data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In \\(\\triangle ABC\\), \\(AC = AB = 25\\) and \\(BC = 40\\). \\(D\\) is a point chosen on \\(BC\\). From \\(D\\), perpendiculars are drawn to meet \\(AC\\) at \\(E\\) and \\(AB\\) at \\(F\\). \\(DE + DF\\) equals: Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 24449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the general term of a sequence ${a_n}$ is $a_n = -n^2 + 12n - 32$, and the sum of its first $n$ terms is $S_n$, for any $m, n \\in \\mathbb{N}^*$ with $m < n$, the maximum value of $S_n - S_m$ is __________________. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 24450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $E(n)$ denote the sum of the even digits of $n$. For example, $E(5681) = 6+8 = 14$. Find $E(1)+E(2)+E(3)+\\cdots+E(100)$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 24451,
"solution": "1. **Understanding the Problem**: We need to find the sum of the even digits in all numbers from 1 to 100. The function $E(n)$ represents the sum of even digits in the number $n$.\n\n2. **Simplifying the Range**: We can consider the numbers from 00 to 99 instead of... |
deepscaler | [
{
"role": "user",
"content": "Given the complex numbers \\( z_{1}, z_{2}, z_{3} \\) satisfying:\n\\[\n\\begin{array}{l}\n\\left|z_{1}\\right| \\leq 1, \\left|z_{2}\\right| \\leq 2, \\\\\n\\left|2z_{3} - z_{1} - z_{2}\\right| \\leq \\left|z_{1} - z_{2}\\right|.\n\\end{array}\n\\]\nWhat is the maximum value o... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 24452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In tetrahedron $S\\-(ABC)$, $SA$ is perpendicular to plane $ABC$, $\\angle BAC=120^{\\circ}$, $SA=AC=2$, $AB=1$, find the surface area of the circumscribed sphere of the tetrahedron. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{40\\pi}{3}"
} | {
"split": "train",
"index": 24453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "306"
} | {
"split": "train",
"index": 24454,
"solution": "Refer to the diagram in solution 2; let $a^2=[E'EP]$, $b^2=[D'DP]$, and $c^2=[F'FP]$. Now, note that $[E'BD]$, $[D'DP]$, and $[E'EP]$ are similar, so through some similarities we find that $\\frac{E'P}{PD}=\\frac{a}{b}\\implies\\frac{E'D}{PD}=\\frac{a+b}{b}\\implie... |
deepscaler | [
{
"role": "user",
"content": "Convert the binary number $1101100_{(2)}$ to a decimal number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 24455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $g(x)=\\sqrt[3]{\\frac{x+3}{4}}$, for what value of $x$ will $g(2x)=2(g(x))$? Express your answer in simplest form. 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": 24456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert the point $(\\rho,\\theta,\\phi) = \\left( 12, \\frac{7 \\pi}{6}, \\frac{\\pi}{3} \\right)$ in spherical coordinates to rectangular coordinates. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-9, -3 \\sqrt{3}, 6)"
} | {
"split": "train",
"index": 24457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a parlor game, the magician asks one of the participants to think of a three digit number $(abc)$ where $a$, $b$, and $c$ represent digits in base $10$ in the order indicated. The magician then asks this person to form the numbers $(acb)$, $(bca)$, $(bac)$, $(cab)$, and $(cb... | math | {
"style": "rule",
"ground_truth": "358"
} | {
"split": "train",
"index": 24458,
"solution": "Let $m$ be the number $100a+10b+c$. Observe that $3194+m=222(a+b+c)$ so\n\\[m\\equiv -3194\\equiv -86\\equiv 136\\pmod{222}\\]\n This reduces $m$ to one of $136, 358, 580, 802$. But also $a+b+c=\\frac{3194+m}{222}>\\frac{3194}{222}>14$ so $a+b+c\\geq 15$. Of the fo... |
deepscaler | [
{
"role": "user",
"content": "In the plane rectangular coordinate system $O-xy$, if $A(\\cos\\alpha, \\sin\\alpha)$, $B(\\cos\\beta, \\sin\\beta)$, $C\\left(\\frac{1}{2}, \\frac{\\sqrt{3}}{2}\\right)$, then one possible value of $\\beta$ that satisfies $\\overrightarrow{OC}=\\overrightarrow{OB}-\\overrighta... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 24459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The real numbers $x$ and $y$ satisfy the equation $2\\cos ^{2}(x+y-1)= \\frac {(x+1)^{2}+(y-1)^{2}-2xy}{x-y+1}$. Find the minimum value of $xy$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 24460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors \\(\\overrightarrow{O P}=\\left(2 \\cos \\left(\\frac{\\pi}{2}+x\\right),-1\\right)\\) and \\(\\overrightarrow{O Q}=\\left(-\\sin \\left(\\frac{\\pi}{2}- x\\right), \\cos 2 x\\right)\\), and the function \\(f(x)=\\overrightarrow{O P} \\cdot \\overrightarrow{O Q}\\... | math | {
