data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In right triangle $DEF$ with $\\angle D = 90^\\circ$, we have $DE = 8$ and $EF = 17$. Find $\\cos F$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{17}"
} | {
"split": "train",
"index": 22949,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles touch each other at a common point $A$. Through point $B$, which lies on their common tangent passing through $A$, two secants are drawn. One secant intersects the first circle at points $P$ and $Q$, and the other secant intersects the second circle at points $M$ an... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 22950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $x$, $y$ satisfying $x > y > 0$, and $x + y \\leqslant 2$, the minimum value of $\\dfrac{2}{x+3y}+\\dfrac{1}{x-y}$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {3+2 \\sqrt {2}}{4}"
} | {
"split": "train",
"index": 22951,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nA firecracker was thrown vertically upwards with a speed of 20 m/s. Three seconds after the start of its flight, it exploded into two unequal parts, the mass ratio of which is $1: 2$. The smaller fragment immediately after the explosion flew horizontally at a speed of $16 \\m... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 22952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As usual, let \\( n! \\) denote the product of the integers from 1 to \\( n \\) inclusive. Determine the largest integer \\( m \\) such that \\( m! \\) divides \\( 100! + 99! + 98! \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 22953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α$ is an obtuse angle, if $\\sin (α- \\frac{3π}{4})= \\frac{3}{5}$, find $\\cos (α+ \\frac{π}{4})=\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{4}{5}"
} | {
"split": "train",
"index": 22954,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $R$ be the region in the Cartesian plane of points $(x, y)$ satisfying $x \\geq 0, y \\geq 0$, and $x+y+\\lfloor x\\rfloor+\\lfloor y\\rfloor \\leq 5$. Determine the area of $R$. 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": 22955,
"solution": "We claim that a point in the first quadrant satisfies the desired property if the point is below the line $x+y=3$ and does not satisfy the desired property if it is above the line. To see this, for a point inside the region, $x+y<3$ and $\\lfloor x\\rfloor+\\lfloor... |
deepscaler | [
{
"role": "user",
"content": "How many ways can we arrange 4 math books, 6 English books, and 2 Science books on a shelf if:\n1. All books of the same subject must stay together.\n2. The Science books can be placed in any order, but cannot be placed next to each other.\n(The math, English, and Science books... | math | {
"style": "rule",
"ground_truth": "207360"
} | {
"split": "train",
"index": 22956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $ABCD$ with $AD\\parallel BC$ , $AB=6$ , $AD=9$ , and $BD=12$ . If $\\angle ABD=\\angle DCB$ , find the perimeter of the trapezoid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 22957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right trapezoid \\(ABCD\\), the sum of the lengths of the bases \\(AD\\) and \\(BC\\) is equal to its height \\(AB\\). In what ratio does the angle bisector of \\(\\angle ABC\\) divide the lateral side \\(CD\\)? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "1:1"
} | {
"split": "train",
"index": 22958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to place $3$ indistinguishable red chips, $3$ indistinguishable blue chips, and $3$ indistinguishable green chips in the squares of a $3 \\times 3$ grid so that no two chips of the same color are directly adjacent to each other, either vertically or hori... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 22959,
"solution": "We are tasked with placing $3$ indistinguishable red chips, $3$ indistinguishable blue chips, and $3$ indistinguishable green chips in a $3 \\times 3$ grid such that no two chips of the same color are directly adjacent either vertically or horizontally.\n\n#### Ste... |
deepscaler | [
{
"role": "user",
"content": "Kolya started playing WoW when the hour and minute hands were opposite each other. He finished playing after a whole number of minutes, at which point the minute hand coincided with the hour hand. How long did he play (assuming he played for less than 12 hours)? Let's think ste... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 22960,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line passing through point $P(-2,2)$ intersects the hyperbola $x^2-2y^2=8$ such that the midpoint of the chord $MN$ is exactly at $P$. Find the length of $|MN|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{30}"
