data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given a $6 \\times 6$ square of $36$ square blocks, find the number of different combinations of $4$ blocks that can be selected so that no two are in the same row or column."
}
] | math | {
"style": "rule",
"ground_truth": "5400"
} | {
"index": 13100,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a $9 \\times 9$ chess board, we consider all the rectangles whose edges lie along grid lines (the board consists of 81 unit squares, and the grid lines lie on the borders of the unit squares). For each such rectangle, we put a mark in every one of the unit squares inside ... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 13101,
"solution": "56. Consider the rectangles which contain the square in the $i$th row and $j$th column. There are $i$ possible positions for the upper edge of such a rectangle, $10-i$ for the lower edge, $j$ for the left edge, and $10-j$ for the right edge; thus we have $i(10-i) j(10-j)$ rectangles a... |
DeepScaleR | [
{
"role": "user",
"content": "Given a 10cm×10cm×10cm cube cut into 1cm×1cm×1cm small cubes, determine the maximum number of small cubes that can be left unused when reassembling the small cubes into a larger hollow cube with no surface voids."
}
] | math | {
"style": "rule",
"ground_truth": "134"
} | {
"index": 13102,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a 12-hour digital clock with a glitch where every '2' is displayed as a '7', determine the fraction of the day that the clock shows the correct time."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{55}{72}"
} | {
"index": 13103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a 2x3 rectangle with six unit squares, the lower left corner at the origin, find the value of $c$ such that a slanted line extending from $(c,0)$ to $(4,4)$ divides the entire region into two regions of equal area."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 13104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a 3x3 matrix where each row and each column forms an arithmetic sequence, and the middle element $a_{22} = 5$, find the sum of all nine elements."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 13105,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a 4-inch cube constructed from 64 smaller 1-inch cubes, with 50 red and 14 white cubes, arrange these cubes such that the white surface area exposed on the larger cube is minimized, and calculate the fraction of the total surface area of the 4-inch cube that is white."
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"index": 13106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a 50-term sequence $(b_1, b_2, \\dots, b_{50})$, the Cesaro sum is 500. What is the Cesaro sum of the 51-term sequence $(2, b_1, b_2, \\dots, b_{50})$?"
}
] | math | {
"style": "rule",
"ground_truth": "492"
} | {
"index": 13107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a [rational number](https://artofproblemsolving.com/wiki/index.php/Rational_number), write it as a [fraction](https://artofproblemsolving.com/wiki/index.php/Fraction) in lowest terms and calculate the product of the resulting [numerator](https://artofproblemsolving.com/wi... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"index": 13108,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a bag contains 28 red balls, 20 green balls, 12 yellow balls, 20 blue balls, 10 white balls, and 10 black balls, determine the minimum number of balls that must be drawn to ensure that at least 15 balls of the same color are selected."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 13109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a bag with 1 red ball and 2 black balls of the same size, two balls are randomly drawn. Let $\\xi$ represent the number of red balls drawn. Calculate $E\\xi$ and $D\\xi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"index": 13110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a ball with a diameter of 6 inches rolling along the path consisting of four semicircular arcs, with radii $R_1 = 100$ inches, $R_2 = 60$ inches, $R_3 = 80$ inches, and $R_4 = 40$ inches, calculate the distance traveled by the center of the ball from the start to the end ... | math | {
"style": "rule",
"ground_truth": "280\\pi"
} | {
"index": 13111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a basketball player has a probability of $a$ for scoring 3 points in a shot, $b$ for scoring 2 points, and $c$ for not scoring any points, where $a, b, c \\in (0, 1)$, and the mathematical expectation for scoring points in one shot is 2, determine the minimum value of $\\... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{3}"
} | {
"index": 13112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a basketball player made 8 baskets during a game, each worth 1, 2, or 3 points, calculate the total number of different numbers that could represent the total points scored by the player."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 13113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a biased coin with probabilities of $\\frac{3}{4}$ for heads and $\\frac{1}{4}$ for tails, and outcomes of tosses being independent, calculate the probabilities of winning Game A and Game B."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 13114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a biased coin with probabilities of $\\frac{3}{4}$ for heads and $\\frac{1}{4}$ for tails, determine the difference between the probability of winning Game A, which involves 4 coin tosses and at least three heads, and the probability of winning Game B, which involves 5 co... | math | {
"style": "rule",
"ground_truth": "\\frac{89}{256}"
} | {
"index": 13115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a bicycle's front tire lasts for 5000km and the rear tire lasts for 3000km, determine the maximum distance the bicycle can travel if the tires are swapped reasonably during use."