"style": "rule",
"ground_truth": "15/2"
} | {
"split": "train",
"index": 24461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\ln(3x)+8x$, calculate the value of $\\lim_{\\triangle x \\to 0} \\frac{f(1-2\\triangle x)-f(1)}{\\triangle x}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"split": "train",
"index": 24462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain school arranges $5$ senior high school teachers to visit $3$ schools for exchange and learning. One school has $1$ teacher, one school has $2$ teachers, and one school has $2$ teachers. The total number of different arrangements is _______. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 24463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the roots of\n\\[6x^4 - 35x^3 + 62x^2 - 35x + 6 = 0.\\]Enter the roots, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2, 3, \\frac{1}{2}, \\frac{1}{3}"
} | {
"split": "train",
"index": 24464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the result of the expression:\n\n\\[ 2013 \\times \\frac{5.7 \\times 4.2 + \\frac{21}{5} \\times 4.3}{\\frac{14}{73} \\times 15 + \\frac{5}{73} \\times 177 + 656} \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 24465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Dolly, Molly and Polly each can walk at $6 \\mathrm{~km} / \\mathrm{h}$. Their one motorcycle, which travels at $90 \\mathrm{~km} / \\mathrm{h}$, can accommodate at most two of them at once (and cannot drive by itself!). Let $t$ hours be the time taken for all three of them to ... | math | {
"style": "rule",
"ground_truth": "t<3.9"
} | {
"split": "train",
"index": 24466,
"solution": "First, we note that the three people are interchangeable in this problem, so it does not matter who rides and who walks at any given moment. We abbreviate the three people as D, M and P. We call their starting point $A$ and their ending point $B$. Here is a strateg... |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\sin^2 6^\\circ + \\sin^2 12^\\circ + \\sin^2 18^\\circ + \\dots + \\sin^2 174^\\circ.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 24467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a rectangle $ABCD$ with $BC = 2 \\cdot AB$ . Let $\\omega$ be the circle that touches the sides $AB$ , $BC$ , and $AD$ . A tangent drawn from point $C$ to the circle $\\omega$ intersects the segment $AD$ at point $K$ . Determine the ratio $\\frac{AK}{KD... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 24468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ satisfies $a_1=1$, $a_2=1$, $a_{n+2}=(1+\\sin^2 \\frac{n\\pi}{2})a_n+2\\cos^2 \\frac{n\\pi}{2}$. Find the sum of the first $20$ terms of this sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1123"
} | {
"split": "train",
"index": 24469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Select 5 different numbers from $0,1,2,3,4,5,6,7,8,9$ to form a five-digit number such that this five-digit number is divisible by $3$, $5$, $7$, and $13$. What is the largest such five-digit number? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "94185"
} | {
"split": "train",
"index": 24470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $(a_1, b_1), (a_2, b_2), \\ldots , (a_{100}, b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i, j)$ satisfying $1 \\le i < j \\le 100$ and $|a_ib_j - a_j b_i|=1$ . Determine the largest possible value o... | math | {
"style": "rule",
"ground_truth": "N=197"
} | {
"split": "train",
"index": 24471,
"solution": "Let's start off with just $(a_1, b_1), (a_2, b_2)$ and suppose that it satisfies the given condition. We could use $(1, 1), (1, 2)$ for example. We should maximize the number of conditions that the third pair satisfies. We find out that the third pair should equal ... |
