data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Find 100 times the area of a regular dodecagon inscribed in a unit circle. Round your answer to the nearest integer if necessary.\n[asy]\ndefaultpen(linewidth(0.7)); real theta = 17; pen dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1);\ndraw(unitcircle,dg);... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 32049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\log_{a} \\frac{2-x}{b+x} (0 < a < 1)$ is an odd function, when $x \\in (-2,2a)$, the range of the function $f(x)$ is $(-\\infty,1)$, then the sum of the real numbers $a+b=$ __________. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}+1"
} | {
"split": "train",
"index": 32050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has $AB=2 \\cdot AC$. Let $D$ and $E$ be on $\\overline{AB}$ and $\\overline{BC}$, respectively, such that $\\angle BAE = \\angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose that $\\triangle CFE$ is equilateral. What is $\\angle ACB$?... | math | {
"style": "rule",
"ground_truth": "90^{\\circ}"
} | {
"split": "train",
"index": 32051,
"solution": "1. **Given Information and Setup**: \n - Triangle $ABC$ with $AB = 2 \\cdot AC$.\n - Points $D$ and $E$ are on $\\overline{AB}$ and $\\overline{BC}$ respectively, such that $\\angle BAE = \\angle ACD = x$.\n - $F$ is the intersection of segments $AE$ and $CD$... |
deepscaler | [
{
"role": "user",
"content": "For what value of $x$ will $\\frac{3+x}{5+x}$ and $\\frac{1+x}{2+x}$ be equal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 32052,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The isosceles triangle and the square shown here have the same area in square units. What is the height of the triangle, $h$, in terms of the side length of the square, $s$?\n\n[asy]\ndraw((0,0)--(0,10)--(10,10)--(10,0)--cycle);\nfill((0,0)--(17,5)--(0,10)--cycle,white);\ndraw(... | math | {
"style": "rule",
"ground_truth": "2s"
} | {
"split": "train",
"index": 32053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pyramid with a triangular base has edges of unit length, and the angles between its edges are \\(60^{\\circ}, 90^{\\circ},\\) and \\(120^{\\circ}\\). What is the volume of the pyramid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{12}"
} | {
"split": "train",
"index": 32054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\sum_{j = 0}^\\infty \\sum_{k = 0}^\\infty 2^{-3k - j - (k + j)^2}.\\] 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": 32055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Equilateral triangle $T$ is inscribed in circle $A$, which has radius $10$. Circle $B$ with radius $3$ is internally tangent to circle $A$ at one vertex of $T$. Circles $C$ and $D$, both with radius $2$, are internally tangent to circle $A$ at the other two vertices of $T$. Cir... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 32056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain region, the rate of taxation is half the amount of the income in thousands: that is, $\\frac{x}{2}\\%$ tax rate for an income of $x$ thousand dollars. What income, in dollars, will yield the highest take-home pay? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "100000"
} | {
"split": "train",
"index": 32057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $C$: $y^2=2px (p > 0)$ with focus $F$ and directrix $l$. A line perpendicular to $l$ at point $A$ on the parabola $C$ at $A(4,y_0)$ intersects $l$ at $A_1$. If $\\angle A_1AF=\\frac{2\\pi}{3}$, determine the value of $p$. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 32058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\triangle+q=59$ and $(\\triangle+q)+q=106$, what is the value of $\\triangle$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 32059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere is inscribed in a cube. Given that one edge of the cube is 6 inches, how many cubic inches are in the volume of the inscribed sphere? Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36\\pi"
} | {
"split": "train",
"index": 32060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert $\\sqrt{2} e^{11 \\pi i/4}$ to rectangular form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1 + i"
} | {
"split": "train",
"index": 32061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles of radius 10 cm overlap such that each circle passes through the center of the other, as shown. How long, in cm, is the common chord (dotted segment) of the two circles? Express your answer in simplest radical form.\n\n[asy]\ndraw(Circle((0,0),10),linewidth(1));\ndr... | math | {
"style": "rule",
"ground_truth": "10\\sqrt3"
} | {
"split": "train",
"index": 32062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all real values of $x$ that satisfy $x + \\frac{45}{x-4} = -10.$\n\nEnter all the solutions, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1, \\; -5"
