35967,
"solution": "Let the radius of the circle be $r$. It can be seen that $\\Delta FHO_{1}$ and $\\Delta O_{2}GJ$ are similar to $\\Delta ACB$, and the length of the hypotenuses are $\\frac{17}{8}r$ and $\\frac {17}{15}r$, respectively. Then, the entire length of $HJ$ is going to b... |
deepscaler | [
{
"role": "user",
"content": "Given a large circle with a radius of 11 and small circles with a radius of 1, determine the maximum number of small circles that can be placed inside the large circle, such that each small circle is internally tangent to the large circle and the small circles do not overlap. L... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 35968,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $ xyz$ space, find the volume of the solid expressed by the sytem of inequality:\r\n\r $ 0\\leqq x\\leqq 1,\\ 0\\leqq y\\leqq 1,\\ 0\\leqq z\\leqq 1$ \r\n\r $ x^2 \\plus{} y^2 \\plus{} z^2 \\minus{} 2xy \\minus{} 1\\geqq 0$ Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3} - \\left(1 + \\frac{\\sqrt{3}}{4}\\right)"
} | {
"split": "train",
"index": 35969,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\frac{7}{3} + \\frac{11}{5} + \\frac{19}{9} + \\frac{37}{17} - 8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{628}{765}"
} | {
"split": "train",
"index": 35970,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $5.47 + 2.58 + 1.95$ and write the sum as a decimal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10.00"
} | {
"split": "train",
"index": 35971,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can we enter numbers from the set $\\{1,2,3,4\\}$ into a $4 \\times 4$ array so that all of the following conditions hold? (a) Each row contains all four numbers. (b) Each column contains all four numbers. (c) Each \"quadrant\" contains all four numbers. (The q... | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 35972,
"solution": "Call a filled $4 \\times 4$ array satisfying the given conditions cool. There are 4 ! possibilities for the first row; WLOG, let it be 1234. Since each quadrant has to contain all four numbers, we have exactly four possibilities for the second row, namely: (i) 3412... |
deepscaler | [
{
"role": "user",
"content": "Rational Man and Irrational Man both buy new cars, and they decide to drive around two racetracks from time $t = 0$ to $t = \\infty.$ Rational Man drives along the path parameterized by\n\\begin{align*}\nx &= \\cos t, \\\\\ny &= \\sin t,\n\\end{align*}and Irrational Man drives... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{33} - 3}{3}"
} | {
"split": "train",
"index": 35973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a clock, there are two instants between $12$ noon and $1 \\,\\mathrm{PM}$ , when the hour hand and the minute hannd are at right angles. The difference *in minutes* between these two instants is written as $a + \\dfrac{b}{c}$ , where $a, b, c$ are positive integers, wi... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 35974,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the values of the real number \\( a \\) such that all the roots of the polynomial in the variable \\( x \\),\n\n\\[ x^{3}-2x^{2}-25x+a \\]\n\nare integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-50"
} | {
"split": "train",
"index": 35975,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle \\(ABC\\), the legs \\(AB\\) and \\(AC\\) measure 4 and 3 respectively. Point \\(D\\) bisects the hypotenuse \\(BC\\). Find the distance between the centers of the incircles of triangles \\(ADC\\) and \\(ABD\\). Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "\\frac{5 \\sqrt{13}}{12}"
} | {
"split": "train",
"index": 35976,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many values of $a$ is it true that the line $y = x + a$ passes through the vertex of the parabola $y = x^2 + a^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 35977,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $a$ such that there exists a set $X$ of $6$ integers satisfying the following conditions: for every $k=1,2,\\ldots ,36$ there exist $x,y\\in X$ such that $ax+y-k$ is divisible by $37$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "6, 31"
} | {
"split": "train",
"index": 35978,
"solution": "\nFind all positive integers \\( a \\) such that there exists a set \\( X \\) of \\( 6 \\) integers satisfying the following conditions: for every \\( k = 1, 2, \\ldots, 36 \\), there exist \\( x, y \\in X \\) such that \\( ax + y - k \\) is divisible by \\( 37 \\)... |
deepscaler | [
{
"role": "user",