deepscaler | [
{
"role": "user",
"content": "Determine the coefficient of \\(x^{29}\\) in the expansion of \\(\\left(1 + x^{5} + x^{7} + x^{9}\\right)^{16}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65520"
} | {
"split": "train",
"index": 24472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Azmi has four blocks, each in the shape of a rectangular prism and each with dimensions $2 \times 3 \times 6$. She carefully stacks these four blocks on a flat table to form a tower that is four blocks high. What is the number of possible heights for this tower? Let's think ste... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 24473,
"solution": "The height of each block is 2, 3 or 6. Thus, the total height of the tower of four blocks is the sum of the four heights, each of which equals 2, 3 or 6. If 4 blocks have height 6, the total height equals $4 \times 6=24$. If 3 blocks have height 6, the fourth block... |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when $5^{137}$ is divided by 8? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 24474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the convex quadrilateral $ABCD$ angle $\\angle{BAD}=90$ , $\\angle{BAC}=2\\cdot\\angle{BDC}$ and $\\angle{DBA}+\\angle{DCB}=180$ . Then find the angle $\\angle{DBA}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 24475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "By joining four identical trapezoids, each with equal non-parallel sides and bases measuring 50 cm and 30 cm, we form a square with an area of 2500 cm² that has a square hole in the middle. What is the area, in cm², of each of the four trapezoids? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 24476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle having center $(0,k)$, with $k>6$, is tangent to the lines $y=x$, $y=-x$ and $y=6$. What is the radius of this circle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{2}+6"
} | {
"split": "train",
"index": 24477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $\\left(2m+n\\right)\\left(2m-n\\right)-\\left(2m-n\\right)^{2}+2n\\left(m+n\\right)$, where $m=2$, $n=-1^{2023}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"split": "train",
"index": 24478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The base of a pyramid is an isosceles triangle with a base of 6 and a height of 9. Each lateral edge is 13. Find the volume of the pyramid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 24479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all prime numbers $p$ such that $y^{2}=x^{3}+4x$ has exactly $p$ solutions in integers modulo $p$. In other words, determine all prime numbers $p$ with the following property: there exist exactly $p$ ordered pairs of integers $(x, y)$ such that $x, y \\in\\{0,1, \\ldots, p... | math | {
"style": "rule",
"ground_truth": "p=2 \\text{ and } p \\equiv 3(\\bmod 4)"
} | {
"split": "train",
"index": 24480,
"solution": "Clearly $p=2$ works with solutions $(0,0)$ and $(1,1)$ and not $(0,1)$ or $(1,0)$. If $p \\equiv 3(\\bmod 4)$ then -1 is not a quadratic residue, so for $x^{3}+4x \\neq 0$, exactly one of $x^{3}+4x$ and $-x^{3}-4x$ is a square and gives two solutions (for positive ... |
deepscaler | [
{
"role": "user",
"content": "In the non-convex quadrilateral $ABCD$ shown below, $\\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$. What is the area of quadrilateral $ABCD$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 24481,
"solution": "1. **Identify the triangles and their properties:**\n - Connect point $B$ to point $D$.\n - Observe that $\\triangle BCD$ is a right triangle because $\\angle BCD = 90^\\circ$. Given $BC = 4$ and $CD = 3$, we can use the Pythagorean theorem to find $BD$:\n ... |
deepscaler | [
{
"role": "user",
"content": "Given \\((n+1)^{\\alpha+1}-n^{\\alpha+1} < n^{\\alpha}(\\alpha+1) < n^{\\alpha+1}-(n-1)^{\\alpha+1}, -1 < \\alpha < 0\\). Let \\(x = \\sum_{k=4}^{10^{6}} \\frac{1}{\\sqrt[3]{k}}\\), find the integer part of \\(x\\). Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "146"
} | {
"split": "train",