} | {
"split": "train",
"index": 32063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first and twentieth terms of an arithmetic sequence are 3 and 63, respectively. What is the fortieth term? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 32064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two points are randomly selected from the eight vertices of a cube, the probability that the line determined by these two points intersects each face of the cube is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"split": "train",
"index": 32065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the eccentricity $e= \\frac { \\sqrt {3}}{2}$ of an ellipse $C: \\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1 (a>b>0)$ with one of its foci at $F( \\sqrt {3} , 0)$,\n(I) Find the equation of ellipse C;\n(II) Let line $l$, passing through the origin O and not perpendicula... | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}"
} | {
"split": "train",
"index": 32066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the average of all the integer values of $M$ such that $\\frac{M}{42}$ is strictly between $\\frac{5}{14}$ and $\\frac{1}{6}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 32067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form $\\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "79"
} | {
"split": "train",
"index": 32068,
"solution": "Recast the problem entirely as a block-walking problem. Call the respective dice $a, b, c, d$. In the diagram below, the lowest $y$-coordinate at each of $a$, $b$, $c$, and $d$ corresponds to the value of the roll.\n The red path corresponds to the sequence of roll... |
deepscaler | [
{
"role": "user",
"content": "What is the value of the expression $(25 + 8)^2 - (8^2 +25^2)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 32069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\sqrt{3}\\sin 2x - 2\\cos^2x + 1$. \n(1) Find the range of the function $f(x)$ in the interval $\\left[-\\frac{\\pi}{12}, \\frac{\\pi}{2}\\right)$. \n(2) Let $\\alpha, \\beta \\in \\left(0, \\frac{\\pi}{2}\\right)$, $f\\left(\\frac{1}{2}\\alpha + \... | math | {
"style": "rule",
"ground_truth": "-\\frac{33}{65}"
} | {
"split": "train",
"index": 32070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer $n$ is infallible if it is possible to select $n$ vertices of a regular 100-gon so that they form a convex, non-self-intersecting $n$-gon having all equal angles. Find the sum of all infallible integers $n$ between 3 and 100, inclusive. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "262"
} | {
"split": "train",
"index": 32071,
"solution": "Suppose $A_{1} A_{2} \\ldots A_{n}$ is an equiangular $n$-gon formed from the vertices of a regular 100-gon. Note that the angle $\\angle A_{1} A_{2} A_{3}$ is determined only by the number of vertices of the 100-gon between $A_{1}$ and $A_{3}$. Thus in order for $... |
deepscaler | [
{
"role": "user",
"content": "What number must we add to $3-5i$ to get $2+7i$? Here, $i^2 = -1.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1+12i"
} | {
"split": "train",
"index": 32072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mean of three numbers is $10$ more than the least of the numbers and $15$ less than the greatest. The median of the three numbers is $5$. What is their sum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 32073,
"solution": "1. Let the three numbers be $x$, $y$, and $z$. Arrange them in ascending order, so $x \\leq y \\leq z$. Given that the median of the three numbers is $5$, we have $y = 5$.\n\n2. The problem states that the mean of the numbers is $10$ more than the least of the numb... |
deepscaler | [
{
"role": "user",
"content": "On a board the following six vectors are written: \\((1,0,0), \\quad(-1,0,0), \\quad(0,1,0), \\quad(0,-1,0), \\quad(0,0,1), \\quad(0,0,-1)\\). Given two vectors \\(v\\) and \\(w\\) on the board, a move consists of erasing \\(v\\) and \\(w\\) and replacing them with \\(\\frac{1}... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{3}"
} | {
"split": "train",
"index": 32074,
"solution": "For a construction, note that one can change \\((1,0,0),(-1,0,0) \\rightarrow(\\sqrt{2}, 0,0),(0,0,0) \\rightarrow(1,0,0),(1,0,0)\\) and similarly for \\((0,1,0),(0,-1,0)\\) and \\((0,0,1),(0,0,-1)\\). Then \\(u=(2,2,2)\\). For the bound, argue as follows: let the ... |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be positive whole numbers such that $\\frac{4.5}{11}<\\frac{a}{b}<\\frac{5}{11}$ . \nFind the fraction $\\frac{a}{b}$ for which the sum $a+b$ is as small as possible. \nJustify your answer Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "3/7"
} | {
"split": "train",
