data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In a rectangle that is sub-divided into 3 identical squares of side length 1, if \\(\\alpha^{\\circ} = \\angle ABD + \\angle ACD\\), find the value of \\(\\alpha\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 39349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z$ be a complex number. If the equation \\[x^3 + (4-i)x^2 + (2+5i)x = z\\] has two roots that form a conjugate pair, find the absolute value of the real part of $z$ .\n\n*Proposed by Michael Tang* Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "423"
} | {
"split": "train",
"index": 39350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a moving circle that passes through the fixed point $F(1,0)$ and is tangent to the fixed line $l$: $x=-1$.\n(1) Find the equation of the trajectory $C$ of the circle's center;\n(2) The midpoint of the chord $AB$ formed by the intersection of line $l$ and $C$ is $(2,1}$. $... | math | {
"style": "rule",
"ground_truth": "\\sqrt{35}"
} | {
"split": "train",
"index": 39351,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice buys three burgers and two sodas for $\\$3.20$, and Bill buys two burgers and a soda for $\\$2.00$. How many cents does a burger cost? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 39352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An arithmetic sequence consists of positive terms, with the sum of the first $n$ terms denoted by $S_n$, satisfying $2S_2 = a_2(a_2 + 1)$, and given that $a_1 = 1$, find the minimum value of $\\frac{2S_n + 13}{n}$. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac{33}{4}"
} | {
"split": "train",
"index": 39353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Jo climbs a flight of 8 stairs, and Jo can take the stairs 1, 2, 3, or 4 at a time, or a combination of these steps that does not exceed 4 with a friend, determine the number of ways Jo and the friend can climb the stairs together, including instances where either Jo... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 39354,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cone and a sphere possess the same radius, $r$. If the volume of the cone is one-third of the volume of the sphere, determine the ratio of the height of the cone to the radius $r$. 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": 39355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the polynomial \\[ p(x) = x^4 - 3x^3 - 9x^2 + 27x - 8, \\] where $x$ is a positive number such that $x^2 - 3x - 9 = 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{65 + 81\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 39356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $$\\frac {x^{2}}{a^{2}}$$ + $$\\frac {y^{2}}{b^{2}}$$ = 1 with its right focus F, a line passing through the origin O intersects the ellipse C at points A and B. If |AF| = 2, |BF| = 4, and the eccentricity of the ellipse C is $$\\frac {\\sqrt {7}}{3}$$, calcula... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 39357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the scalene triangles with natural number side lengths, a perimeter not exceeding 30, and the sum of the longest and shortest sides exactly equal to twice the third side, there are ____ distinct triangles. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 39358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the maximum and minimum values of the function $y=2\\sin \\left( \\frac{\\pi x}{6}- \\frac{\\pi}{3}\\right)$ where $(0\\leqslant x\\leqslant 9)$ is to be determined. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2-\\sqrt{3}"
} | {
"split": "train",
"index": 39359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $1$, $1$, $3$, $1$, $3$, $5$, $1$, $3$, $5$, $7$, $1$, $3$, $5$, $7$, $9$, $\\ldots$, where the first term is $1$, the next two terms are $1$, $3$, and the next three terms are $1$, $3$, $5$, and so on. Let $S_{n}$ denote the sum of the first $n$ terms of this ... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 39360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The units of length include , and the conversion rate between two adjacent units is . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 39361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a company, employees have a combined monthly salary of $10,000. A kind manager proposes to triple the salaries of those earning up to $500, and to increase the salaries of others by $1,000, resulting in a total salary of $24,000. A strict manager proposes to reduce the salar... | math | {
"style": "rule",
"ground_truth": "7000"
} | {
"split": "train",