} | {
"split": "train",
"index": 22961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos x(\\sin x+\\cos x)-\\frac{1}{2}$.\n(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;\n(2) Find the maximum and minimum values of $f(x)$ on the interval $[-\\frac{\\pi}{4},\\frac{\\pi}{2}]$. Let... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 22962,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first 9 positive odd integers are placed in the magic square so that the sum of the numbers in each row, column and diagonal are equal. Find the value of $A + E$ .\n\n\\[ \\begin{tabular}{|c|c|c|}\\hline A & 1 & B \\hline 5 & C & 13 \\hline D & E & 3 \\hline\\end{tabular... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 22963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $3, 8, 13, a, b, 33$ is arithmetic. What is the sum of values $a$ and $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 22964,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 22965,
"solution": "1. **Determine the relative speed of Bob with respect to John**: Since both are moving east, we subtract John's speed from Bob's speed to find the rate at which the distance between them is closing.\n \\[\n \\text{Relative speed} = 5 \\text{ mph (Bob's speed)} ... |
deepscaler | [
{
"role": "user",
"content": "Find the number of solutions to\n\\[\\sin x = \\left( \\frac{1}{3} \\right)^x\\] on the interval $(0,200 \\pi).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 22966,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A_{1}, A_{2}, \\ldots, A_{m}$ be finite sets of size 2012 and let $B_{1}, B_{2}, \\ldots, B_{m}$ be finite sets of size 2013 such that $A_{i} \\cap B_{j}=\\emptyset$ if and only if $i=j$. Find the maximum value of $m$. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\binom{4025}{2012}"
} | {
"split": "train",
"index": 22967,
"solution": "In general, we will show that if each of the sets $A_{i}$ contain $a$ elements and if each of the sets $B_{j}$ contain $b$ elements, then the maximum value for $m$ is $\\binom{a+b}{a}$. Let $U$ denote the union of all the sets $A_{i}$ and $B_{j}$ and let $|U|=n$. C... |
deepscaler | [
{
"role": "user",
"content": "An ant has one sock and one shoe for each of its six legs, and on one specific leg, both the sock and shoe must be put on last. Find the number of different orders in which the ant can put on its socks and shoes. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "10!"
} | {
"split": "train",
"index": 22968,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three fair, six-sided dice are rolled. What is the probability that the sum of the three numbers showing is less than 16? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{103}{108}"
} | {
"split": "train",
"index": 22969,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $A B=5, B C=8, C A=11$. The incircle $\\omega$ and $A$-excircle $^{1} \\Gamma$ are centered at $I_{1}$ and $I_{2}$, respectively, and are tangent to $B C$ at $D_{1}$ and $D_{2}$, respectively. Find the ratio of the area of $\\triangle A I_{1} D_{1... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 22970,
"solution": "Let $D_{1}^{\\prime}$ and $D_{2}^{\\prime}$ be the points diametrically opposite $D_{1}$ and $D_{2}$ on the incircle and $A$-excircle, respectively. As $I_{x}$ is the midpoint of $D_{x}$ and $D_{x}^{\\prime}$, we have $$\\frac{\\left[A I_{1} D_{1}\\right]}{\\left[A... |
deepscaler | [
{
"role": "user",
"content": "In quadrilateral $EFGH$, $m\\angle F = 100^\\circ, m\\angle G = 140^\\circ$, $EF=6, FG=5,$ and $GH=7$. Calculate the area of $EFGH$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26.02"
} | {
"split": "train",
"index": 22971,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( l \\) and \\( m \\) be two skew lines. On line \\( l \\), there are three points \\( A, B, C \\) such that \\( AB = BC \\). Perpendiculars \\( AD, BE, CF \\) are drawn from points \\( A, B, \\) and \\( C \\) to line \\( m \\) respectively, with feet of perpendiculars at... | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"split": "train",
"index": 22972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Gnollish language consists of 3 words, ``splargh,'' ``glumph,'' and ``amr.'' In a sentence, ``splargh'' cannot come directly before ``glumph''; all other sentences are grammatically correct (including sentences with repeated words). How many valid 3-word sentences are the... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 22973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Write the number in the form of a fraction (if possible):\n\n$$\nx=0.5123412341234123412341234123412341234 \\ldots\n$$\n\nCan you generalize this method to all real numbers with a periodic decimal expansion? And conversely? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{51229}{99990}"