"index": 32075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are eight athletes participating in the men's 100m final. It is known that the stadium has eight tracks numbered from inside to outside as 1, 2, 3, 4, 5, 6, 7, 8. If the track numbers of the specified three athletes must be three consecutive numbers, calculate the total n... | math | {
"style": "rule",
"ground_truth": "4320"
} | {
"split": "train",
"index": 32076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a square \\( ABCD \\). Point \\( N \\) lies on side \\( AD \\) such that \\( AN : ND = 2 : 3 \\), point \\( F \\) lies on side \\( CD \\) such that \\( DF : FC = 1 : 4 \\), and point \\( K \\) lies on side \\( AB \\) such that \\( AK : KB = 1 : 4 \\). Find the angle \\( \... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 32077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the largest natural \\( m \\), when will the product \\( m! \\cdot 2022! \\) be a factorial of a natural number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2022! - 1"
} | {
"split": "train",
"index": 32078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x) = x^4 + x^2 + 5x$, evaluate $f(5) - f(-5)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 32079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a nonnegative integer $n$, let $r_{11}(n)$ represent the remainder when $n$ is divided by $11.$ What is the $15^{\\text{th}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_{11}(7n)\\le 3~?$$ Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 32080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle XYZ$, a triangle $\\triangle MNO$ is inscribed such that vertices $M, N, O$ lie on sides $YZ, XZ, XY$, respectively. The circumcircles of $\\triangle XMO$, $\\triangle YNM$, and $\\triangle ZNO$ have centers $P_1, P_2, P_3$, respectively. Given that $XY = 26, YZ ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 32081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a moving point $M$ whose distance to the point $F(0,1)$ is equal to its distance to the line $y=-1$, the trajectory of point $M$ is denoted as $C$.\n\n(1) Find the equation of the trajectory $C$.\n\n(2) Let $P$ be a point on the line $l: x-y-2=0$. Construct two tangents $... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 32082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A curve is described parametrically by\n\\[(x,y) = (2 \\cos t - \\sin t, 4 \\sin t).\\]The graph of the curve can be expressed in the form\n\\[ax^2 + bxy + cy^2 = 1.\\]Enter the ordered triple $(a,b,c).$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{1}{4}, \\frac{1}{8}, \\frac{5}{64} \\right)"
} | {
"split": "train",
"index": 32083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ , $b$ , $c$ be positive integers with $a \\le 10$ . Suppose the parabola $y = ax^2 + bx + c$ meets the $x$ -axis at two distinct points $A$ and $B$ . Given that the length of $\\overline{AB}$ is irrational, determine, with proof, the smallest possible val... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{13}}{9}"
} | {
"split": "train",
"index": 32084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $(2-w)(2-w^2)\\cdots(2-w^{10})$ where $w=e^{2\\pi i/11}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2047"
} | {
"split": "train",
"index": 32085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\(ABCD\\) is a square with a side length of \\(8 \\, \\text{cm}\\). Point \\(E\\) is on \\(AB\\) and point \\(F\\) is on \\(DC\\) so that \\(\\triangle AEF\\) is right-angled at \\(E\\). If the area of \\(\\triangle AEF\\) is \\(30\\%\\) of the area of \\(ABCD\... | math | {
"style": "rule",
"ground_truth": "4.8"
} | {
"split": "train",
"index": 32086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with center $O$ has radius 25. Chord $\\overline{AB}$ of length 30 and chord $\\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\\frac{m}{n}$, where $m$ and $n$ ar... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 32087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The decimal expansion of $8/11$ is a repeating decimal. What is the least number of digits in a repeating block of 8/11? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 32088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be a set of ten distinct positive numbers (not necessarily integers). Determine the maximum possible number of arithmetic progressions consisting of three distinct numbers from the set $A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 32089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $n!!$ as in the original problem. Evaluate $\\sum_{i=1}^{5} \\frac{(2i-1)!!}{(2i)!!}$, and express the result as a fraction in lowest terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{437}{256}"
} | {
"split": "train",
"index": 32090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all two-digit positive integers whose squares end with the digits 25? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "495"