}
] | math | {
"style": "rule",
"ground_truth": "3750"
} | {
"index": 13116,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a box containing $30$ red balls, $22$ green balls, $18$ yellow balls, $15$ blue balls, and $10$ black balls, determine the minimum number of balls that must be drawn from the box to guarantee that at least $12$ balls of a single color will be drawn."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 13117,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a box contains $4$ shiny pennies and $5$ dull pennies, determine the probability that the third shiny penny appears on the sixth draw."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{21}"
} | {
"index": 13118,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a box contains $5$ shiny pennies and $6$ dull pennies, determine the sum of the numerator and denominator of the probability that it will take exactly six draws to get the fourth shiny penny."
}
] | math | {
"style": "rule",
"ground_truth": "236"
} | {
"index": 13119,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a box contains a total of 180 marbles, 25% are silver, 20% are gold, 15% are bronze, 10% are sapphire, and 10% are ruby, and the remainder are diamond marbles. If 10% of the gold marbles are removed, calculate the number of marbles left in the box."
}
] | math | {
"style": "rule",
"ground_truth": "176"
} | {
"index": 13120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a certain DNA fragment consists of 500 base pairs, with A+T making up 34% of the total number of bases, calculate the total number of free cytosine deoxyribonucleotide molecules required when this DNA fragment is replicated twice."
}
] | math | {
"style": "rule",
"ground_truth": "1320"
} | {
"index": 13121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a certain sunflower cell has 34 chromosomes at the late stage of the second meiotic division when forming pollen grains, determine the number of tetrads that can be produced by this cell during meiosis."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 13122,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a checkerboard with 31 rows and 29 columns, where each corner square is black and the squares alternate between red and black, determine the number of black squares on this checkerboard."
}
] | math | {
"style": "rule",
"ground_truth": "465"
} | {
"index": 13123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $C$ passes through points $A(-1,0)$ and $B(3,0)$, and the center of the circle is on the line $x-y=0$.\n$(1)$ Find the equation of circle $C$;\n$(2)$ If point $P(x,y)$ is any point on circle $C$, find the maximum and minimum distance from point $P$ to the line $x... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}\\sqrt{5}"
} | {
"index": 13124,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $C$ with the equation $(x-a)^2 + (y-2)^2 = 4$ ($a > 0$) and a line $l$ with the equation $x - y + 3 = 0$, which intersects the circle at points $A$ and $B$:\n1. When the chord AB cut by line $l$ on circle $C$ has a length of $2\\sqrt{2}$, find the value of $a$.\n... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} - 1"
} | {
"index": 13125,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $C$: $(x+1)^{2}+y^{2}=r^{2}$ and a parabola $D$: $y^{2}=16x$ intersect at points $A$ and $B$, and $|AB|=8$, calculate the area of circle $C$."