} | {
"split": "train",
"index": 22974,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four of the eight vertices of a cube are the vertices of a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 22975,
"solution": "1. **Assume the side length of the cube**: Let the side length of the cube be $s = 1$.\n\n2. **Identify the vertices of the tetrahedron**: Four vertices of a cube that form a regular tetrahedron can be chosen such that they are not all on the same face and no three... |
deepscaler | [
{
"role": "user",
"content": "Penta chooses 5 of the vertices of a unit cube. What is the maximum possible volume of the figure whose vertices are the 5 chosen points? 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": 22976,
"solution": "Label the vertices of the cube $A, B, C, D, E, F, G, H$, such that $A B C D$ is the top face of the cube, $E$ is directly below $A, F$ is directly below $B, G$ is directly below $C$, and $H$ is directly below $D$. We can obtain a volume of $\\frac{1}{2}$ by taking ... |
deepscaler | [
{
"role": "user",
"content": "If $10$ divides the number $1\\cdot2^1+2\\cdot2^2+3\\cdot2^3+\\dots+n\\cdot2^n$ , what is the least integer $n\\geq 2012$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2014"
} | {
"split": "train",
"index": 22977,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of the magnitudes of two angles of a triangle is 2, and the difference in lengths of the sides opposite these angles is 2 cm; the length of the third side of the triangle is 5 cm. Calculate the area of the triangle. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\frac{15 \\sqrt{7}}{4}"
} | {
"split": "train",
"index": 22978,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ be a function defined on $R$ such that $f(x+3) + f(x+1) = f(2) = 1$. Find $\\sum_{k=1}^{2023} f(k) =$ ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1012"
} | {
"split": "train",
"index": 22979,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)=\\cos x\\cdot\\ln x$, $f(x_{0})=f(x_{1})=0(x_{0}\\neq x_{1})$, find the minimum value of $|x_{0}-x_{1}|$ ___. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {\\pi}{2}-1"
} | {
"split": "train",
"index": 22980,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $a$ and $b$ are the points obtained by throwing a dice in order, and the function is $f(x)=\\frac{1}{2}ax^{2}+bx+1$.\n\n(1) Find the probability that $f(x)$ is a decreasing function in the interval $(-\\infty,-1]$;\n\n(2) Find the probability that the function $f(x)$ ha... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 22981,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A complex quartic polynomial $Q$ is quirky if it has four distinct roots, one of which is the sum of the other three. There are four complex values of $k$ for which the polynomial $Q(x)=x^{4}-k x^{3}-x^{2}-x-45$ is quirky. Compute the product of these four values of $k$. Let's ... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 22982,
"solution": "Let the roots be $a, b, c, d$ with $a+b+c=d$. Since $a+b+c=k-d$ by Vieta's formulas, we have $d=k / 2$. Hence $$0=P\\left(\\frac{k}{2}\\right)=\\left(\\frac{k}{2}\\right)^{4}-k\\left(\\frac{k}{2}\\right)^{3}-\\left(\\frac{k}{2}\\right)^{2}-\\left(\\frac{k}{2}\\righ... |
deepscaler | [
{
"role": "user",
"content": "Let the function $f(x)= \\frac{ \\sqrt{3}}{2}- \\sqrt{3}\\sin^2 \\omega x-\\sin \\omega x\\cos \\omega x$ ($\\omega > 0$) and the graph of $y=f(x)$ has a symmetry center whose distance to the nearest axis of symmetry is $\\frac{\\pi}{4}$.\n\n$(1)$ Find the value of $\\omega$; $... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 22983,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the mean score of the students in the first section is 92 and the mean score of the students in the second section is 78, and the ratio of the number of students in the first section to the number of students in the second section is 5:7, calculate the combined mean ... | math | {
"style": "rule",
"ground_truth": "\\frac{1006}{12}"
} | {
"split": "train",
"index": 22984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sides of a right triangle are $a$ and $b$ and the hypotenuse is $c$. A perpendicular from the vertex divides $c$ into segments $r$ and $s$, adjacent respectively to $a$ and $b$. If $a : b = 1 : 3$, then the ratio of $r$ to $s$ is: Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "1 : 9"
} | {
"split": "train",
"index": 22985,