} | {
"split": "train",
"index": 32091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a \\) and \\( b \\) be the roots of \\( x^2 + 2000x + 1 = 0 \\) and let \\( c \\) and \\( d \\) be the roots of \\( x^2 - 2008x + 1 = 0 \\). Find the value of \\( (a+c)(b+c)(a-d)(b-d) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32064"
} | {
"split": "train",
"index": 32092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $2^x+ 2^x+ 2^x+ 2^x= 128$, what is the value of $(x + 1)(x - 1)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 32093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a 4 by 4 grid of points, a rectangle is formed by connecting four points: (1,1), (1,3), (3,3), and (3,1). What fraction of the larger square's area is inside the rectangle? Express your answer as a common fraction. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 32094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In ancient China, Confucianism required students to master six basic skills (Six Arts): rites, music, archery, charioteering, calligraphy, and mathematics. A school's traditional Chinese culture society held a \"Six Arts\" lecture activity on weekends, with six sessions in a da... | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 32095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\overset{⇀}{m}=(2,1)$, $\\overset{⇀}{n}=(\\sin θ,\\cos θ)$, where $θ∈(0, \\dfrac{π}{2})$ is the inclination angle of line $l$ passing through point $A(1,4)$, if $\\overset{⇀}{m}· \\overset{⇀}{n}$ is at its maximum when line $l$ is tangent to the circle $(x+1)^{2}+(... | math | {
"style": "rule",
"ground_truth": "\\dfrac{2 \\sqrt{5}}{5}"
} | {
"split": "train",
"index": 32096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "China was the first country in the world to use negative numbers. Li Heng, in the book \"Fa Jing\" written during the Warring States period, already used negative numbers. If the year 500 BC is written as $-500$ years, then the year 2024 AD should be written as ______ years. Le... | math | {
"style": "rule",
"ground_truth": "+2024"
} | {
"split": "train",
"index": 32097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of zeros in the quotient $Q = R_{30}/R_6$, where $R_k$ is a number consisting of $k$ repeated digits of 1 in base-ten. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 32098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular garden needs to be enclosed on three sides using a 70-meter rock wall as one of the sides. Fence posts are placed every 10 meters along the fence, including at the ends where the fence meets the rock wall. If the area of the garden is 2100 square meters, calculate... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 32099,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the numbers 1, 2, 3, 4, 5, and 6, two distinct numbers are taken out each time and denoted as $a$ and $b$. The total number of distinct values obtained for $3^{\\frac{a}{b}}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 32100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 3 small balls of each of the red, yellow, and blue colors, with the same size. Each of the 3 balls of each color is marked with numbers 1, 2, and 3. If 3 balls are randomly drawn, find the probability that neither their colors nor their numbers are the same. Let's thi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{14}"
} | {
"split": "train",
"index": 32101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three friends, Rowan, Sara, and Tim, are playing a monetary game. Each starts with $3. A bell rings every 20 seconds, and with each ring, any player with money chooses one of the other two players independently at random and gives them $1. The game continues for 2020 rounds. Wh... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 32102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nLet \\( P(x) = x^{4} + a x^{3} + b x^{2} + c x + d \\), where \\( a, b, c, \\) and \\( d \\) are real coefficients. Given that\n\n\\[ P(1) = 7, \\quad P(2) = 52, \\quad P(3) = 97, \\]\n\nfind the value of \\(\\frac{P(9) + P(-5)}{4}\\). Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "1202"
} | {
"split": "train",
"index": 32103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a class of 38 students, we need to randomly select 5 people to participate in a survey. The number of possible selections where student A is chosen but student B is not is ______. (Express the answer numerically) Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "58905"
} | {
"split": "train",
"index": 32104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle is tangent to the extensions of two sides \\(AB\\) and \\(AD\\) of a square \\(ABCD\\), and the point of tangency cuts off a segment of length \\(6 - 2\\sqrt{5}\\) cm from vertex \\(A\\). Two tangents are drawn to this circle from point \\(C\\). Find the side length of... | math | {