}
] | math | {
"style": "rule",
"ground_truth": "25\\pi"
} | {
"index": 13126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $C$: $x^{2}+y^{2}-2x-2ay+a^{2}-24=0$ ($a\\in\\mathbb{R}$) whose center lies on the line $2x-y=0$.\n\n$(1)$ Find the value of the real number $a$; \n$(2)$ Find the minimum length of the chord formed by the intersection of circle $C$ and line $l$: $(2m+1)x+(m+1)y-... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{5}"
} | {
"index": 13127,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $C: (x-1)^2+(y-2)^2=25$, and a line $l: (2m+1)x+(m+1)y-7m-4=0$. If the chord intercepted by line $l$ on circle $C$ is the shortest, then the value of $m$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{4}"
} | {
"index": 13128,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $C: (x-1)^{2} + (y-2)^{2} = 25$ and a line $l: (2m+1)x + (m+1)y - 7m-4 = 0$, where $m \\in \\mathbb{R}$. Find the minimum value of the chord length $|AB|$ cut by line $l$ on circle $C$."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{5}"
} | {
"index": 13129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $C: (x-1)^{2} + (y-2)^{2} = 25$ and a line $l: mx-y-3m+1=0$ intersect at points $A$ and $B$. Find the minimum value of $|AB|$."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{5}"
} | {
"index": 13130,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $C: (x-3)^2 + (y-4)^2 = 25$, the shortest distance from a point on circle $C$ to line $l: 3x + 4y + m = 0 (m < 0)$ is $1$. If point $N(a, b)$ is located on the part of line $l$ in the first quadrant, find the minimum value of $\\frac{1}{a} + \\frac{1}{b}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7 + 4\\sqrt{3}}{55}"
} | {
"index": 13131,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $C: (x-3)^{2}+(y-4)^{2}=1$, and points $A(-1,0)$, $B(1,0)$, let $P$ be a moving point on the circle, then the maximum and minimum values of $d=|PA|^{2}+|PB|^{2}$ are \\_\\_\\_\\_\\_\\_ and \\_\\_\\_\\_\\_\\_ respectively."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 13132,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $C: (x-3)^{2}+y^{2}=25$ and a line $l: (m+1)x+(m-1)y-2=0$ (where $m$ is a parameter), the minimum length of the chord intercepted by the circle $C$ and the line $l$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{5}"
} | {
"index": 13133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $C: x^2+y^2-2x+4y-4=0$, and a line $l$ with a slope of 1 intersects the circle $C$ at points $A$ and $B$.\n\n(1) Express the equation of the circle in standard form, and identify the center and radius of the circle;\n\n(2) Does there exist a line $l$ such that th... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 13134,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $O$ with radius $1$, $PA$ and $PB$ are two tangents to the circle, and $A$ and $B$ are the points of tangency. The minimum value of $\\overrightarrow{PA} \\cdot \\overrightarrow{PB}$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "-3+2\\sqrt{2}"
} | {
"index": 13135,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $O: x^2 + y^2 = 1$ and a point $A(-2, 0)$, if there exists a fixed point $B(b, 0)$ ($b \\neq -2$) and a constant $\\lambda$ such that for any point $M$ on the circle $O$, it holds that $|MB| = \\lambda|MA|$. The maximum distance from point $P(b, \\lambda)$ to the... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{2}"
} | {
"index": 13136,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $O: x^2 + y^2 = 6$, and $P$ is a moving point on circle $O$. A perpendicular line $PM$ is drawn from $P$ to the x-axis at $M$, and $N$ is a point on $PM$ such that $\\overrightarrow{PM} = \\sqrt{2} \\overrightarrow{NM}$.\n\n(Ⅰ) Find the equation of the trajectory... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 13137,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $O: x^{2}+y^{2}=4$.<br/>$(1)$ A tangent line is drawn from point $P(2,1)$ to circle $O$, find the equation of the tangent line $l$;<br/>$(2)$ Let $A$ and $B$ be the points where circle $O$ intersects the positive $x$-axis and positive $y$-axis, respectively. A mo... | math | {
"style": "rule",
"ground_truth": "\\frac{8\\sqrt{5}}{5}"
} | {
"index": 13138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $x^2 + (y-1)^2 = 1$ with its tangent line $l$, which intersects the positive x-axis at point A and the positive y-axis at point B. Determine the y-intercept of the tangent line $l$ when the distance AB is minimized."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3+\\sqrt{5}}{2}"
} | {
"index": 13139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle $⊙O$ with equation $x^{2}+y^{2}=6$, let $P$ be a moving point on the circle. Draw a line $PM$ perpendicular to the x-axis at point $M$, and let $N$ be a point on $PM$ such that $\\overrightarrow{PM}= \\sqrt {2} \\overrightarrow{NM}$.\n\n(I) Find the equation for ... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 13140,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle C that passes through points A(1, 4) and B(3, -2), and the distance from the center of the circle C to line AB is $\\sqrt{10}$, find the equation of circle C."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 13141,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle C with its center C on the positive x-axis and a radius of 5, the chord intercepted by the line $x-y+3=0$ has a length of $2\\sqrt{17}$.\n(1) Find the equation of circle C;\n(2) Suppose the line $ax-y+5=0$ intersects circle C at points A and B, find the range of ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 13142,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle O with a radius of 6, the length of chord AB is 6.\n(1) Find the size of the central angle α corresponding to chord AB;\n(2) Find the arc length l and the area S of the sector where α is located."