"index": 39362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The set of points $(x,y)$ such that $|x - 3| \\le y \\le 4 - |x - 1|$ defines a region in the $xy$-plane. Compute the area of this region. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 39363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a convex quadrilateral with $AB=2$, $AD=7$, and $CD=3$ such that the bisectors of acute angles $\\angle{DAB}$ and $\\angle{ADC}$ intersect at the midpoint of $\\overline{BC}$. Find the square of the area of $ABCD$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 39364,
"solution": "2022 AIME II Q11(Hand-drawn picture)\n According to the problem, we have $AB=AB'=2$, $DC=DC'=3$, $MB=MB'$, $MC=MC'$, and $B'C'=7-2-3=2$\n Because $M$ is the midpoint of $BC$, we have $BM=MC$, so: \\[MB=MB'=MC'=MC.\\]\n Then, we can see that $\\bigtriangleup{MB'C'}$... |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a} = (\\sin x, \\cos x)$, $\\overrightarrow{b} = (\\sin x, \\sin x)$, and $f(x) = \\overrightarrow{a} \\cdot \\overrightarrow{b}$\n\n(1) If $x \\in \\left[-\\frac{\\pi}{4}, \\frac{\\pi}{4}\\right]$, find the range of the function $f(x)$.\n\n(2) L... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6} + \\sqrt{2}}{2}"
} | {
"split": "train",
"index": 39365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $B$ lies on line segment $\\overline{AC}$ with $AB=16$ and $BC=4$. Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\\triangle ABD$ and $\\triangle BCE$. Let $M$ be the midpoint of $\\overline{AE}$, and $N$ be the midpoint of $\\overline... | math | {
"style": "rule",
"ground_truth": "507"
} | {
"split": "train",
"index": 39366,
"solution": "Let $A$ be the origin, so $B=(16,0)$ and $C=(20,0).$ Using equilateral triangle properties tells us that $D=(8,8\\sqrt3)$ and $E=(18,2\\sqrt3)$ as well. Therefore, $M=(9,\\sqrt3)$ and $N=(14,4\\sqrt3).$ Applying the Shoelace Theorem to triangle $BMN$ gives\n\\[x=\\... |
deepscaler | [
{
"role": "user",
"content": "From an arbitrary tetrahedron, four smaller tetrahedra are separated by four planes passing through the midpoints of the edges emanating from each vertex. Calculate the ratio of the volume of the remaining body to the volume of the original tetrahedron. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 39367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the round table, $10$ people are sitting, some of them are knights, and the rest are liars (knights always say pride, and liars always lie) . It is clear thath I have at least one knight and at least one liar. \nWhat is the largest number of those sitting at the table can s... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 39368,
"solution": "\nTo solve this problem, we need to maximize the number of people at a round table who can truthfully say: \"Both of my neighbors are knights.\" Considering the rules:\n- Knights always tell the truth.\n- Liars always lie.\n- At least one knight and one liar are pr... |
deepscaler | [
{
"role": "user",
"content": "If $\\tan \\theta = 4,$ then find $\\tan 3 \\theta.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{52}{47}"
} | {
"split": "train",
"index": 39369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the domain of the function $f(x)$ is $(4a-3,3-2a^{2})$, where $a\\in \\mathbb{R}$, and $y=f(2x-3)$ is an even function. If $B_{n}=1\\times a^{1}+4\\times a^{2}+7\\times a^{3}+\\cdots +(3n-2)a^{n}$, then $B_{50}=$ ? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 39370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos (2x-φ)- \\sqrt {3}\\sin (2x-φ)(|φ| < \\dfrac {π}{2})$, its graph is shifted to the right by $\\dfrac {π}{12}$ units and is symmetric about the $y$-axis. Find the minimum value of $f(x)$ in the interval $\\[- \\dfrac {π}{2},0\\]$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "- \\sqrt {3}"
} | {
"split": "train",
"index": 39371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of minutes before Jack arrives at the park that Jill arrives at the park, given that they are 2 miles apart, Jill cycles at a constant speed of 12 miles per hour, and Jack jogs at a constant speed of 5 miles per hour. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 39372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a street, there are 10 lamps numbered 1, 2, 3, …, 10. Now, we need to turn off four of them, but we cannot turn off two or three adjacent lamps, nor can we turn off the two lamps at both ends. How many ways are there to turn off the lamps under these conditions? Let's think ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 39373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In \\\\(\\triangle ABC\\\\), the sides opposite to angles \\\\(A\\\\), \\\\(B\\\\), and \\\\(C\\\\) are \\\\(a\\\\), \\\\(b\\\\), and \\\\(c\\\\) respectively. Given that \\\\(a=2\\\\), \\\\(c=3\\\\), and \\\\(\\cos B= \\dfrac {1}{4}\\\\),\n\\\\((1)\\\\) find the value of \\\\(... | math | {