"content": "In a village where Glafira lives, there is a small pond that is filled by springs at the bottom. Curious Glafira found out that a herd of 17 cows completely drank this pond in 3 days. After some time, the springs filled the pond again, and then 2 cows drank it in 30 days. How m... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 35979,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given three points in space \\\\(A(0,2,3)\\\\), \\\\(B(-2,1,6)\\\\), and \\\\(C(1,-1,5)\\\\):\n\\\\((1)\\\\) Find \\\\(\\cos < \\overrightarrow{AB}, \\overrightarrow{AC} >\\\\).\n\\\\((2)\\\\) Find the area of the parallelogram with sides \\\\(AB\\\\) and \\\\(AC\\\\). Let's th... | math | {
"style": "rule",
"ground_truth": "7\\sqrt{3}"
} | {
"split": "train",
"index": 35980,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The set of points with spherical coordinates of the form\n\\[(\\rho, \\theta, \\phi) = \\left( 1, \\theta, \\frac{\\pi}{6} \\right)\\]forms a circle. Find the radius of this circle. 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": 35981,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ that satisfy $|\\overrightarrow{a}| = |\\overrightarrow{b}| = 1$ and $|3\\overrightarrow{a} - 2\\overrightarrow{b}| = \\sqrt{7}$,\n\n(I) Find the magnitude of the angle between $\\overrightarrow{a}$ and $\\overrighta... | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"split": "train",
"index": 35982,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the numbers from the set $\\{1, 2, 3, \\ldots, 100\\}$ have a perfect square factor other than one? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 35983,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cone has a volume of $12\\pi$ cubic centimeters. The height of the cone is 4 cm. How many centimeters is the circumference of the base of the cone, in terms of $\\pi$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\pi"
} | {
"split": "train",
"index": 35984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share costs equally, find the difference between the amount of money Tom gave Sammy and the amount of money Doro... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 35985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An infinite geometric series has a first term of $12$ and a second term of $4.$ A second infinite geometric series has the same first term of $12,$ a second term of $4+n,$ and a sum of four times that of the first series. Find the value of $n.$ Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 35986,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain TV station randomly selected $100$ viewers to evaluate a TV program in order to understand the evaluation of the same TV program by viewers of different genders. It is known that the ratio of the number of \"male\" to \"female\" viewers selected is $9:11$. The evaluat... | math | {
"style": "rule",
"ground_truth": "212.5"
} | {
"split": "train",
"index": 35987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(xy) = \\frac{f(x)}{y}\\]for all positive real numbers $x$ and $y.$ If $f(30) = 20,$ find $f(40).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 35988,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 2019 students in a school, and some of these students are members of different student clubs. Each student club has an advisory board consisting of 12 students who are members of that particular club. An {\\em advisory meeting} (for a particular club) can be realized ... | math | {
"style": "rule",
"ground_truth": "\\binom{2003}{11}"
} | {
"split": "train",
"index": 35989,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a whole number $n$ is not prime, then the whole number $n-2$ is not prime. A value of $n$ which shows this statement to be false is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 35990,
"solution": "To disprove the statement \"If a whole number $n$ is not prime, then the whole number $n-2$ is not prime,\" we need to find a counterexample where $n$ is not prime but $n-2$ is prime.\n\n1. **Check each option:**\n - **Option A: $n = 9$**\n - $9$ is not prime... |
deepscaler | [
{
"role": "user",
"content": "Given externally tangent circles with centers at points $A$ and $B$ and radii of lengths $6$ and $4$, respectively, a line externally tangent to both circles intersects ray $AB$ at point $C$. Calculate $BC$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 35991,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular pyramid V-ABCD with a base edge length of 4 and a lateral edge length of $\\sqrt{13}$, its surface area is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 35992,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)$ with the domain $[1, +\\infty)$, and $f(x) = \\begin{cases} 1-|2x-3|, & 1\\leq x<2 \\\\ \\frac{1}{2}f\\left(\\frac{1}{2}x\\right), & x\\geq 2 \\end{cases}$, then the number of zeros of the function $y=2xf(x)-3$ in the interval $(1, 2017)$ is \\_\\_\\_\... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 35993,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $D$ lies on side $\\overline{BC}$ of $\\triangle ABC$ so that $\\overline{AD}$ bisects $\\angle BAC.$ The perpendicular bisector of $\\overline{AD}$ intersects the bisectors of $\\angle ABC$ and $\\angle ACB$ in points $E$ and $F,$ respectively. Given that $AB=4,BC=5,$ an... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 35994,