}
] | math | {
"style": "rule",
"ground_truth": "6\\pi"
} | {
"index": 13143,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle \\\\(O: x^2 + y^2 = 2\\\\) and a line \\\\(l: y = kx - 2\\\\).\n\n\\\\((1)\\\\) If line \\\\(l\\\\) intersects circle \\\\(O\\\\) at two distinct points \\\\(A\\\\) and \\\\(B\\\\), and \\\\(\\angle AOB = \\frac{\\pi}{2}\\\\), find the value of \\\\(k\\\\).\n\n\\... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 13144,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle and $2006$ points lying on this circle. Albatross colors these $2006$ points in $17$ colors. After that, Frankinfueter joins some of the points by chords such that the endpoints of each chord have the same color and two different chords have no common point... | math | {
"style": "rule",
"ground_truth": "117"
} | {
"index": 13145,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle circumscribed around triangle \\(FDC\\), a tangent \\(FK\\) is drawn such that \\(\\angle KFC = 58^\\circ\\). Points \\(K\\) and \\(D\\) lie on opposite sides of line \\(FC\\) as shown in the diagram. Find the acute angle between the angle bisectors of \\(\\angle... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 13146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle described by the equation $x^{2}-2x+y^{2}-2y+1=0$, find the cosine value of the angle between the two tangent lines drawn from an external point $P(3,2)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 13147,
"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 $u:v$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 13148,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle of radius $2$, there are many line segments of length $2$ that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"index": 13149,
"solution": "1. **Identify the Geometry of the Problem:**\n Let $AB$ be a line segment of length $2$, tangent to a circle $O$ at its midpoint $P$. The radius of circle $O$, $OP$, is $2$. Since $P$ is the midpoint of $AB$, we have $AP = PB = 1$.\n\n2. **Use the Right Triangle Property:**\n Sinc... |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle of radius $3$ units, find the area of the region consisting of all line segments of length $6$ units that are tangent to the circle at their midpoints."
}
] | math | {
"style": "rule",
"ground_truth": "9\\pi"
} | {
"index": 13150,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle of radius $3$, there are multiple line segments of length $6$ that are tangent to the circle at their midpoints. Calculate the area of the region occupied by all such line segments."
}
] | math | {
"style": "rule",
"ground_truth": "9\\pi"
} | {
"index": 13151,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle of radius $\\sqrt{13}$, let $A$ be a point at a distance $4 + \\sqrt{13}$ from the center $O$ of the circle. Let $B$ be the point on the circle nearest to point $A$. A line passing through the point $A$ intersects the circle at points $K$ and $L$. The maximum pos... | math | {
"style": "rule",
"ground_truth": "146"
} | {
"index": 13152,
"solution": "Let $N,M$ les on $AL$ such that $BM\\bot AL, ON\\bot AL$, call $BM=h, ON=k,LN=KN=d$ We call $\\angle{LON}=\\alpha$ By similar triangle, we have $\\frac{h}{k}=\\frac{4}{4+\\sqrt{13}}, h=\\frac{4k}{4+\\sqrt{13}}$. Then, we realize the area is just $dh=d\\cdot \\frac{4K}{4+\\sqrt{13}}$ A... |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle of radius 3, find the area of the region consisting of all line segments of length 6 that are tangent to the circle at their midpoints.\nA) $3\\pi$ \nB) $6\\pi$ \nC) $9\\pi$ \nD) $12\\pi$ \nE) $15\\pi$"
}
] | math | {
"style": "rule",
"ground_truth": "9\\pi"
} | {
"index": 13153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle of radius 3, there are many line segments of length 4 that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.\nA) $9\\pi$\nB) $\\pi$\nC) $4\\pi$\nD) $13\\pi$\nE) $16\\pi$"
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"index": 13154,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle that is tangent to the side \\(DC\\) of a regular pentagon \\(ABCDE\\) at point \\(D\\) and tangent to the side \\(AB\\) at point \\(A\\), what is the degree measure of the minor arc \\(AD\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 13155,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle with 2018 points, each point is labeled with an integer. Each integer must be greater than the sum of the two integers immediately preceding it in a clockwise direction.\n\nDetermine the maximum possible number of positive integers among the 2018 integers."