"solution": "Given a right triangle with sides $a$, $b$, and hypotenuse $c$, where $a:b = 1:3$. Let's denote $a = k$ and $b = 3k$ for some positive constant $k$. By the Pythagorean theorem, we have:\n\\[ c^2 = a^2 + b^2 = k^2 + (3k)^2 = k^2 + 9k^2 = 10k^2 \\]\nThus, the hypoten... |
deepscaler | [
{
"role": "user",
"content": "When $x = -1$, the value of the algebraic expression $2ax^{3} - 3bx + 8$ is $18$. Express the value of the algebraic expression $9b - 6a + 2$ in terms of the given information. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 22986,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the $8$ vertices of a cube, select $4$ vertices. The probability that these $4$ vertices lie in the same plane is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{35}"
} | {
"split": "train",
"index": 22987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the sum of all real numbers $x$ that are not in the domain of the function $$g(x) = \\frac{1}{2+\\frac{1}{3+\\frac{1}{x}}}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{13}{21}"
} | {
"split": "train",
"index": 22988,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x)=2x^3+4$, find $f^{-1}(58)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 22989,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each ordered pair of real numbers $(x,y)$ satisfying \\[\\log_2(2x+y) = \\log_4(x^2+xy+7y^2)\\]there is a real number $K$ such that \\[\\log_3(3x+y) = \\log_9(3x^2+4xy+Ky^2).\\]Find the product of all possible values of $K$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "189"
} | {
"split": "train",
"index": 22990,
"solution": "Using the logarithmic property $\\log_{a^n}b^n = \\log_{a}b$, we note that \\[(2x+y)^2 = x^2+xy+7y^2\\] That gives \\[x^2+xy-2y^2=0\\] upon simplification and division by $3$. Factoring $x^2+xy-2y^2=0$ by Simon's Favorite Factoring Trick gives \\[(x+2y)(x-y)=0\\] T... |
deepscaler | [
{
"role": "user",
"content": "Calculate the lengths of the arcs of the curves given by the equations in polar coordinates.\n\n$$\n\\rho = 2 \\varphi, \\; 0 \\leq \\varphi \\leq \\frac{4}{3}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{20}{9} + \\ln 3"
} | {
"split": "train",
"index": 22991,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m, n, p$ be fixed positive real numbers which satisfy $mnp = 8$. Depending on these constants, find the minimum of $$x^2+y^2+z^2+ mxy + nxz + pyz,$$\nwhere $x, y, z$ are arbitrary positive real numbers satisfying $xyz = 8$. When is the equality attained?\nSolve the problem... | math | {
"style": "rule",
"ground_truth": "z=\\sqrt[3]{4m}"
} | {
"split": "train",
"index": 22992,
"solution": "\nTo minimize the expression \\( x^2 + y^2 + z^2 + mxy + nxz + pyz \\) with the constraint \\( xyz = 8 \\), we will follow a systematic approach rooted in mathematical optimization techniques.\n\n### Case \\( (a) \\): \\( m = n = p = 2 \\)\n\n1. **Substitute for \\... |
deepscaler | [
{
"role": "user",
"content": "In a regular quadrilateral prism $ABCDA'A'B'C'D'$ with vertices on the same sphere, $AB = 1$ and $AA' = \\sqrt{2}$, calculate the spherical distance between points $A$ and $C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 22993,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A telephone number has the form \\text{ABC-DEF-GHIJ}, where each letter represents\na different digit. The digits in each part of the number are in decreasing\norder; that is, $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore,\n$D$, $E$, and $F$ are consecutive even di... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 22994,
"solution": "1. **Identify the constraints**: The telephone number is in the form $\\text{ABC-DEF-GHIJ}$ where each segment has digits in decreasing order. Additionally, $D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, and $J$ are consecutive odd digits; and $A + B... |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\{a_n\\}$ is an arithmetic sequence with a common difference greater than $0$, and it satisfies $a_1+a_5=4$, $a_2a_4=-5$, calculate the sum of the first $10$ terms of the sequence $\\{a_n\\}$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "95"
} | {
"split": "train",
"index": 22995,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let us call a number \\( \\mathrm{X} \\) \"50-podpyirayushchim\" if for any 50 real numbers \\( a_{1}, \\ldots, a_{50} \\) whose sum is an integer, there exists at least one \\( a_i \\) such that \\( \\left|a_{i}-\\frac{1}{2}\\right| \\geq X \\).\n\nFind the greatest 50-podpyir... | math | {
"style": "rule",
"ground_truth": "0.01"
} | {
"split": "train",