"solution": "Points are defined as shown. It is pretty easy to show that $\\triangle AFE \\sim \\triangle AGH$ by spiral similarity at $A$ by some short angle chasing. Now, note that $AD$ is the altitude of $\\triangle AFE$, as the altitude of $AGH$. We need to compare these al... |
deepscaler | [
{
"role": "user",
"content": "A particle is located on the coordinate plane at $(5,0)$. Define a ''move'' for the particle as a counterclockwise rotation of $\\frac{\\pi}{4}$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction. Find the particle's position after $... | math | {
"style": "rule",
"ground_truth": "(-5 \\sqrt{2}, 5 + 5 \\sqrt{2})"
} | {
"split": "train",
"index": 35995,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A, B, C, D$ lie on a circle in that order such that $\\frac{A B}{B C}=\\frac{D A}{C D}$. If $A C=3$ and $B D=B C=4$, find $A D$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 35996,
"solution": "By Ptolemy's theorem, we have $A B \\cdot C D+B C \\cdot D A=A C \\cdot B D=3 \\cdot 4=12$. Since the condition implies $A B \\cdot C D=B C \\cdot D A$, we have $D A=\\frac{6}{B C}=\\frac{3}{2}$."
} |
deepscaler | [
{
"role": "user",
"content": "The expression $x^2 + 17x + 70$ can be rewritten as $(x + a)(x + b)$, and the expression $x^2 - 18x + 80$ written as $(x - b)(x - c)$, where a, b, and c are integers. Calculate the value of $a + b + c$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 35997,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence \\(\\{a_n\\}\\) with the sum of the first \\(n\\) terms \\(S_n\\) such that \\(S_n = 2^n + r\\) (where \\(r\\) is a constant), let \\(b_n = 2(1 + \\log_2 a_n)\\) for \\(n \\in \\mathbb{N}^*\\).\n\n1. Find the sum of the first \\(n\\) terms of the sequ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4} \\sqrt{2}"
} | {
"split": "train",
"index": 35998,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) The definite integral $\\int_{-1}^{1}(x^{2}+\\sin x)dx=$ ______.\n\n(2) There are 2 red balls, 1 white ball, and 1 blue ball in a box. The probability of drawing two balls with at least one red ball is ______.\n\n(3) Given the function $f(x)=\\begin{cases}1-\\log_{a}(x+2), ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 35999,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x (x+y) = 27$ and $y (x+y) = 54$, what is the value of $(x+y)^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 36000,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If\n\\[(1 + \\tan 0^\\circ)(1 + \\tan 1^\\circ)(1 + \\tan 2^\\circ) \\dotsm (1 + \\tan 30^\\circ) = 2^m,\\]\nfind the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 36001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(A\\) and \\(G\\) be two opposite vertices of a cube with unit edge length. What is the distance between the plane determined by the vertices adjacent to \\(A\\), denoted as \\(S_{A}\\), and the plane determined by the vertices adjacent to \\(G\\), denoted as \\(S_{G}\\)?... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 36002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $19^2-17^2+15^2-13^2+11^2-9^2+7^2-5^2+3^2-1^2?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 36003,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\frac{x^2+y^2}{x^2-y^2} + \\frac{x^2-y^2}{x^2+y^2} = k$ . Compute the following expression in terms of $k$ : \\[E(x,y) = \\frac{x^8 + y^8}{x^8-y^8} - \\frac{ x^8-y^8}{x^8+y^8}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{(k^2 - 4)^2}{4k(k^2 + 4)}"
} | {
"split": "train",
"index": 36004,
"solution": "To start, we add the two fractions and simplify. \\begin{align*} k &= \\frac{(x^2+y^2)^2 + (x^2-y^2)^2}{x^4-y^4} \\\\ &= \\frac{2x^4 + 2y^4}{x^4 - y^4}. \\end{align*} Dividing both sides by two yields \\[\\frac{k}{2} = \\frac{x^4 + y^4}{x^4 - y^4}.\\] That means \\... |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of\n\\[\\frac{2x + 3y + 4}{\\sqrt{x^2 + y^2 + 4}}\\]\nover all real numbers $x$ and $y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{29}"
} | {
"split": "train",
"index": 36005,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x = \\frac{3}{4}$ is a solution to the equation $108x^2 + 61 = 145x - 7,$ what is the other value of $x$ that solves the equation? 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{68}{81}"