}
] | math | {
"style": "rule",
"ground_truth": "1009"
} | {
"index": 13156,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle with 800 points labeled in sequence clockwise as \\(1, 2, \\ldots, 800\\), dividing the circle into 800 arcs. Initially, one point is painted red, and subsequently, additional points are painted red according to the following rule: if the \\(k\\)-th point is alre... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 13157,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle with a diameter of $1$ unit, if a point $P$ on the circle starts from point $A$ representing $3$ on the number line and rolls one round to the left along the number line, find the real number represented by the point $B$ where $P$ arrives on the number line."
}... | math | {
"style": "rule",
"ground_truth": "3 - \\pi"
} | {
"index": 13158,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle with a radius of 5, its center lies on the x-axis with an integer horizontal coordinate and is tangent to the line 4x + 3y - 29 = 0.\n\n(1) Find the equation of the circle;\n\n(2) If the line ax - y + 5 = 0 (a ≠ 0) intersects the circle at points A and B, does th... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 13159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle with center $C$ and equation $x^{2}+y^{2}+2x-4y+m=0$, which intersects with the line $2x+y-3=0$ at points $A$ and $B$:\n\n$(1)$ If $\\triangle ABC$ is an equilateral triangle, find the value of $m$; \n$(2)$ Is there a constant $m$ such that the circle with diame... | math | {
"style": "rule",
"ground_truth": "-\\frac{18}{5}"
} | {
"index": 13160,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle with center \\(O\\) and radius \\(OD\\) perpendicular to chord \\(AB\\), intersecting \\(AB\\) at point \\(C\\). Line segment \\(AO\\) is extended to intersect the circle at point \\(E\\). If \\(AB = 8\\) and \\(CD = 2\\), calculate the area of \\(\\triangle BCE\... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 13161,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle with center \\(O\\) and radius \\(OD\\) perpendicular to chord \\(AB\\), intersecting \\(AB\\) at point \\(C\\). Line segment \\(AO\\) is extended to intersect the circle at point \\(E\\). If \\(AB = 8\\) and \\(CD = 2\\), what is the area of \\(\\triangle BCE\\)... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 13162,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle with radius $4$, find the area of the region formed by all line segments of length $4$ that are tangent to this circle at their midpoints.\nA) $2\\pi$\nB) $4\\pi$\nC) $8\\pi$\nD) $16\\pi$"
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"index": 13163,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle with radius $8$, two intersecting chords $PQ$ and $RS$ intersect at point $T$, where $PQ$ is bisected by $RS$. Assume $RS=10$ and the point $P$ is on the minor arc $RS$. Further, suppose that $PQ$ is the only chord starting at $P$ which is bisected by $RS$. Deter... | math | {
"style": "rule",
"ground_truth": "8\\sqrt{39}"
} | {
"index": 13164,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circle with radius $R$ and a fixed point $A$ on the circumference, a point is randomly chosen on the circumference and connected to point $A$. The probability that the length of the chord formed is between $R$ and $\\sqrt{3}R$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 13165,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a circular park 20 feet in diameter with a straight walking path 5 feet wide passing through its center, calculate the area of the park covered by grass after the path is laid."
}
] | math | {
"style": "rule",
"ground_truth": "100\\pi - 100"
} | {
"index": 13166,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a class of 50 students with exam scores following a normal distribution $N(100,10^2)$, and $P(90 ≤ ξ ≤ 100) = 0.3$, estimate the number of students who scored above 110 points."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13167,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a class with 21 students, such that at least two of any three students are friends, determine the largest possible value of k."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 13168,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cluster of circles ${C_n}:{({x-n})^2}+{({y-2n})^2}={n^2}(n≠0)$, the line $l:y=kx+b$ is a common tangent to them, then $k+b=$____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 13169,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a company needs to select 8 engineering and technical personnel from its 6 subsidiaries to form a task force, with each subsidiary contributing at least one person, calculate the total number of ways to allocate these 8 positions."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 13170,
"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 $2^x + 4^y$."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"index": 13171,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a complex number $z = (a^2 - 4) + (a + 2)i$ where $a \\in \\mathbb{R}$: \n(Ⅰ) If $z$ is a pure imaginary number, find the value of the real number $a$; \n(Ⅱ) If the point corresponding to $z$ in the complex plane lies on the line $x + 2y + 1 = 0$, find the value of the ... | math | {
"style": "rule",
"ground_truth": "a = -1"
} | {
"index": 13172,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a complex number $z$ satisfying $z+ \\bar{z}=6$ and $|z|=5$.\n\n$(1)$ Find the imaginary part of the complex number $z$;\n\n$(2)$ Find the real part of the complex number $\\dfrac{z}{1-i}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{2}"