"index": 24482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "About 40% of students in a certain school are nearsighted, and about 30% of the students in the school use their phones for more than 2 hours per day, with a nearsighted rate of about 50% among these students. If a student who uses their phone for no more than 2 hours per day i... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{14}"
} | {
"split": "train",
"index": 24483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mrs. Riley recorded this information from a recent test taken by all of her students. Using the data, what was the average percent score for these $100$ students?\n\n\\begin{tabular}{|c|c|}\n\\multicolumn{2}{c}{}\\\\\\hline\n\\textbf{$\\%$ Score}&\\textbf{Number of Students}\\\... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"split": "train",
"index": 24484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among all right triangles \\(ABC\\) with \\( \\angle C = 90^\\circ\\), find the maximum value of \\( \\sin A + \\sin B + \\sin^2 A \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + \\frac{1}{2}"
} | {
"split": "train",
"index": 24485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a point $P(x,y)$ moving on the ellipse $\\frac{x^{2}}{4} + \\frac{y^{2}}{3} = 1$, let $d = \\sqrt{x^{2} + y^{2} + 4y + 4} - \\frac{x}{2}$. Find the minimum value of $d$.\nA) $\\sqrt{5} - 2$\nB) $2\\sqrt{2} - 1$\nC) $\\sqrt{5} - 1$\nD) $\\sqrt{6} - 1$ Let's think step by s... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} - 1"
} | {
"split": "train",
"index": 24486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(x) = b_0 + b_1x + \\dots + b_nx^n$ be a polynomial with integer coefficients, and $0 \\le b_i < 5$ for all $0 \\le i \\le n$.\n\nGiven that $P(\\sqrt{5}) = 40 + 31\\sqrt{5}$, compute $P(3)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "381"
} | {
"split": "train",
"index": 24487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\sqrt{25000}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50\\sqrt{10}"
} | {
"split": "train",
"index": 24488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagonals of a regular hexagon have two possible lengths. What is the ratio of the shorter length to the longer length? Express your answer as a common fraction in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 24489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\lfloor (12.1)^2 \\rfloor - \\lfloor 12.1 \\rfloor \\cdot \\lfloor 12.1 \\rfloor .$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 24490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{P}$ be the parabola given by the equation \\( y = x^2 \\). Suppose a circle $\\mathcal{C}$ intersects $\\mathcal{P}$ at four distinct points. If three of these points are \\((-4,16)\\), \\((1,1)\\), and \\((6,36)\\), find the sum of the distances from the directr... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 24491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the common difference of the arithmetic sequence $\\{a_n\\}$ is $d$, and the sum of the first $n$ terms is $S_n$, with $a_1=d=1$, find the minimum value of $\\frac{S_n+8}{a_n}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 24492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest integer that can be placed in the box so that $\\frac{1}{2} < \\frac{\\square}{9}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 24493,
"solution": "We know that $\\frac{1}{2} = 0.5$. Since $\\frac{4}{9} \\approx 0.44$ is less than $\\frac{1}{2} = 0.5$, then 4 cannot be placed in the box. (No integer smaller than 4 can be placed in the box either.) Since $\\frac{5}{9} \\approx 0.56$ is greater than $\\frac{1}{2... |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $C$: $\\frac{x^{2}}{a^{2}}-y^{2}=1 (a > 0)$ and the line $l$: $x+y=1$ intersect at two distinct points $A$ and $B$.\n(I) Find the range of the eccentricity $e$ of the hyperbola $C$.\n(II) Let $P$ be the intersection point of line $l$ and the $y$-axis, and $\... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{13}"
} | {
"split": "train",
"index": 24494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two fair dice are tossed, what is the probability that their sum is divisible by 5 ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 24495,
"solution": "$\\frac{1}{4}$."