} | {
"split": "train",
"index": 36006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ratio of the distance saved by taking a shortcut along the diagonal of a rectangular field to the longer side of the field is $\\frac{1}{2}$, determine the ratio of the shorter side to the longer side of the rectangle. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 36007,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a whole number, $M$, such that $\\frac{M}{5}$ is strictly between 9.5 and 10.5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 36008,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let a three-digit number \\( n = \\overline{a b c} \\). If the digits \\( a, b, c \\) can form an isosceles (including equilateral) triangle, calculate how many such three-digit numbers \\( n \\) are there. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 36009,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add $452_8$ and $167_8$ in base $8$, then subtract $53_8$ from the result. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "570_8"
} | {
"split": "train",
"index": 36010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x) = x^{3} + 3xf''(2)$, then $f(2) = \\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-28"
} | {
"split": "train",
"index": 36011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number $A$ that satisfies the following conditions:\n\na) Its notation ends with the digit 6;\n\nb) By moving the digit 6 from the end of the number to its beginning, the number increases fourfold. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "153846"
} | {
"split": "train",
"index": 36012,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A Martian traffic light consists of six identical bulbs arranged in two horizontal rows (one below the other) with three bulbs in each row. A rover driver in foggy conditions can distinguish the number and relative positions of the lit bulbs on the traffic light (for example, i... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 36013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be a real number such that $\\sin^{10}x+\\cos^{10} x = \\tfrac{11}{36}$. Then $\\sin^{12}x+\\cos^{12} x = \\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 36014,
"solution": "We can substitute $y = \\sin^2{x}$. Since we know that $\\cos^2{x}=1-\\sin^2{x}$, we can do some simplification.\n This yields $y^5+(1-y)^5=\\frac{11}{36}$. From this, we can substitute again to get some cancellation through binomials. If we let $z=\\frac{1}{2}-y$,... |
deepscaler | [
{
"role": "user",
"content": "There are 8 Olympic volunteers, among them volunteers $A_{1}$, $A_{2}$, $A_{3}$ are proficient in Japanese, $B_{1}$, $B_{2}$, $B_{3}$ are proficient in Russian, and $C_{1}$, $C_{2}$ are proficient in Korean. One volunteer proficient in Japanese, Russian, and Korean is to be sel... | math | {
"style": "rule",
"ground_truth": "\\dfrac {5}{6}"
} | {
"split": "train",
"index": 36015,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all the roots of $\\left(x^{2}+3 x+2\\right)\\left(x^{2}-7 x+12\\right)\\left(x^{2}-2 x-1\\right)+24=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0, 2, 1 \\pm \\sqrt{6}, 1 \\pm 2 \\sqrt{2}"
} | {
"split": "train",
"index": 36016,
"solution": "We re-factor as $(x+1)(x-3)(x+2)(x-4)\\left(x^{2}-2 x-1\\right)+24$, or $\\left(x^{2}-2 x-3\\right)\\left(x^{2}-2 x-8\\right)\\left(x^{2}-2 x-1\\right)+24$, and this becomes $(y-4)(y-9)(y-2)+24$ where $y=(x-1)^{2}$. Now, $(y-4)(y-9)(y-2)+24=(y-8)(y-6)(y-1)$, so $y$... |
deepscaler | [
{
"role": "user",
"content": "What is the least integer a greater than $14$ so that the triangle with side lengths $a - 1$ , $a$ , and $a + 1$ has integer area? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 36017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer $n\\geq 4$ is called *interesting* if there exists a complex number $z$ such that $|z|=1$ and \\[1+z+z^2+z^{n-1}+z^n=0.\\] Find how many interesting numbers are smaller than $2022.$ Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "404"
} | {
"split": "train",
"index": 36018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), the sides opposite to angles \\( A \\), \\( B \\), and \\( C \\) are \\( a \\), \\( b \\), and \\( c \\) respectively. If the angles \\( A \\), \\( B \\), and \\( C \\) form a geometric progression, and \\( b^{2} - a^{2} = ac \\), then the ra... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{7}"
} | {
"split": "train",