} | {
"index": 13173,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a complex number $z$ satisfying the equation $|z-1|=|z+2i|$ (where $i$ is the imaginary unit), find the minimum value of $|z-1-i|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{5}}{10}"
} | {
"index": 13174,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a complex number $z$ that satisfies the following two conditions:\n\n① $1 < z + \\frac{2}{z} \\leqslant 4$.\n\n② The real part and the imaginary part of $z$ are both integers, and the corresponding point in the complex plane is located in the fourth quadrant.\n\n(I) Find ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{65}}{5}"
} | {
"index": 13175,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a complex number $z=3+bi$ ($b\\in\\mathbb{R}$), and $(1+3i) \\cdot z$ is a pure imaginary number. \n(1) Find the complex number $z$; \n(2) If $w= \\frac{z}{2+i}$, find the modulus of the complex number $w$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 13176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a complex number $z=3+bi\\left(b=R\\right)$, and $\\left(1+3i\\right)\\cdot z$ is an imaginary number.<br/>$(1)$ Find the complex number $z$;<br/>$(2)$ If $ω=\\frac{z}{{2+i}}$, find the complex number $\\omega$ and its modulus $|\\omega|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 13177,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cone and a cylinder made of rubber, the cone has a base radius of $5$ and a height of $4$, while the cylinder has a base radius of $2$ and a height of $8$. If they are remade into a new cone and a new cylinder with the same base radius, while keeping the total volume an... | math | {
"style": "rule",
"ground_truth": "r = \\sqrt{7}"
} | {
"index": 13178,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cone and a cylinder with equal base radii and heights, if the axis section of the cone is an equilateral triangle, calculate the ratio of the lateral surface areas of this cone and cylinder."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 13179,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cone whose vertex and the circumference of its base are both on the same sphere, if the radius of the sphere is $1$, then when the volume of the cone is maximized, the height of the cone is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{4}{3}"
} | {
"index": 13180,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cone with a base radius of $1$ and a height of $\\sqrt{3}$, both the apex of the cone and the base circle are on the surface of a sphere $O$, calculate the surface area of this sphere."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16\\pi}{3}"
} | {
"index": 13181,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cone with a base radius of $5$ and a lateral area of $65π$, let the angle between the slant height and the height of the cone be $θ$. Find the value of $\\sinθ$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{13}"
} | {
"index": 13182,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cone with vertex $S$, and generatrices $SA$, $SB$ perpendicular to each other, and the angle between $SA$ and the base of the cone is $30^{\\circ}$. If the area of $\\triangle SAB$ is $8$, then the volume of this cone is ______."
}
] | math | {
"style": "rule",
"ground_truth": "8\\pi"
} | {
"index": 13183,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cone-shaped island with a total height of 12000 feet, where the top $\\frac{1}{4}$ of its volume protrudes above the water level, determine how deep the ocean is at the base of the island."
}
] | math | {
"style": "rule",
"ground_truth": "1092"
} | {
"index": 13184,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a configuration of four unit squares arranged in a 2x2 grid, find the area of triangle $\\triangle ABC$, where $A$ is the midpoint of the top side of the top-left square, $B$ is the bottom-right corner of the bottom-right square, and $C$ is the midpoint of the right side ... | math | {
"style": "rule",
"ground_truth": "0.375"
} | {
"index": 13185,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a connected simple graph \\( G \\) with \\( e \\) edges and pieces placed on each vertex of \\( G \\) (where each piece can only be placed on a single vertex of \\( G \\)), you are allowed to perform the following operation: if the number of pieces on a vertex \\( v \\) i... | math | {
"style": "rule",
"ground_truth": "e"
} | {
"index": 13186,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a connected simple graph \\( G \\) with a known number of edges \\( e \\), where each vertex has some number of pieces placed on it (each piece can only be placed on one vertex of \\( G \\)). The only operation allowed is when a vertex \\( v \\) has a number of pieces not... | math | {
"style": "rule",
"ground_truth": "e"
} | {
"index": 13187,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a constant function on the interval $(0,1)$, $f(x)$, which satisfies: when $x \\notin \\mathbf{Q}$, $f(x)=0$; and when $x=\\frac{p}{q}$ (with $p, q$ being integers, $(p, q)=1, 0<p<q$), $f(x)=\\frac{p+1}{q}$. Determine the maximum value of $f(x)$ on the interval $\\left(\\... | math | {
"style": "rule",
"ground_truth": "$\\frac{16}{17}$"
} | {
"index": 13188,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a convex hexagon $A B C D E F$ with all six side lengths equal, and internal angles $\\angle A$, $\\angle B$, and $\\angle C$ are $134^{\\circ}$, $106^{\\circ}$, and $134^{\\circ}$ respectively. Find the measure of the internal angle $\\angle E$."