"style": "rule",
"ground_truth": "\\dfrac {3 \\sqrt {6}}{8}"
} | {
"split": "train",
"index": 39374,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given eight cubes with volumes \\(1, 8, 27, 64, 125, 216, 343,\\) and \\(512\\) cubic units, and each cube having its volume decreasing upwards, determine the overall external surface area of this arrangement. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "1021"
} | {
"split": "train",
"index": 39375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 3 numbers A, B, and C, such that $1001C - 2002A = 4004$, and $1001B + 3003A = 5005$. What is the average of A, B, and C? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 39376,
"solution": "1. **Isolate $C$ and $B$ in terms of $A$:**\n - From the equation $1001C - 2002A = 4004$, we can isolate $C$:\n \\[\n 1001C = 2002A + 4004 \\implies C = 2A + 4\n \\]\n - From the equation $1001B + 3003A = 5005$, we can isolate $B$:\n \\[\n 1... |
deepscaler | [
{
"role": "user",
"content": "Jennifer wants to do origami, and she has a square of side length $ 1$ . However, she would prefer to use a regular octagon for her origami, so she decides to cut the four corners of the square to get a regular octagon. Once she does so, what will be the side length of the oct... | math | {
"style": "rule",
"ground_truth": "1 - \\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 39377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ has $AB = 8$ and $BC = 13$ . Points $P_1$ and $P_2$ lie on $AB$ and $CD$ with $P_1P_2 \\parallel BC$ . Points $Q_1$ and $Q_2$ lie on $BC$ and $DA$ with $Q_1Q_2 \\parallel AB$ . Find the area of quadrilateral $P_1Q_1P_2Q_2$ . Let's think ... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 39378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Doug and Ryan are competing in the 2005 Wiffle Ball Home Run Derby. In each round, each player takes a series of swings. Each swing results in either a home run or an out, and an out ends the series. When Doug swings, the probability that he will hit a home run is $1 / 3$. When... | math | {
"style": "rule",
"ground_truth": "1/5"
} | {
"split": "train",
"index": 39379,
"solution": "Denote this probability by $p$. Doug hits more home runs if he hits a home run on his first try when Ryan does not, or if they both hit home runs on their first try and Doug hits more home runs thereafter. The probability of the first case occurring is $\\frac{1}{3... |
deepscaler | [
{
"role": "user",
"content": "If $x$ is a positive number such that \\[\\sqrt{8x}\\cdot\\sqrt{10x}\\cdot\\sqrt{3x}\\cdot\\sqrt{15x}=15,\\]find all possible values for $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 39380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A burger at Ricky C's now weighs 180 grams, of which 45 grams are filler. What percent of the burger is not filler? Additionally, what percent of the burger is filler? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25\\%"
} | {
"split": "train",
"index": 39381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the three-dimensional Cartesian coordinate system, the equation of the plane passing through $P(x_{0}, y_{0}, z_{0})$ with normal vector $\\overrightarrow{m}=(a, b, c)$ is $a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0$, and the equation of the line passing through $P(x_{0}, y_{0}, z_{... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{55}}{11}"
} | {
"split": "train",
"index": 39382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given non-negative real numbers $a$, $b$, $c$ satisfy $\\frac{a-1}{2}=\\frac{b-2}{3}=\\frac{3-c}{4}$, let the maximum value of $S=a+2b+c$ be $m$, and the minimum value be $n$. Then the value of $\\frac{n}{m}$ is ______. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{11}"
} | {
"split": "train",
"index": 39383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is \\( 110\\% \\) of 500? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "550"
} | {
"split": "train",
"index": 39384,
"solution": "Solution 1: \\( 10\\% \\) of 500 is \\( \\frac{1}{10} \\) or 0.1 of 500, which equals 50. \\( 100\\% \\) of 500 is 500. Thus, \\( 110\\% \\) of 500 equals \\( 500 + 50 \\), which equals 550. Solution 2: \\( 110\\% \\) of 500 is equal to \\( \\frac{110}{100} \\times... |
deepscaler | [
{
"role": "user",
"content": "When plotted in the standard rectangular coordinate system, trapezoid $EFGH$ has vertices $E(2, -3)$, $F(2, 2)$, $G(7, 8)$, and $H(7, 3)$. What is the area of trapezoid $EFGH$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 39385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given: In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, and it is known that $\\frac {\\cos A-2\\cos C}{\\cos B}= \\frac {2c-a}{b}$. \n$(1)$ Find the value of $\\frac {\\sin C}{\\sin A}$; \n$(2)$ If $\\cos B= \\frac {1}{4... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {15}}{4}"