"index": 36019,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $(2x-1)^{5}=a\\_{0}+a\\_{1}x+a\\_{2}x^{2}+…+a\\_{5}x^{5}$, find the values of:\n\n(1) $a\\_{0}+a\\_{1}+a\\_{2}+a\\_{3}+a\\_{4}$,\n\n(2) $|a\\_{0}|+|a\\_{1}|+|a\\_{2}|+|a\\_{3}|+|a\\_{4}|+|a\\_{5}|$,\n\n(3) $a\\_{1}+a\\_{3}+a\\_{5}$,\n\n(4) $(a\\_{0}+a\\_{2}+a\\_{4})^{2}-(... | math | {
"style": "rule",
"ground_truth": "-243"
} | {
"split": "train",
"index": 36020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{55}"
} | {
"split": "train",
"index": 36021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first term of a given sequence is 2, and each successive term is the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 10000? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16384"
} | {
"split": "train",
"index": 36022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfying $x^{2}+y^{2}-4x-2y-4=0$, calculate the maximum value of $x-y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1+3\\sqrt{2}"
} | {
"split": "train",
"index": 36023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The variables \\(a, b, c, d, e\\), and \\(f\\) represent the numbers 4, 12, 15, 27, 31, and 39 in some order. Suppose that\n\n\\[\n\\begin{aligned}\n& a + b = c, \\\\\n& b + c = d, \\\\\n& c + e = f,\n\\end{aligned}\n\\]\n\nwhat is the value of \\(a + c + f\\)? Let's think step... | math | {
"style": "rule",
"ground_truth": "73"
} | {
"split": "train",
"index": 36024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $M$ be the number of $8$-digit positive integers such that the digits are in both increasing order and even. Determine the remainder obtained when $M$ is divided by $1000$. (Repeated digits are allowed.) Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 36025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the total number of cards in a stack where cards are numbered consecutively from 1 through $2n$ and rearranged such that, after a similar process of splitting into two piles and restacking alternately (starting with pile B), card number 252 retains its original position. L... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 36026,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by $5$ minutes over the preceding day... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 36027,
"solution": "1. **Understanding the problem**: Linda travels for one hour each day at a speed such that the time to travel one mile is an integer number of minutes. Each subsequent day, the time to travel one mile increases by 5 minutes. The distance she travels each day is als... |
deepscaler | [
{
"role": "user",
"content": "There is a strip with a length of 100, and each cell of the strip contains a chip. You can swap any two adjacent chips for 1 ruble, or you can swap any two chips that have exactly three chips between them for free. What is the minimum number of rubles needed to rearrange the ch... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 36028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a square array of 25 dots arranged in a 5x5 grid, what is the probability that five randomly chosen dots will be collinear? 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{2}{8855}"
} | {
"split": "train",
"index": 36029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{D}$ be the $2 \\times 2$ matrix corresponding to the dilation, centered at the origin, with scale factor 7. Find $\\det \\mathbf{D}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 36030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cyclic pentagon \\( A B C D E \\) has a right angle \\( \\angle A B C = 90^\\circ \\) and side lengths \\( A B = 15 \\) and \\( B C = 20 \\). Supposing that \\( A B = D E = E A \\), find \\( C D \\). Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 36031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the expression $\\left(100 - \\left(5000 - 500\\right)\\right) \\times \\left(5000 - \\left(500 - 100\\right)\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-20240000"
} | {
"split": "train",
"index": 36032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $M$ be the number of positive integers that are less than or equal to $2048$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $M$ is divided by $1000$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 36033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the Fibonacci numbers by $F_{0}=0, F_{1}=1, F_{n}=F_{n-1}+F_{n-2}$ for $n \\geq 2$. For how many $n, 0 \\leq n \\leq 100$, is $F_{n}$ a multiple of 13? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 36034,
"solution": "The sequence of remainders modulo 13 begins $0,1,1,2,3,5,8,0$, and then we have $F_{n+7} \\equiv 8 F_{n}$ modulo 13 by a straightforward induction. In particular, $F_{n}$ is a multiple of 13 if and only if $7 \\mid n$, so there are 15 such $n$."