} |
deepscaler | [
{
"role": "user",
"content": "Each of the three aluminum cans holds an integer number of liters of water. The second can holds 1.5 times more than the first, and the volume of the third can is equal to \\(\\frac{4^3}{3}\\) times the volume of the first can. What is the total number of liters of water that t... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 24496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The complex number $2+i$ and the complex number $\\frac{10}{3+i}$ correspond to points $A$ and $B$ on the complex plane, calculate the angle $\\angle AOB$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 24497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set $A=\\{(x,y) \\,|\\, |x| \\leq 1, |y| \\leq 1, x, y \\in \\mathbb{R}\\}$, and $B=\\{(x,y) \\,|\\, (x-a)^2+(y-b)^2 \\leq 1, x, y \\in \\mathbb{R}, (a,b) \\in A\\}$, then the area represented by set $B$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "12 + \\pi"
} | {
"split": "train",
"index": 24498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer with exactly 20 positive divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 24499,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, it is known that $\\frac {a_{11}}{a_{10}} + 1 < 0$, and the sum of the first $n$ terms of the sequence, $S_n$, has a maximum value. Find the maximum value of $n$ for which $S_n > 0$. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 24500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two integers \\( m \\) and \\( n \\) which are coprime, calculate the GCD of \\( 5^m + 7^m \\) and \\( 5^n + 7^n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 24501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the limit, when $n$ tends to the infinity, of $$ \\frac{\\sum_{k=0}^{n} {{2n} \\choose {2k}} 3^k} {\\sum_{k=0}^{n-1} {{2n} \\choose {2k+1}} 3^k} $$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 24502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For an even integer positive integer $n$ Kevin has a tape of length $4 n$ with marks at $-2 n,-2 n+1, \\ldots, 2 n-1,2 n$. He then randomly picks $n$ points in the set $-n,-n+1,-n+2, \\ldots, n-1, n$, and places a stone on each of these points. We call a stone 'stuck' if it is ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{n-1}"
} | {
"split": "train",
"index": 24503,
"solution": "After we have selected the positions of the initial $n$ stones, we number their positions: $a_{1}<a_{2}<\\ldots<a_{n}$. The conditions on how we move the stones imply that the expected value of $\\left(a_{i}-a_{j}\\right)$ after $t$ minutes is still equal to $a_{i}... |
deepscaler | [
{
"role": "user",
"content": "One dimension of a cube is increased by $1$, another is decreased by $1$, and the third is left unchanged. The volume of the new rectangular solid is $5$ less than that of the cube. What was the volume of the cube? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 24504,
"solution": "1. **Define the dimensions and volume of the original cube:**\n Let the edge length of the original cube be $a$. Therefore, the volume of the cube is given by:\n \\[\n V_{\\text{cube}} = a^3\n \\]\n\n2. **Define the dimensions and volume of the new rectangu... |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ and $z$ be nonzero complex numbers such that $x + y + z = 20$ and\n\\[(x - y)^2 + (x - z)^2 + (y - z)^2 = xyz.\\]Find $\\frac{x^3 + y^3 + z^3}{xyz}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 24505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the lateral side \\( CD \\) of the trapezoid \\( ABCD \\) (\\( AD \\parallel BC \\)), a point \\( M \\) is marked. A perpendicular \\( AH \\) is dropped from vertex \\( A \\) to segment \\( BM \\). It turns out that \\( AD = HD \\). Find the length of segment \\( AD \\), giv... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 24506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When f(t)=0.1, solve for t in the function model: $f(t)=\\frac{1}{{1+{e^{-0.22(3t-40)}}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 24507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "8 singers are participating in a festival and are scheduled to perform \\( m \\) times, with 4 singers performing in each show. Design a plan to minimize the number of performances \\( m \\) so that any two singers perform together the same number of times. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 24508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the tangent line of the curve $y=\\ln x$ at point $P(x_{1}, y_{1})$ is tangent to the curve $y=e^{x}$ at point $Q(x_{2}, y_{2})$, then $\\frac{2}{{x_1}-1}+x_{2}=$____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 24509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $DEF$ has vertices $D(0,10)$, $E(4,0)$, $F(10,0)$. A vertical line intersects $DF$ at $P$ and $\\overline{EF}$ at $Q$, forming triangle $PQF$. If the area of $\\triangle PQF$ is 16, determine the positive difference of the $x$ and $y$ coordinates of point $P$. Let's th... | math | {