} | {
"split": "train",
"index": 39386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $COB$ is a sector of a circle with $\\angle COB=45^\\circ.$ $OZ$ is drawn perpendicular to $CB$ and intersects $CB$ at $W.$ What is the length of $WZ$? Assume the radius of the circle is 10 units. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "10 - 5 \\sqrt{2+\\sqrt{2}}"
} | {
"split": "train",
"index": 39387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of polygon $ABCDEF$ is 52 with $AB=8$, $BC=9$ and $FA=5$. What is $DE+EF$? [asy]\npair a=(0,9), b=(8,9), c=(8,0), d=(4,0), e=(4,4), f=(0,4);\ndraw(a--b--c--d--e--f--cycle);\ndraw(shift(0,-.25)*a--shift(.25,-.25)*a--shift(.25,0)*a);\ndraw(shift(-.25,0)*b--shift(-.25,-.2... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 39388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the quadrilateral pyramid $S-ABCD$ with a right trapezoid as its base, where $\\angle ABC = 90^\\circ$, $SA \\perp$ plane $ABCD$, $SA = AB = BC = 1$, and $AD = \\frac{1}{2}$, find the tangent of the angle between plane $SCD$ and plane $SBA$. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 39389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number obtained from the last two nonzero digits of $80!$ is equal to $n$. Find the value of $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 39390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the circle with center $O$, the measure of $\\angle RIP$ is $36^\\circ$ and $OR=10$ cm. Find the number of centimeters in the length of arc $RP$. Express your answer in terms of $\\pi$. [asy]\ndraw((1,0)..(0,1)..(-1,0)..(0,-1)..cycle);\ndraw((0,0)--(4/5,3/5)--(-1,0)--(4/5,-3... | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 39391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$, where the sum of the first $n$ terms is denoted as $S_n$, and $S_n = a\\left(\\frac{1}{4}\\right)^{n-1} + 6$, find the value of $a$. 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": 39392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many whole numbers between 1 and 2000 do not contain the digits 1 or 2? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "511"
} | {
"split": "train",
"index": 39393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $D$ and $E$ are points on sides $BC$ and $AC$, respectively, such that $\\frac{BD}{DC} = \\frac{2}{3}$ and $\\frac{AE}{EC} = \\frac{3}{4}$. Find the value of $\\frac{AF}{FD} \\cdot \\frac{BF}{FE}$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{35}{12}"
} | {
"split": "train",
"index": 39394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Trapezoid $A B C D$ is inscribed in the parabola $y=x^{2}$ such that $A=\\left(a, a^{2}\\right), B=\\left(b, b^{2}\\right)$, $C=\\left(-b, b^{2}\\right)$, and $D=\\left(-a, a^{2}\\right)$ for some positive reals $a, b$ with $a>b$. If $A D+B C=A B+C D$, and $A B=\\frac{3}{4}$, w... | math | {
"style": "rule",
"ground_truth": "\\frac{27}{40}"
} | {
"split": "train",
"index": 39395,
"solution": "Let $t=\\frac{3}{4}$, so $2 a+2 b=2 \\sqrt{(a-b)^{2}+\\left(a^{2}-b^{2}\\right)^{2}}=2 t$ gives $a=b=t$ and $t^{2}=(a-b)^{2}\\left[1+(a+b)^{2}\\right]=(a-b)^{2}\\left[1+t^{2}\\right]$. Thus $a=\\frac{t+\\frac{t}{\\sqrt{1+t^{2}}}}{2}=\\frac{\\frac{3}{4}+\\frac{3}{5}... |
deepscaler | [
{
"role": "user",
"content": "Altitudes $B E$ and $C F$ of acute triangle $A B C$ intersect at $H$. Suppose that the altitudes of triangle $E H F$ concur on line $B C$. If $A B=3$ and $A C=4$, then $B C^{2}=\\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$. Let's thi... | math | {
"style": "rule",
"ground_truth": "33725"
} | {
"split": "train",
"index": 39396,
"solution": "Let $P$ be the orthocenter of $\\triangle E H F$. Then $E H \\perp F P$ and $E H \\perp A C$, so $F P$ is parallel to $A C$. Similarly, $E P$ is parallel to $A B$. Using similar triangles gives $$1=\\frac{B P}{B C}+\\frac{C P}{B C}=\\frac{A E}{A C}+\\frac{A F}{A B}... |
deepscaler | [
{
"role": "user",
"content": "Find the curve defined by the equation\n\\[r = 4 \\tan \\theta \\sec \\theta.\\](A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{(C)}"
} | {
"split": "train",
"index": 39397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle PAT,$ $\\angle P=36^{\\circ},$ $\\angle A=56^{\\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\\overline{TP}$ and $\\overline{TA},$ respectively, so that $PU=AG=1.$ Let $M$ and $N$ be the midpoints of segments $\\overline{PA}$ and $\\overline{UG},$ respec... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 39398,