"index": 22996,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A stock investment increased by $15\\%$ in the first year. At the start of the next year, by what percent must the stock now decrease to return to its original price at the beginning of the first year? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "13.04\\%"
} | {
"split": "train",
"index": 22997,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vasya, Petya, and Kolya are in the same class. Vasya always lies in response to any question, Petya alternates between lying and telling the truth, and Kolya lies in response to every third question but tells the truth otherwise. One day, each of them was asked six consecutive ... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 22998,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 22999,
"solution": "The AMC 8 scoring system is structured as follows:\n- Each correct answer awards 1 point.\n- Each incorrect answer awards 0 points.\n- Questions left unanswered also award 0 points.\n\nGiven the problem statement:\n- Billy answers 13 questions correctly.\n- Billy a... |
deepscaler | [
{
"role": "user",
"content": "Set $u_0 = \\frac{1}{4}$, and for $k \\ge 0$ let $u_{k+1}$ be determined by the recurrence\n\\[u_{k+1} = 2u_k - 2u_k^2.\\]This sequence tends to a limit; call it $L$. What is the least value of $k$ such that\n\\[|u_k-L| \\le \\frac{1}{2^{1000}}?\\] Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 23000,
"solution": "1. **Define the sequence and transformation**: Given the recurrence relation:\n \\[\n u_{k+1} = 2u_k - 2u_k^2\n \\]\n We start with $u_0 = \\frac{1}{4}$. To simplify the recurrence relation, we perform a transformation by defining $v_k = 2u_k - 1$. Substitu... |
deepscaler | [
{
"role": "user",
"content": "How many integers $-11 \\leq n \\leq 11$ satisfy $(n-2)(n+4)(n + 8)<0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 23001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $A+B$ (in base 10), given the following addition problem \\[ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c}& & & 4 & A & B_{6}\\\\ &+& & & 4 & 1_{6}\\\\ \\cline{2-6}& & & 5 & 3 & A_{6}\\\\ \\end{array} \\] Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 23002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Milly has twelve socks, three of each color: red, blue, green, and yellow. She randomly draws five socks. What is the probability that she has exactly two pairs of socks of different colors and one sock of a third color? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{22}"
} | {
"split": "train",
"index": 23003,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Our basketball team has 12 members, each of whom can play any position. In how many ways can we choose a starting lineup consisting of a center, a power forward, a shooting forward, a point guard, and a shooting guard? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "95,\\!040"
} | {
"split": "train",
"index": 23004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A two-digit positive integer $x$ has the property that when 109 is divided by $x$, the remainder is 4. What is the sum of all such two-digit positive integers $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 23005,
"solution": "Suppose that the quotient of the division of 109 by $x$ is $q$. Since the remainder is 4, this is equivalent to $109=q x+4$ or $q x=105$. Put another way, $x$ must be a positive integer divisor of 105. Since $105=5 \times 21=5 \times 3 \times 7$, its positive integ... |
deepscaler | [
{
"role": "user",
"content": "If the least common multiple of two 6-digit integers has 10 digits, then their greatest common divisor has at most how many digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 23006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^{2}}{8} + \\frac{y^{2}}{2} = 1$, and a point $A(2,1)$ on the ellipse. The slopes of the lines $AB$ and $AC$ connecting point $A$ to two moving points $B$ and $C$ on the ellipse are $k_{1}$ and $k_{2}$, respectively, with $k_{1} + k_{2} = 0$. Determine... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 23007,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average of 15, 30, $x$, and $y$ is 25. What are the values of $x$ and $y$ if $x = y + 10$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"split": "train",
"index": 23008,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what values of the constant $c$ does the graph of $f(x) = \\frac{x^2-x+c}{x^2+x-6}$ have exactly one vertical asymptote?\n\nEnter the possible values of $c,$ separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2 \\text{ or } -12"
} | {
"split": "train",