"style": "rule",
"ground_truth": "8\\sqrt{2}-10"
} | {
"split": "train",
"index": 24510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the graph of the function $f(x) = (x^2 - ax - 5)(x^2 - ax + 3)$ is symmetric about the line $x=2$, then the minimum value of $f(x)$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"split": "train",
"index": 24511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many miles can a car travel in 20 minutes if it travels $ \\, \\frac{3}{4} \\, $ as fast as a train going 80 miles per hour? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20\\text{ miles}"
} | {
"split": "train",
"index": 24512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 24513,
"solution": "First, we find the probability both are green, then the probability both are blue, and add the two probabilities. The sum should be equal to $0.58$.\n The probability both are green is $\\frac{4}{10}\\cdot\\frac{16}{16+N}$, and the probability both are blue is $\\f... |
deepscaler | [
{
"role": "user",
"content": "\nAn isosceles right triangle has a leg length of 36 units. Starting from the right angle vertex, an infinite series of equilateral triangles is drawn consecutively on one of the legs. Each equilateral triangle is inscribed such that their third vertices always lie on the hypot... | math | {
"style": "rule",
"ground_truth": "324"
} | {
"split": "train",
"index": 24514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Example: The ancients used Heavenly Stems and Earthly Branches to keep track of order. There are 10 Heavenly Stems: Jia, Yi, Bing, Ding, Wu, Ji, Geng, Xin, Ren, Gui; and 12 Earthly Branches: Zi, Chou, Yin, Mao, Chen, Si, Wu, Wei, Shen, You, Xu, Hai. The 10 characters of the Hea... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 24515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\frac{2}{10}+\\frac{4}{100}+\\frac{6}{1000}=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": ".246"
} | {
"split": "train",
"index": 24516,
"solution": "1. **Convert fractions to a common denominator**: We start by converting each fraction to have a common denominator, which is 1000 in this case. This is done by multiplying the numerator and denominator of each fraction by the appropriate factor:\n \\[\n \\frac... |
deepscaler | [
{
"role": "user",
"content": "Given that z and w are complex numbers with a modulus of 1, and 1 ≤ |z + w| ≤ √2, find the minimum value of |z - w|. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 24517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The interior angles of a convex polygon form an arithmetic sequence, with the smallest angle being $120^\\circ$ and the common difference being $5^\\circ$. Determine the number of sides $n$ for the polygon. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "n = 9"
} | {
"split": "train",
"index": 24518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\ln x+ \\frac {1}{2}ax^{2}-x-m$ ($m\\in\\mathbb{Z}$). \n(I) If $f(x)$ is an increasing function, find the range of values for $a$; \n(II) If $a < 0$, and $f(x) < 0$ always holds, find the minimum value of $m$. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 24519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the trapezoid shown, the ratio of the area of triangle $ABC$ to the area of triangle $ADC$ is $7:3$. If $AB + CD = 210$ cm, how long is segment $\\overline{AB}$? [asy]\nimport olympiad; size(150); defaultpen(linewidth(0.8));\npair A = (0,0), B = (5,0), C = (3,2), D = (1,2);\... | math | {
"style": "rule",
"ground_truth": "147\\text{ cm}"
} | {
"split": "train",
"index": 24520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the coefficient of the $x^3$ term in the expansion of the product $$(3x^3 + 2x^2 + 4x + 5)(4x^2 + 5x + 6).$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 24521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Camille the snail lives on the surface of a regular dodecahedron. Right now he is on vertex $P_{1}$ of the face with vertices $P_{1}, P_{2}, P_{3}, P_{4}, P_{5}$. This face has a perimeter of 5. Camille wants to get to the point on the dodecahedron farthest away from $P_{1}$. T... | math | {
"style": "rule",
"ground_truth": "\\frac{17+7 \\sqrt{5}}{2}"
} | {
"split": "train",
"index": 24522,
"solution": "Consider the net of the dodecahedron. It suffices to look at three pentagons $A B C D E, E D F G H$, and $G F I J K$, where $A J=L$. This can be found by the law of cosines on triangle $A E J$. We have $A E=1$, $E J=\\tan 72^{\\circ}$, and $\\angle A E J=162^{\\cir... |
deepscaler | [
{
"role": "user",
"content": "For a natural number \\( N \\), if at least five of the natural numbers from 1 to 9 can divide \\( N \\), then \\( N \\) is called a \"five-rule number.\" What is the smallest \"five-rule number\" greater than 2000? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"split": "train",