} |
deepscaler | [
{
"role": "user",
"content": "What integer is closest to the value of $\\sqrt[3]{6^3+8^3}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 36035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $M$ be the greatest five-digit number whose digits have a product of $180$. Find the sum of the digits of $M$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 36036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Grandma gave Bryce and Carter some raisins. Bryce received 6 more raisins than Carter, and Carter received half the number of raisins Bryce received. How many raisins did Bryce receive? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 36037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product $(8)(888\\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "991"
} | {
"split": "train",
"index": 36038,
"solution": "\n1. **Understanding the Problem:**\n We need to find the value of $k$ such that the product of $8$ and a number consisting of $k$ eights, i.e., $(8)(888\\ldots8)$, results in a number whose digits sum to $1000$.\n\n2. **Exploring the Pattern:**\n Let's examine... |
deepscaler | [
{
"role": "user",
"content": "The curve $C$ is given by the equation $xy=1$. The curve $C'$ is the reflection of $C$ over the line $y=2x$ and can be written in the form $12x^2+bxy+cy^2+d=0$. Find the value of $bc$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 36039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $i$ is the imaginary unit, $a\\in\\mathbb{R}$, if $\\frac{1-i}{a+i}$ is a pure imaginary number, calculate the modulus of the complex number $z=(2a+1)+ \\sqrt{2}i$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{11}"
} | {
"split": "train",
"index": 36040,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x)=\\dfrac{5x+1}{x-1}$, find the value of $f(7)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 36041,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(ABCD\\) be a square of side length 1. \\(P\\) and \\(Q\\) are two points on the plane such that \\(Q\\) is the circumcentre of \\(\\triangle BPC\\) and \\(D\\) is the circumcentre of \\(\\triangle PQA\\). Find the largest possible value of \\(PQ^2\\). Express the answer ... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{3}"
} | {
"split": "train",
"index": 36042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Place each of the digits 6, 7, 8 and 9 in exactly one square to make the smallest possible product. What is this product? [asy]draw((0,.5)--(10,.5),linewidth(1));\ndraw((4,1)--(6,1)--(6,3)--(4,3)--(4,1),linewidth(1));\ndraw((7,1)--(9,1)--(9,3)--(7,3)--(7,1),linewidth(1));\ndraw... | math | {
"style": "rule",
"ground_truth": "5372"
} | {
"split": "train",
"index": 36043,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set $M$ consisting of all functions $f(x)$ that satisfy the property: there exist real numbers $a$ and $k$ ($k \\neq 0$) such that for all $x$ in the domain of $f$, $f(a+x) = kf(a-x)$. The pair $(a,k)$ is referred to as the \"companion pair\" of the function $f(x)$.\n... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 36044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the plane vectors $\\boldsymbol{\\alpha}$ and $\\boldsymbol{\\beta}$ satisfy $|\\boldsymbol{\\alpha} + 2\\boldsymbol{\\beta}| = 3$ and $|2\\boldsymbol{\\alpha} + 3\\boldsymbol{\\beta}| = 4$, find the minimum value of $\\boldsymbol{\\alpha} \\cdot \\boldsymbol{\\beta}... | math | {
"style": "rule",
"ground_truth": "-170"
} | {
"split": "train",
"index": 36045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Amelia has a coin that lands heads with probability $\\frac{1}{3}$, and Blaine has a coin that lands on heads with probability $\\frac{2}{5}$. Amelia and Blaine alternately toss their coins until someone gets a head; the first one to get a head wins. All coin tosses are indepen... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 36046,
"solution": "To find the probability that Amelia wins, we consider the following cases:\n\n1. **Amelia wins on her first turn.**\n2. **Amelia wins on her second turn.**\n3. **Amelia wins on her third turn.**\n4. **And so on.**\n\n#### Case Analysis:\n- **Case 1:** Amelia wins o... |
deepscaler | [
{
"role": "user",
"content": "Distribute 5 students into dormitories A, B, and C, with each dormitory having at least 1 and at most 2 students. Among these, the number of different ways to distribute them without student A going to dormitory A is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 36047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the digits 3, 4, 7, and 8, form two two-digit numbers (each digit can only be used once and must be used) such that their product is maximized. What is the maximum product? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6142"
} | {
"split": "train",
"index": 36048,
"solution": ""
} |
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