"index": 23009,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Last year 100 adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was 4. What was the total number of cats and kittens received by ... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 23010,
"solution": "1. **Determine the number of female cats:**\n Given that there are 100 adult cats and half of them are female, we calculate the number of female cats as follows:\n \\[\n \\frac{100}{2} = 50 \\text{ female cats}\n \\]\n\n2. **Calculate the number of litters:... |
deepscaler | [
{
"role": "user",
"content": "Given that the function $y=f(x)$ is an odd function defined on $R$, when $x\\leqslant 0$, $f(x)=2x+x^{2}$. If there exist positive numbers $a$ and $b$ such that when $x\\in[a,b]$, the range of $f(x)$ is $[\\frac{1}{b}, \\frac{1}{a}]$, find the value of $a+b$. Let's think step b... | math | {
"style": "rule",
"ground_truth": "\\frac{3+ \\sqrt{5}}{2}"
} | {
"split": "train",
"index": 23011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $A$ and $C$ coincide, forming the pentagon $ABEFD$. What is the length of segment $EF$? Express your answer in simplest radical form. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 23012,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The measure of angle $ACB$ is 45 degrees. If ray $CA$ is rotated 510 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 23013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many arithmetic sequences, where the common difference is a natural number greater than 2, satisfy the conditions that the first term is 1783, the last term is 1993, and the number of terms is at least 3? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 23014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C: \\frac{y^{2}}{a^{2}}+ \\frac{x^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\\frac{\\sqrt{2}}{2}$ and the sum of the distances from a point on the ellipse to the two foci is $2\\sqrt{2}$. A line $l$ with slope $k(k\\neq 0)$ passes through the upper fo... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{6}}{16}"
} | {
"split": "train",
"index": 23015,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all angles $x \\in [0^\\circ, 360^\\circ]$ that satisfy\n\\[\\sin^5 x - \\cos^5 x = \\frac{1}{\\cos x} - \\frac{1}{\\sin x}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "270^\\circ"
} | {
"split": "train",
"index": 23016,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a sequence, all natural numbers from 1 to 2017 inclusive were written down. How many times was the digit 7 written? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "602"
} | {
"split": "train",
"index": 23017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pablo has 27 solid $1 \\times 1 \\times 1$ cubes that he assembles in a larger $3 \\times 3 \\times 3$ cube. If 10 of the smaller cubes are red, 9 are blue, and 8 are yellow, what is the smallest possible surface area of the larger cube that is red? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 23018,
"solution": "The 27 small cubes that make up the larger $3 \\times 3 \\times 3$ can be broken into 4 categories: 1 small cube in the very centre of the larger cube (not seen in the diagram), 8 small cubes at the vertices of larger cube (an example is marked with $V$), 12 small ... |
deepscaler | [
{
"role": "user",
"content": "Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next $365$-day period will... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 23019,
"solution": "1. **Identify the Periodicity**: \n Alice, Beatrix, and Claire visit every 3, 4, and 5 days respectively. The least common multiple (LCM) of 3, 4, and 5 is:\n \\[\n \\text{LCM}(3, 4, 5) = 60\n \\]\n This means every 60 days, all three friends visit Daphne... |
deepscaler | [
{
"role": "user",
"content": "A club has between 150 and 250 members. Every month, all the members meet up for a group activity that requires the members to be divided into seven distinct groups. If one member is unable to attend, the remaining members can still be evenly divided into the seven groups. Calc... | math | {
"style": "rule",
"ground_truth": "2807"
} | {
"split": "train",
"index": 23020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the parabola defined by the equation $y=(x-2)^2+3$ is rotated 180 degrees about its vertex, then shifted 3 units to the left, then shifted 2 units down. The resulting parabola has zeros at $x=a$ and $x=b$. What is $a+b$? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 23021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the difference between $1234_5$ and $432_5$ in base $5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "302_5"
} | {
"split": "train",