"index": 24523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain electronic device contains three components, with probabilities of failure for each component being $0.1, 0.2, 0.3$, respectively. If the probabilities of the device failing when one, two, or three components fail are $0.25, 0.6, 0.9$, respectively, find the probabili... | math | {
"style": "rule",
"ground_truth": "0.1601"
} | {
"split": "train",
"index": 24524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many unordered pairs of edges in a regular tetrahedron determine a plane? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 24525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( p \\) be an integer such that both roots of the equation\n\\[ 5x^2 - 5px + (66p - 1) = 0 \\]\nare positive integers. Find the value of \\( p \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "76"
} | {
"split": "train",
"index": 24526,
"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": 24527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the fraction $\\frac{987654321}{2^{30}\\cdot 5^6}$, determine the minimum number of digits to the right of the decimal point required to express this fraction as a decimal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 24528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle, whose center lies on the line \\( y = b \\), intersects the parabola \\( y = \\frac{12}{5} x^2 \\) at least at three points; one of these points is the origin, and two of the remaining points lie on the line \\( y = \\frac{12}{5} x + b \\). Find all values of \\( b \\... | math | {
"style": "rule",
"ground_truth": "169/60"
} | {
"split": "train",
"index": 24529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest product obtainable from two integers whose sum is 246? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15129"
} | {
"split": "train",
"index": 24530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Team A has a probability of $$\\frac{2}{3}$$ of winning each set in a best-of-five set match, and Team B leads 2:0 after the first two sets. Calculate the probability of Team B winning the match. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{19}{27}"
} | {
"split": "train",
"index": 24531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many values of \\( k \\) is \\( 18^{18} \\) the least common multiple of the positive integers \\( 9^9 \\), \\( 12^{12} \\), and \\( k \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 24532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number for which there exist that many natural numbers such that the sum of the squares of their squares is equal to $ 1998. $ *Gheorghe Iurea* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 24533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a cyclic quadrilateral where the side lengths are distinct primes less than $20$ with $AB\\cdot BC + CD\\cdot DA = 280$. What is the largest possible value of $BD$?\nA) 17 \nB) $\\sqrt{310}$ \nC) $\\sqrt{302.76}$ \nD) $\\sqrt{285}$ Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{302.76}"
} | {
"split": "train",
"index": 24534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$A B C$ is a right triangle with $\\angle A=30^{\\circ}$ and circumcircle $O$. Circles $\\omega_{1}, \\omega_{2}$, and $\\omega_{3}$ lie outside $A B C$ and are tangent to $O$ at $T_{1}, T_{2}$, and $T_{3}$ respectively and to $A B, B C$, and $C A$ at $S_{1}, S_{2}$, and $S_{3}... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}+1}{2}"
} | {
"split": "train",
"index": 24535,
"solution": "Let $[P Q R]$ denote the area of $\\triangle P Q R$. The key to this problem is following fact: $[P Q R]=\\frac{1}{2} P Q \\cdot P R \\sin \\angle Q P R$. Assume that the radius of $O$ is 1. Since $\\angle A=30^{\\circ}$, we have $B C=1$ and $A B=\\sqrt{3}$. So $[A... |
deepscaler | [
{
"role": "user",
"content": "Find the distance between the foci of the hyperbola\n\\[\\frac{y^2}{18} - \\frac{x^2}{2} = 1.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{5}"
} | {
"split": "train",
"index": 24536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence \\(\\{a_n\\}\\) with the sum of its first \\(n\\) terms denoted by \\(S_n\\), let \\(T_n = \\frac{S_1 + S_2 + \\cdots + S_n}{n}\\). \\(T_n\\) is called the \"mean\" of the sequence \\(a_1, a_2, \\cdots, a_n\\). It is known that the \"mean\" of the sequence \\... | math | {
"style": "rule",
"ground_truth": "2009"
} | {
"split": "train",
"index": 24537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b$ be real numbers. If the complex number $\\frac{1+2i}{a+bi} \\= 1+i$, then $a=\\_\\_\\_\\_$ and $b=\\_\\_\\_\\_$. 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": 24538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $c$: $$\\sqrt{4+\\sqrt{8+4c}}+ \\sqrt{2+\\sqrt{2+c}} = 2+2\\sqrt{2}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 24539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n\\ge3$, $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outs... | math | {