"style": "rule",
"ground_truth": "(\\sqrt{5} - 1)(2\\sqrt{2} - \\sqrt{5} + 1)"
} | {
"split": "train",
"index": 32105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A, B$ and $C$ on a circle of radius $r$ are situated so that $AB=AC, AB>r$, and the length of minor arc $BC$ is $r$. If angles are measured in radians, then $AB/BC=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}\\csc{\\frac{1}{4}}"
} | {
"split": "train",
"index": 32106,
"solution": "1. **Identify the central angle corresponding to arc $BC$**: Given that the length of minor arc $BC$ is $r$, and knowing that the arc length of a circle is given by $r\\theta$ where $\\theta$ is the central angle in radians, we have:\n \\[\n r\\theta = r \\impl... |
deepscaler | [
{
"role": "user",
"content": "Jane and her brother each spin this spinner once. The spinner has five congruent sectors. If the non-negative difference of their numbers is less than 3, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins? Express your answer as a common fraction.\n\... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{25}"
} | {
"split": "train",
"index": 32107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of a regular octagonal prism is 8 cubic meters, and its height is 2.2 meters. Find the lateral surface area of the prism. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16 \\sqrt{2.2 (\\sqrt{2} - 1)}"
} | {
"split": "train",
"index": 32108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a plane with 100 seats, there are 100 passengers, each with an assigned seat. The first passenger ignores the assigned seat and randomly sits in one of the 100 seats. After that, each subsequent passenger either sits in their assigned seat if it is available or chooses a ran... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 32109,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A teacher gave a test to a class in which $10\\%$ of the students are juniors and $90\\%$ are seniors. The average score on the test was $84.$ The juniors all received the same score, and the average score of the seniors was $83.$ What score did each of the juniors receive on t... | math | {
"style": "rule",
"ground_truth": "93"
} | {
"split": "train",
"index": 32110,
"solution": "1. **Assume the total number of students**: Let's assume there are $n$ students in the class. According to the problem, $10\\%$ are juniors and $90\\%$ are seniors. Therefore, the number of juniors is $0.1n$ and the number of seniors is $0.9n$.\n\n2. **Calculate th... |
deepscaler | [
{
"role": "user",
"content": "Given $\\tan \\alpha=4$, calculate the value of $\\frac{1+\\cos 2\\alpha+8\\sin^2\\alpha}{\\sin 2\\alpha}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{65}{4}"
} | {
"split": "train",
"index": 32111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that x > 0, y > 0, z > 0, and x + $\\sqrt{3}$y + z = 6, find the minimum value of x³ + y² + 3z. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{37}{4}"
} | {
"split": "train",
"index": 32112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$ is an odd multiple of $7767$, find the greatest common divisor of $6a^2+49a+108$ and $2a+9$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 32113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On grid paper, a step-like right triangle was drawn with legs equal to 6 cells. Then all the grid lines inside the triangle were traced. What is the maximum number of rectangles that can be found in this drawing? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 32114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sides of triangle $DEF$ are in the ratio of $3:4:5$. Segment $EG$ is the angle bisector drawn to the shortest side, dividing it into segments $DG$ and $GE$. What is the length, in inches, of the longer subsegment of side $DE$ if the length of side $DE$ is $12$ inches? Expre... | math | {
"style": "rule",
"ground_truth": "\\frac{48}{7}"
} | {
"split": "train",
"index": 32115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest possible positive integer \\( n \\) with the following property: For all positive integers \\( x, y, \\) and \\( z \\) such that \\( x \\mid y^{3} \\), \\( y \\mid z^{3} \\), and \\( z \\mid x^{3} \\), it always holds that \\( x y z \\mid (x+y+z)^{n} \\).... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 32116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b, c$ be the three roots of $p(x)=x^{3}+x^{2}-333 x-1001$. Find $a^{3}+b^{3}+c^{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2003"
} | {
"split": "train",
"index": 32117,
"solution": "We know that $x^{3}+x^{2}-333 x-1001=(x-a)(x-b)(x-c)=x^{3}-(a+b+c) x^{2}+(a b+b c+c a) x-a b c$. Also, $(a+b+c)^{3}-3(a+b+c)(a b+b c+c a)+3 a b c=a^{3}+b^{3}+c^{3}$. Thus, $a^{3}+b^{3}+c^{3}=(-1)^{3}-3(-1)(-333)+3 \\cdot 1001=2003$."