"solution": "1. **Identify Given Information and Setup:**\n - In $\\triangle PAT$, we have $\\angle P = 36^\\circ$, $\\angle A = 56^\\circ$, and $PA = 10$.\n - Points $U$ and $G$ are on $\\overline{TP}$ and $\\overline{TA}$ respectively, such that $PU = AG = 1$.\n - $M$ a... |
deepscaler | [
{
"role": "user",
"content": "A school band found they could arrange themselves in rows of 6, 7, or 8 with no one left over. What is the minimum number of students in the band? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 39399,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles right triangle $ABC$ with $\\angle A = 90^{\\circ}$ and $AB = AC = 2$, calculate the projection of the vector $\\vec{AB}$ in the direction of $\\vec{BC}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{2}"
} | {
"split": "train",
"index": 39400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For distinct positive integers $a$ , $b < 2012$ , define $f(a,b)$ to be the number of integers $k$ with $1 \\le k < 2012$ such that the remainder when $ak$ divided by 2012 is greater than that of $bk$ divided by 2012. Let $S$ be the minimum value of $f(a,b)$ , where $a$ and $b... | math | {
"style": "rule",
"ground_truth": "S = 502"
} | {
"split": "train",
"index": 39401,
"solution": "Solution 1\nFirst we'll show that $S \\geq 502$ , then we'll find an example $(a, b)$ that have $f(a, b)=502$ .\nLet $x_k$ be the remainder when $ak$ is divided by 2012, and let $y_k$ be defined similarly for $bk$ . First, we know that, if $x_k > y_k >0$ , then $x_... |
deepscaler | [
{
"role": "user",
"content": "Let $A$, $B$, $R$, $M$, and $L$ be positive real numbers such that\n\\begin{align*}\n\\log_{10} (AB) + \\log_{10} (AM) &= 2, \\\\\n\\log_{10} (ML) + \\log_{10} (MR) &= 3, \\\\\n\\log_{10} (RA) + \\log_{10} (RB) &= 5.\n\\end{align*}\nCompute the value of the product $ABRML$. Let... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 39402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation of the directrix of the parabola $y^{2}=6x$ is $x=\\frac{3}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{3}{2}"
} | {
"split": "train",
"index": 39403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a certain grocery store, cookies may be bought in boxes of $10$ or $21.$ What is the minimum positive number of cookies that must be bought so that the cookies may be split evenly among $13$ people?\n\n*Author: Ray Li* Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 39404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two lines $l_{1}$: $(a-1)x+2y+1=0$, $l_{2}$: $x+ay+1=0$, find the value of $a$ that satisfies the following conditions:\n\n$(1) l_{1} \\parallel l_{2}$ \n$(2) l_{1} \\perp l_{2}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 39405,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a square-based pyramid (with base vertices $A, B, C, D$) with equal side edges, and let the apex be $E$. Let $P$ be the point that divides the side edge $A E$ in a ratio of 3:1, such that $E P : P A = 3$, and let $Q$ be the midpoint of the side edge $C E$. In what rati... | math | {
"style": "rule",
"ground_truth": "4/3"
} | {
"split": "train",
"index": 39406,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose the function \\( y= \\left| \\log_{2} \\frac{x}{2} \\right| \\) has a domain of \\([m, n]\\) and a range of \\([0,2]\\). What is the minimum length of the interval \\([m, n]\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3/2"
} | {
"split": "train",
"index": 39407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many values of $a$ is it true that:\n(1) $a$ is a positive integer such that $a \\le 50$.\n(2) the quadratic equation $x^2 + (2a+1)x + a^2 = 0$ has two integer solutions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 39408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $ABC$, where $\\angle A = \\angle B$ and $AB = 10$. What is the area of $\\triangle ABC$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 39409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function \\( f(x) \\) satisfies the equation \\( 2 f(x) + x^{2} f\\left(\\frac{1}{x}\\right) = \\frac{3 x^{3} - x^{2} + 4 x + 3}{x + 1} \\) and \\( g(x) = \\frac{5}{x + 1} \\), determine the minimum value of \\( f(x) + g(x) \\). Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{4}"
} | {
"split": "train",
"index": 39410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A total of 120 is divided into four parts proportional to 3, 2, 4, and 5. What is the second largest part? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{240}{7}"
} | {
"split": "train",