"style": "rule",
"ground_truth": "761"
} | {
"split": "train",
"index": 24540,
"solution": "To solve the problem, we need to understand the pattern of how diamonds are added in each figure $F_n$.\n\n1. **Observing the pattern:**\n - $F_1$ has 1 diamond.\n - $F_2$ surrounds $F_1$ with a square and adds 4 diamonds (one on each side).\n - $F_3$ surroun... |
deepscaler | [
{
"role": "user",
"content": "Stacy has $d$ dollars. She enters a mall with 10 shops and a lottery stall. First she goes to the lottery and her money is doubled, then she goes into the first shop and spends 1024 dollars. After that she alternates playing the lottery and getting her money doubled (Stacy alwa... | math | {
"style": "rule",
"ground_truth": "1023"
} | {
"split": "train",
"index": 24541,
"solution": "Work backwards. Before going into the last shop she had $\\$ 1024$, before the lottery she had $\\$ 512$, then $\\$ 1536, \\$ 768, \\ldots$. We can easily prove by induction that if she ran out of money after $n$ shops, $0 \\leq n \\leq 10$, she must have started w... |
deepscaler | [
{
"role": "user",
"content": "The increasing sequence $1,3,4,9,10,12,13\\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $100^{\\mbox{th}}$ term of this sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "981"
} | {
"split": "train",
"index": 24542,
"solution": "Notice that the first term of the sequence is $1$, the second is $3$, the fourth is $9$, and so on. Thus the $64th$ term of the sequence is $729$. Now out of $64$ terms which are of the form $729$ + $'''S'''$, $32$ of them include $243$ and $32$ do not. The smalles... |
deepscaler | [
{
"role": "user",
"content": "A box contains 5 white balls and 3 black balls. What is the probability that when drawing the balls one at a time, all draws alternate in color starting with a white ball? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{56}"
} | {
"split": "train",
"index": 24543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle is inscribed in a triangle with side lengths $9, 12,$ and $15$. Let the segments of the side of length $9$, made by a point of tangency, be $u$ and $v$, with $u<v$. Find the ratio $u:v$. 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": 24544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest possible value of $x+y$ where $x, y \\geq 1$ and $x$ and $y$ are integers that satisfy $x^{2}-29y^{2}=1$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11621"
} | {
"split": "train",
"index": 24545,
"solution": "Continued fraction convergents to $\\sqrt{29}$ are $5, \\frac{11}{2}, \\frac{16}{3}, \\frac{27}{5}, \\frac{70}{13}$ and you get $70^{2}-29 \\cdot 13^{2}=-1$ so since $(70+13\\sqrt{29})^{2}=9801+1820\\sqrt{29}$ the answer is $9801+1820=11621$"
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon $A B C D E F$ has side length 1 and center $O$. Parabolas $P_{1}, P_{2}, \\ldots, P_{6}$ are constructed with common focus $O$ and directrices $A B, B C, C D, D E, E F, F A$ respectively. Let $\\chi$ be the set of all distinct points on the plane that lie on a... | math | {
"style": "rule",
"ground_truth": "35 \\sqrt{3}"
} | {
"split": "train",
"index": 24546,
"solution": "Recall the focus and the directrix are such that the parabola is the locus of points equidistant from the focus and the directrix. We will consider pairs of parabolas and find their points of intersections (we label counterclockwise): (1): $P_{1} \\cap P_{2}$, two ... |
deepscaler | [
{
"role": "user",
"content": "The number of ounces of water needed to reduce $9$ ounces of shaving lotion containing $50$ % alcohol to a lotion containing $30$ % alcohol is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 24547,
"solution": "1. **Identify the amount of alcohol in the original lotion**: \n The original shaving lotion is 9 ounces with 50% alcohol. Therefore, the amount of alcohol in the lotion is:\n \\[\n \\frac{50}{100} \\times 9 = \\frac{9}{2} \\text{ ounces}\n \\]\n\n2. **Set... |
deepscaler | [
{
"role": "user",
"content": "For any positive integer $n$, we define the integer $P(n)$ by :\n\n$P(n)=n(n+1)(2n+1)(3n+1)...(16n+1)$.\n\nFind the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3),...,P(2016)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "510510"
} | {
"split": "train",
"index": 24548,
"solution": "\nTo find the greatest common divisor (GCD) of the integers \\( P(1), P(2), P(3), \\ldots, P(2016) \\), where \\( P(n) = n(n+1)(2n+1)(3n+1)\\cdots(16n+1) \\), we will first consider each part of the product and determine if there is a consistent factor across all \... |
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