"index": 23022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two non-zero vectors $\\overrightarrow{m}$ and $\\overrightarrow{n}$ with an angle of $\\frac{\\pi}{3}$ between them, and the magnitude of $\\overrightarrow{n}$ is a positive scalar multiple of the magnitude of $\\overrightarrow{m}$, i.e., $|\\overrightarrow{n}| = λ|\\ove... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"split": "train",
"index": 23023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_{n}\\}$ such that $a_{1}+2a_{2}+\\cdots +na_{n}=n$, and a sequence $\\{b_{n}\\}$ such that ${b_{m-1}}+{b_m}=\\frac{1}{{{a_m}}}({m∈N,m≥2})$. Find:<br/>\n$(1)$ The general formula for $\\{a_{n}\\}$;<br/>\n$(2)$ The sum of the first $20$ terms of $\\{b_{n}\\... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 23024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fenced, rectangular field measures $24$ meters by $52$ meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the square... | math | {
"style": "rule",
"ground_truth": "702"
} | {
"split": "train",
"index": 23025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the $2001$ students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between $80$ percent and $85$ percent of the school population, and the number who study French is between $30$ percent and $40$ percent. Let $m$ ... | math | {
"style": "rule",
"ground_truth": "298"
} | {
"split": "train",
"index": 23026,
"solution": "Let $S$ be the percent of people who study Spanish, $F$ be the number of people who study French, and let $S \\cup F$ be the number of students who study both. Then $\\left\\lceil 80\\% \\cdot 2001 \\right\\rceil = 1601 \\le S \\le \\left\\lfloor 85\\% \\cdot 2001 ... |
deepscaler | [
{
"role": "user",
"content": "Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their place to the spectator (there were no draws). As a result, Egor participated in 13 games, and Nikita participated in 27 games. How many games ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 23027,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let $N$ be the original number of the room, and let $M$ be the ro... | math | {
"style": "rule",
"ground_truth": "2031"
} | {
"split": "train",
"index": 23028,
"solution": "Let $A$ represent the portion of $N$ to the right of the deleted zero, and $B$ represent the rest of $N$. For example, if the unique zero in $N=12034$ is removed, then $A=34$ and $B=12000$. Then, $\\frac{M}{N}=\\frac{A+B / 10}{A+B}=1-\\frac{9}{10} \\frac{B}{N}$. Th... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $2^{0^{1^9}} + (2^0)^{1^9}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 23029,
"solution": "To solve the expression \\[2^{\\left(0^{\\left(1^9\\right)}\\right)}+\\left(\\left(2^0\\right)^1\\right)^9,\\] we need to evaluate each term separately.\n\n1. **Evaluate the first term: $2^{\\left(0^{\\left(1^9\\right)}\\right)}$**\n - First, simplify the exponen... |
deepscaler | [
{
"role": "user",
"content": "Twenty-eight 4-inch wide square posts are evenly spaced with 4 feet between adjacent posts to enclose a rectangular field. The rectangle has 6 posts on each of the longer sides (including the corners). What is the outer perimeter, in feet, of the fence? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 23030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sqrt{(31)(30)(29)(28)+1}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "869"
} | {
"split": "train",
"index": 23031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the textbook, students were once asked to explore the coordinates of the midpoint of a line segment: In a plane Cartesian coordinate system, given two points $A(x_{1}, y_{1})$ and $B(x_{2}, y_{2})$, the midpoint of the line segment $AB$ is $M$, then the coordinates of $M$ ar... | math | {
"style": "rule",
"ground_truth": "4 \\text{ or } 0"
} | {
"split": "train",
"index": 23032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many divisors of \\(88^{10}\\) leave a remainder of 4 when divided by 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 23033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$, and $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$, where $a,b,c,d$ are distinct real numbers, find $a+b+c+d$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 23034,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Equilateral triangles $A B F$ and $B C G$ are constructed outside regular pentagon $A B C D E$. Compute $\\angle F E G$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48^{\\circ}"
} | {
"split": "train",
"index": 23035,
"solution": "We have $\\angle F E G=\\angle A E G-\\angle A E F$. Since $E G$ bisects $\\angle A E D$, we get $\\angle A E G=54^{\\circ}$. Now, $\\angle E A F=108^{\\circ}+60^{\\circ}=168^{\\circ}$. Since triangle $E A F$ is isosceles, this means $\\angle A E F=6^{\\circ}$, so ... |