}
] | math | {
"style": "rule",
"ground_truth": "134"
} | {
"index": 13189,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a convex quadrilateral \\( ABCD \\) with \\( X \\) being the midpoint of the diagonal \\( AC \\). It is found that \\( CD \\parallel BX \\). Find \\( AD \\) given that \\( BX = 3 \\), \\( BC = 7 \\), and \\( CD = 6 \\)."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 13190,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a convex quadrilateral \\( ABCD \\) with an interior point \\( P \\) such that \\( P \\) divides \\( ABCD \\) into four triangles \\( ABP, BCP, CDP, \\) and \\( DAP \\). Let \\( G_1, G_2, G_3, \\) and \\( G_4 \\) denote the centroids of these triangles, respectively. Dete... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"index": 13191,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a convex quadrilateral \\(ABCD\\) with \\(X\\) as the midpoint of the diagonal \\(AC\\), it turns out that \\(CD \\parallel BX\\). Find \\(AD\\) if it is known that \\(BX = 3\\), \\(BC = 7\\), and \\(CD = 6\\)."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 13192,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a convex quadrilateral \\(ABCD\\) with \\(\\angle C = 57^{\\circ}\\), \\(\\sin \\angle A + \\sin \\angle B = \\sqrt{2}\\), and \\(\\cos \\angle A + \\cos \\angle B = 2 - \\sqrt{2}\\), find the measure of angle \\(D\\) in degrees."
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"index": 13193,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a convex quadrilateral \\(ABCD\\), \\(X\\) is the midpoint of the diagonal \\(AC\\). It is known that \\(CD \\parallel BX\\). Find \\(AD\\), given that \\(BX = 3\\), \\(BC = 7\\), and \\(CD = 6\\)."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 13194,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length $1$, let $P$ be a moving point on the space diagonal $B C_{1}$ and $Q$ be a moving point on the base $A B C D$. Find the minimum value of $D_{1} P + P Q$."
}
] | math | {
"style": "rule",
"ground_truth": "1 + \\frac{\\sqrt{2}}{2}"
} | {
"index": 13195,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cube $ABCD$-$A\\_1B\\_1C\\_1D\\_1$ with edge length $1$, point $M$ is the midpoint of $BC\\_1$, and $P$ is a moving point on edge $BB\\_1$. Determine the minimum value of $AP + MP$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{2}"
} | {
"index": 13196,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cube $ABCDEFGH$ with an edge length of $6$, let $M$ and $N$ be points on $BB_1$ and $B_1C_1$ respectively, such that $B_1M = B_1N = 2$. Let $S$ and $P$ be the midpoints of segments $AD$ and $MN$ respectively. Find the distance between the skew lines $SP$ and $AC_1$."
... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{114}}{38}"
} | {
"index": 13197,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cube \\( A B C D A_{1} B_{1} C_{1} D_{1} \\), \\( M \\) is the center of the face \\( A B B_{1} A_{1} \\), \\( N \\) is a point on the edge \\( B_{1} C_{1} \\), \\( L \\) is the midpoint of \\( A_{1} B_{1} \\); \\( K \\) is the foot of the perpendicular dropped from \\(... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + 1"
} | {
"index": 13198,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given a cube \\( ABCD A_1 B_1 C_1 D_1 \\) with an edge length of 1. A sphere passes through vertices \\( A \\) and \\( C \\) and the midpoints \\( F \\) and \\( E \\) of edges \\( B_1 C_1 \\) and \\( C_1 D_1 \\) respectively. Find the radius \\( R \\) of this sphere."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{41}}{8}"
} | {
"index": 13199,
"solution": ""
} |
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