"index": 39411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Outstanding Brazilian footballer Ronaldinho Gaúcho will be $X$ years old in the year $X^{2}$. How old will he be in 2018, when the World Cup is held in Russia? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 39412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a WeChat group, five members simultaneously grab for four red envelopes, each person can grab at most one, and all red envelopes are claimed. Among the four red envelopes, there are two containing 2 yuan and two containing 3 yuan. Determine the number of scenarios in which b... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 39413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the set \\( \\{1, 2, 3, \\ldots, 999, 1000\\} \\), select \\( k \\) numbers. If among the selected numbers, there are always three numbers that can form the side lengths of a triangle, what is the smallest value of \\( k \\)? Explain why. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 39414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of solutions to the equation\n\\[\\sin x = \\left( \\frac{1}{3} \\right)^x\\] on the interval $(0,150 \\pi)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 39415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 0's are located to the right of the decimal point and before the first non-zero digit in the terminating decimal representation of $\\frac{1}{2^5\\cdot5^8}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 39416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "2 distinct vertices of an octagon are chosen at random. What is the probability that they are adjacent? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{7}"
} | {
"split": "train",
"index": 39417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In quadrilateral $ABCD$, $\\angle{BAD}\\cong\\angle{ADC}$ and $\\angle{ABD}\\cong\\angle{BCD}$, $AB = 8$, $BD = 10$, and $BC = 6$. The length $CD$ may be written in the form $\\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. Let's think ste... | math | {
"style": "rule",
"ground_truth": "69"
} | {
"split": "train",
"index": 39418,
"solution": "Draw a line from $B$, parallel to $\\overline{AD}$, and let it meet $\\overline{CD}$ at $M$. Note that $\\triangle{DAB}$ is similar to $\\triangle{BMC}$ by AA similarity, since $\\angle{ABD}=\\angle{MCB}$ and since $BM$ is parallel to $CD$ then $\\angle{BMC}=\\angl... |
deepscaler | [
{
"role": "user",
"content": "In a gumball machine containing 13 red, 5 blue, 1 white, and 9 green gumballs, what is the least number of gumballs that must be bought to guarantee receiving 3 gumballs of the same color? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 39419,
"solution": "It is possible that after buying 7 gumballs, Wally has received 2 red, 2 blue, 1 white, and 2 green gumballs. This is the largest number of each color that he could receive without having three gumballs of any one color. If Wally buys another gumball, he will recei... |
deepscaler | [
{
"role": "user",
"content": "The length of a rectangular playground exceeds twice its width by 25 feet, and the perimeter of the playground is 650 feet. What is the area of the playground in square feet? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22,500"
} | {
"split": "train",
"index": 39420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sides of triangle $PQR$ are in the ratio of $3:4:5$. Segment $QS$ is the angle bisector drawn to the shortest side, dividing it into segments $PS$ and $SR$. What is the length, in inches, of the longer subsegment of side $PR$ if the length of side $PR$ is $15$ inches? Expre... | math | {
"style": "rule",
"ground_truth": "\\frac{60}{7}"
} | {
"split": "train",
"index": 39421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How does the area of the quadrilateral formed by connecting the midpoints of the sides of a concave quadrilateral compare to the area of the original quadrilateral? 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": 39422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the figure, $ABCD$ is a rectangle, $AZ=WC=6$ units, $AB=12$ units and the area of trapezoid $ZWCD$ is 120 square units. What is the area of triangle $BQW$? [asy]\ndraw((0,0)--(12,0)--(12,20)--(0,20)--(0,0)--(12,20));\ndraw((0,14)--(12,6));\nlabel(\"$A$\",(0,20),W);\nlabel(\"... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 39423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the arithmetic sequence $\\{a_n\\}$, we have $a_2=4$, and $a_4+a_7=15$. \n(Ⅰ) Find the general term formula for the sequence $\\{a_n\\}$. \n(Ⅱ) Let $b_n= \\frac{1}{a_n a_{n+1}}$, calculate the value of $b_1+b_2+b_3+\\dots+b_{10}$. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{39}"
} | {
"split": "train",