deepscaler | [
{
"role": "user",
"content": "Arrange the sequence $\\{2n+1\\}$ ($n\\in\\mathbb{N}^*$), sequentially in brackets such that the first bracket contains one number, the second bracket two numbers, the third bracket three numbers, the fourth bracket four numbers, the fifth bracket one number, and so on in a cyc... | math | {
"style": "rule",
"ground_truth": "403"
} | {
"split": "train",
"index": 23036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $q(x) = x^5 - 4x^3 + 5$, then find the coefficient of the $x^3$ term in the polynomial $(q(x))^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 23037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eyes are the windows of the soul. In order to protect students' eyesight, Qihang High School conducts eye examinations for students every semester. The table below shows the results of the right eye vision examination for 39 students in a certain class at the school. In this se... | math | {
"style": "rule",
"ground_truth": "4.6"
} | {
"split": "train",
"index": 23038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Charlie and Daisy each arrive at a cafe at a random time between 1:00 PM and 3:00 PM. Each stays for 20 minutes. What is the probability that Charlie and Daisy are at the cafe at the same time? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 23039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $(x+1)^{42}$, what is the coefficient of the $x^2$ term? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "861"
} | {
"split": "train",
"index": 23040,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Line segment $\\overline{AB}$ has perpendicular bisector $\\overline{CD}$ , where $C$ is the midpoint of $\\overline{AB}$ . The segments have lengths $AB = 72$ and $CD = 60$ . Let $R$ be the set of points $P$ that are midpoints of line segments $\\overline{XY}$ ,... | math | {
"style": "rule",
"ground_truth": "1080"
} | {
"split": "train",
"index": 23041,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the length of the chord that the line given by the parametric equations\n$$\\begin{cases} x=1+ \\frac {4}{5}t \\\\ y=-1- \\frac {3}{5}t \\end{cases}$$\n(where t is the parameter) cuts off from the curve whose polar equation is $\\rho= \\sqrt {2}\\cos\\left(\\theta+ \\frac ... | math | {
"style": "rule",
"ground_truth": "\\frac {7}{5}"
} | {
"split": "train",
"index": 23042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a complex number $Z = x + yi$ ($x, y \\in \\mathbb{R}$) such that $|Z - 4i| = |Z + 2|$, find the minimum value of $2^x + 4^y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"split": "train",
"index": 23043,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Alex, Jamie, and Casey play a game over 8 rounds, and for each round, the probability Alex wins is $\\frac{1}{3}$, and Jamie is three times as likely to win as Casey, calculate the probability that Alex wins four rounds, Jamie wins three rounds, and Casey wins one round. ... | math | {
"style": "rule",
"ground_truth": "\\frac{35}{486}"
} | {
"split": "train",
"index": 23044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( m \\) be an integer greater than 1, and let's define a sequence \\( \\{a_{n}\\} \\) as follows:\n\\[\n\\begin{array}{l}\na_{0}=m, \\\\\na_{1}=\\varphi(m), \\\\\na_{2}=\\varphi^{(2)}(m)=\\varphi(\\varphi(m)), \\\\\n\\vdots \\\\\na_{n}=\\varphi^{(n)}(m)=\\varphi\\left(\\v... | math | {
"style": "rule",
"ground_truth": "1944"
} | {
"split": "train",
"index": 23045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the plane rectangular coordinate system $xOy$, the parameter equation of the line $l$ with an inclination angle $\\alpha = 60^{\\circ}$ is $\\left\\{\\begin{array}{l}{x=2+t\\cos\\alpha}\\\\{y=t\\sin\\alpha}\\end{array}\\right.$ (where $t$ is the parameter). Taking the coordi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{3}"
} | {
"split": "train",
"index": 23046,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each positive integer number $n \\ ge 1$ is assigned the number $p_n$ which is the product of all its non-zero digits. For example, $p_6 = 6$ , $p_ {32} = 6$ , $p_ {203} = 6$ . Let $S = p_1 + p_2 + p_3 + \\dots + p_ {999}$ . Find the largest prime that divides $S $ . L... | math | {
"style": "rule",
"ground_truth": "103"
} | {
"split": "train",
"index": 23047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct four-digit numbers are divisible by 5 and have 45 as their last two digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 23048,
"solution": ""
} |
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