} |
deepscaler | [
{
"role": "user",
"content": "How many degrees are in the measure of the smaller angle that is formed by the hour-hand and minute-hand of a clock when it is 5 o'clock? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150^\\circ"
} | {
"split": "train",
"index": 32118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sita and Geeta are two sisters. If Sita's age is written after Geeta's age a four digit perfect square (number) is obtained. If the same exercise is repeated after 13 years another four digit perfect square (number) will be obtained. What is the sum of the present ages of Sita ... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 32119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the polar equation of curve $C$ is $\\rho-4\\sin \\theta=0$. With the pole as the origin and the positive half-axis of the $x$-axis as the polar axis, a Cartesian coordinate system is established. Line $l$ passes through point $M(1,0)$ with an inclination angle of $\\dfra... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 32120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate\n\\[\n\\left(c^c - c (c - 1)^{c-1}\\right)^c\n\\]\nwhen \\( c = 4 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "148^4"
} | {
"split": "train",
"index": 32121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a modified design of Wei's logo, nine identical circles are placed within a square of side length 24 inches. Each circle is tangent to two sides of the square and to its adjacent circles wherever possible. How many square inches will be shaded in this new design? Let's think... | math | {
"style": "rule",
"ground_truth": "576 - 144\\pi"
} | {
"split": "train",
"index": 32122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the last two digits of the sum $$6! + 1 + 12! + 1 + 18! + 1 + \\cdots + 96! + 1.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 32123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside the cube \\(A B C D A_{1} B_{1} C_{1} D_{1}\\) there is a center \\(O\\) of a sphere with radius 10. The sphere intersects the face \\(A A_{1} D_{1} D\\) creating a circle with radius 1, the face \\(A_{1} B_{1} C_{1} D_{1}\\) creating a circle with radius 1, and the face... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 32124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sally has six red cards numbered 1 through 6 and seven blue cards numbered 2 through 8. She attempts to create a stack where the colors alternate, and the number on each red card divides evenly into the number on each neighboring blue card. Can you determine the sum of the numb... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 32125,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the set $\\{1, a, \\frac{b}{a}\\}$ equals $\\{0, a^2, a + b\\}$, find the value of $a - b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 32126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A math professor stands up in front of a room containing 100 very smart math students and says, 'Each of you has to write down an integer between 0 and 100, inclusive, to guess 'two-thirds of the average of all the responses.' Each student who guesses the highest integer that i... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 32127,
"solution": "Since the average cannot be greater than 100, no student will write down a number greater than $\\frac{2}{3} \\cdot 100$. But then the average cannot be greater than $\\frac{2}{3} \\cdot 100$, and, realizing this, each student will write down a number no greater th... |
deepscaler | [
{
"role": "user",
"content": "The solutions to the equation $(z-4)^6 = 64$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labelled $D, E,$ and $F$. What is the least possible area of triangle $DEF$? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 32128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Winnie wrote all the integers from 1 to 2017 inclusive on a board. She then erased all the integers that are a multiple of 3. Next, she reinstated all those integers that are a multiple of 6. Finally, she erased all integers then on the board which are a multiple of 27. Of the ... | math | {
"style": "rule",
"ground_truth": "373"
} | {
"split": "train",
"index": 32129,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Ming observed a faucet continuously dripping water due to damage. To investigate the waste caused by the water leakage, Xiao Ming placed a graduated cylinder under the faucet to collect water. He recorded the total amount of water in the cylinder every minute, but due to a... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 32130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a^2 = b(b + c)$. Find the value of $\\frac{B}{A}$. 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": 32131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a hyperbola $C$ that shares the same foci with the ellipse $\\frac{x^{2}}{27}+ \\frac{y^{2}}{36}=1$ and passes through the point $(\\sqrt{15},4)$.\n\n(I) Find the equation of the hyperbola $C$.\n\n(II) If $F\\_1$ and $F\\_2$ are the two foci of the hyperbola $C$, and poin... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 32132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of three different numbers is 100. The two larger numbers differ by 8 and the two smaller numbers differ by 5. What is the value of the largest number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{121}{3}"