"index": 39424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f$ be a polynomial such that, for all real number $x$ , $f(-x^2-x-1) = x^4 + 2x^3 + 2022x^2 + 2021x + 2019$ .\nCompute $f(2018)$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2019"
} | {
"split": "train",
"index": 39425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each segment with endpoints at the vertices of a regular 100-sided polygon is colored red if there is an even number of vertices between the endpoints, and blue otherwise (in particular, all sides of the 100-sided polygon are red). Numbers are placed at the vertices such that t... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 39426,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $y = ax^2 + bx + c$ is described, where $a$, $b$, and $c$ are integers. Given that the vertex of this parabola is at $(-2, 3)$ and one point on the graph is $(1, 6)$, determine the value of $a$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 39427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A_1,B_1,C_1,D_1$ be the midpoints of the sides of a convex quadrilateral $ABCD$ and let $A_2, B_2, C_2, D_2$ be the midpoints of the sides of the quadrilateral $A_1B_1C_1D_1$ . If $A_2B_2C_2D_2$ is a rectangle with sides $4$ and $6$ , then what is the product o... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 39428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many decreasing sequences $a_1, a_2, \\ldots, a_{2019}$ of positive integers are there such that $a_1\\le 2019^2$ and $a_n + n$ is even for each $1 \\le n \\le 2019$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\binom{2037171}{2019}"
} | {
"split": "train",
"index": 39429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The difference of the roots of $x^2-7x-9=0$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{85}"
} | {
"split": "train",
"index": 39430,
"solution": "1. **Identify the coefficients of the quadratic equation**: Given the quadratic equation $x^2 - 7x - 9 = 0$, we can identify the coefficients as $a = 1$, $b = -7$, and $c = -9$.\n\n2. **Apply Vieta's formulas**: Vieta's formulas tell us that for a quadratic equatio... |
deepscaler | [
{
"role": "user",
"content": "A group of dancers are arranged in a rectangular formation. When they are arranged in 12 rows, there are 5 positions unoccupied in the formation. When they are arranged in 10 rows, there are 5 positions unoccupied. How many dancers are in the group if the total number is betwee... | math | {
"style": "rule",
"ground_truth": "295"
} | {
"split": "train",
"index": 39431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " \nPeter Ivanovich, along with 49 other men and 50 women, are seated in a random order around a round table. We call a man satisfied if a woman is sitting next to him. Find:\n\na) The probability that Peter Ivanovich is satisfied.\n\nb) The expected number of satisfied men. Let... | math | {
"style": "rule",
"ground_truth": "\\frac{1250}{33}"
} | {
"split": "train",
"index": 39432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a class of 120 students, the teacher recorded the following scores for an exam. Calculate the average score for the class.\n\n\\begin{tabular}{|c|c|}\n\\multicolumn{2}{c}{}\\\\\\hline\n\\textbf{Score (\\%)}&\\textbf{Number of Students}\\\\\\hline\n95&12\\\\\\hline\n85&24\\\\... | math | {
"style": "rule",
"ground_truth": "69.83"
} | {
"split": "train",
"index": 39433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Provided $x$ is a multiple of $27720$, determine the greatest common divisor of $g(x) = (5x+3)(11x+2)(17x+7)(3x+8)$ and $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 39434,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of subsets of $\\{1,2,3,\\ldots,10\\}$ that contain exactly one pair of consecutive integers. Examples of such subsets are $\\{\\mathbf{1},\\mathbf{2},5\\}$ and $\\{1,3,\\mathbf{6},\\mathbf{7},10\\}.$ Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "235"
} | {
"split": "train",
"index": 39435,
"solution": "The problem is the same as laying out a line of polynomoes to cover spots $0,1,...10$: 1 triomino ($RGG$), $n$ dominoes ($RG$), and $8-2n$ monominoes ($R$). The $G$ spots cover the members of the subset. The total number spots is 11, because one $R$ spot always cov... |
deepscaler | [
{
"role": "user",
"content": "In a regular hexagon $ABCDEF$, points $P$, $Q$, $R$, and $S$ are chosen on sides $\\overline{AB}$, $\\overline{CD}$, $\\overline{DE}$, and $\\overline{FA}$ respectively, so that lines $PC$ and $RA$, as well as $QS$ and $EB$, are parallel. Moreover, the distances between these p... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 39436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\\vec{m}=(a,c)$ and $\\vec{n}=(\\cos C,\\cos A)$.\n1. If $\\vec{m}\\parallel \\vec{n}$ and $a= \\sqrt {3}c$, find angle $A$;\n2. If $\\vec{m}\\cdot \\vec{n}=3b... | math | {