} | {
"split": "train",
"index": 32133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\( PQR \\) is a straight line segment and \\( QS = QT \\). Also, \\( \\angle PQS = x^\\circ \\) and \\( \\angle TQR = 3x^\\circ \\). If \\( \\angle QTS = 76^\\circ \\), find the value of \\( x \\). Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 32134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer with eight positive odd integer divisors and sixteen positive even integer divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 32135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all primes $p$ and $q$ such that $3p^{q-1}+1$ divides $11^p+17^p$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(3, 3)"
} | {
"split": "train",
"index": 32136,
"solution": "\nTo solve the problem, we need to identify all pairs of primes \\( p \\) and \\( q \\) such that the expression \\( 3p^{q-1} + 1 \\) divides \\( 11^p + 17^p \\). The reference answer indicates that the only solution is the pair \\((3, 3)\\). Let's go through the p... |
deepscaler | [
{
"role": "user",
"content": "There are 4 different digits that can form 18 different four-digit numbers arranged in ascending order. The first four-digit number is a perfect square, and the second-last four-digit number is also a perfect square. What is the sum of these two numbers? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "10890"
} | {
"split": "train",
"index": 32137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During the Qingming Festival, a certain school, in order to commemorate the revolutionary martyrs, requires students to participate in the \"Qingming Sacrifice to the Martyrs\" activity by either visiting the Revolutionary Martyrs Memorial Hall or participating online. Students... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{21}"
} | {
"split": "train",
"index": 32138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"split": "train",
"index": 32139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f$ satisfies $f(x)+f(2 x+y)+5 x y=f(3 x-y)+2 x^{2}+1$ for all real numbers $x, y$. Determine the value of $f(10)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-49"
} | {
"split": "train",
"index": 32140,
"solution": "Setting $x=10$ and $y=5$ gives $f(10)+f(25)+250=f(25)+200+1$, from which we get $f(10)=-49$. Remark: By setting $y=\\frac{x}{2}$, we see that the function is $f(x)=-\\frac{1}{2} x^{2}+1$, and it can be checked that this function indeed satisfies the given equation.... |
deepscaler | [
{
"role": "user",
"content": "A bus, a truck, and a car are driving in the same direction on a straight road at constant speeds. At a certain moment, the bus is in front, the car is at the back, and the truck is exactly in the middle between the bus and the car. After 10 minutes, the car catches up with the... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 32141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\(a\\) and \\(b\\) are positive numbers such that \\(a^b=b^a\\) and \\(b=27a\\), then find the value of \\(a\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[26]{27}"
} | {
"split": "train",
"index": 32142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle{ABC}$ with side lengths $AB = 15$, $AC = 8$, and $BC = 17$, let $O$ and $I$ denote the circumcenter and incenter, respectively. A circle with center $M$ is tangent to the legs $AC$ and $BC$ and to the circumcircle of $\\triangle{ABC}$. What is the area of $\\tria... | math | {
"style": "rule",
"ground_truth": "3.4"
} | {
"split": "train",
"index": 32143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the numbers - 2, 5, 8, 11, and 14 are arranged in a specific cross-like structure, find the maximum possible sum for the numbers in either the row or the column. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 32144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hollow glass sphere with uniform wall thickness and an outer diameter of $16 \\mathrm{~cm}$ floats in water in such a way that $\\frac{3}{8}$ of its surface remains dry. What is the wall thickness, given that the specific gravity of the glass is $s = 2.523$? Let's think step ... | math | {
"style": "rule",
"ground_truth": "0.8"
} | {
"split": "train",
"index": 32145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A club has 30 members, which includes 12 females and 18 males. In how many ways can a 5-person executive committee be formed such that at least one member must be female? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "133,938"
} | {
"split": "train",
"index": 32146,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the center of the square $PQRS$? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "\\frac{32}{5}"
} | {
"split": "train",
"index": 32147,
"solution": "1. **Identify the lines and their slopes**: \n - Since $SP$ and $RQ$ are opposite sides of square $PQRS$, and points $(3,0)$ and $(5,0)$ lie on $SP$ and $RQ$ respectively, lines $SP$ and $RQ$ are parallel with some positive slope $m$.\n - Similarly, points $(7,... |
deepscaler | [
{
"role": "user",
"content": "Let $a_1 , a_2 , \\dots$ be a sequence for which $a_1=2$ , $a_2=3$, and $a_n=\\frac{a_{n-1}}{a_{n-2}}$ for each positive integer $n \\ge 3$. What is $a_{2006}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 32148,
"solution": ""
} |
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