"style": "rule",
"ground_truth": "\\frac {4-6 \\sqrt {2}}{15}"
} | {
"split": "train",
"index": 39437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a cabinet, there are 3 pairs of different shoes. If 2 shoes are randomly taken out, let event A denote \"the taken out shoes do not form a pair\"; event B denote \"both taken out shoes are for the same foot\"; event C denote \"one shoe is for the left foot and the other is f... | math | {
"style": "rule",
"ground_truth": "\\dfrac{2}{5}"
} | {
"split": "train",
"index": 39438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the center of a circular field, there is a geologists' hut. From it, 6 straight roads radiate out, dividing the field into 6 equal sectors. Two geologists leave their hut at a speed of 4 km/h, each choosing a road at random. Determine the probability that the distance betwee... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 39439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[e^{2 \\pi i/13} + e^{4 \\pi i/13} + e^{6 \\pi i/13} + \\dots + e^{24 \\pi i/13}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 39440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABCD$ is a rectangle (see the accompanying diagram) with $P$ any point on $\\overline{AB}$. $\\overline{PS} \\perp \\overline{BD}$ and $\\overline{PR} \\perp \\overline{AC}$. $\\overline{AF} \\perp \\overline{BD}$ and $\\overline{PQ} \\perp \\overline{AF}$. Then $PR + PS$ is e... | math | {
"style": "rule",
"ground_truth": "AF"
} | {
"split": "train",
"index": 39441,
"solution": "1. **Identify Parallel Lines**: Since $\\overline{PQ}$ and $\\overline{BD}$ are both perpendicular to $\\overline{AF}$, it follows that $\\overline{PQ} \\parallel \\overline{BD}$.\n\n2. **Angle Relationships**: Because $\\overline{PQ} \\parallel \\overline{BD}$ and... |
deepscaler | [
{
"role": "user",
"content": "Given circles ${C}_{1}:{x}^{2}+{y}^{2}=1$ and ${C}_{2}:(x-4)^{2}+(y-2)^{2}=1$, a moving point $M\\left(a,b\\right)$ is used to draw tangents $MA$ and $MB$ to circles $C_{1}$ and $C_{2}$ respectively, where $A$ and $B$ are the points of tangency. If $|MA|=|MB|$, calculate the mi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 39442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the series of activities with the theme \"Labor Creates a Better Life\" carried out at school, Class 1 of Grade 8 is responsible for the design, planting, and maintenance of a green corner on campus. The students agreed that each person would take care of one pot of green pl... | math | {
"style": "rule",
"ground_truth": "$369"
} | {
"split": "train",
"index": 39443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 39444,
"solution": "Let the three numbers be $a$, $b$, and $c$, where $a < b < c$. We are given the sums of these numbers taken in pairs:\n1. $a + b = 12$\n2. $a + c = 17$\n3. $b + c = 19$\n\nWe aim to find the value of $b$, the middle number. To do this, we can manipulate the given e... |
deepscaler | [
{
"role": "user",
"content": "In the trapezoid \\(ABCD\\) with bases \\(AD \\parallel BC\\), the diagonals intersect at point \\(E\\). Given the areas \\(S(\\triangle ADE) = 12\\) and \\(S(\\triangle BCE) = 3\\), find the area of the trapezoid. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 39445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A Moskvich car was sent to transport mail from the post office to the airfield. The plane carrying the mail landed earlier than expected, and the delivered mail was sent to the post office by a passing truck. After driving for 30 minutes, the truck met the Moskvich on the road,... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 39446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A small bottle of shampoo can hold $35$ milliliters of shampoo, whereas a large bottle can hold $500$ milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy? Let's think step by... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 39447,
"solution": "To find the minimum number of small bottles necessary to completely fill a large bottle, we need to determine how many times $35$ milliliters (the capacity of one small bottle) goes into $500$ milliliters (the capacity of one large bottle).\n\n1. **Calculate the nu... |
deepscaler | [
{
"role": "user",
"content": "Given the set $M=\\{x|2x^{2}-3x-2=0\\}$ and the set $N=\\{x|ax=1\\}$. If $N \\subset M$, what is the value of $a$? 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": 39448,
"solution": ""
} |
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