data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Find the shortest distance from the line $3 x+4 y=25$ to the circle $x^{2}+y^{2}=6 x-8 y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7 / 5"
} | {
"split": "train",
"index": 20549,
"solution": "The circle is $(x-3)^{2}+(y+4)^{2}=5^{2}$. The center $(3,-4)$ is a distance of $$ \\frac{|3 \\cdot 3+4 \\cdot-4-25|}{\\sqrt{3^{2}+4^{2}}}=\\frac{32}{5} $$ from the line, so we subtract 5 for the radius of the circle and get $7 / 5$."
} |
deepscaler | [
{
"role": "user",
"content": "Let $B_{k}(n)$ be the largest possible number of elements in a 2-separable $k$-configuration of a set with $2n$ elements $(2 \\leq k \\leq n)$. Find a closed-form expression (i.e. an expression not involving any sums or products with a variable number of terms) for $B_{k}(n)$. ... | math | {
"style": "rule",
"ground_truth": "\\binom{2n}{k} - 2\\binom{n}{k}"
} | {
"split": "train",
"index": 20550,
"solution": "First, a lemma: For any \\( a \\) with \\( 0 \\leq a \\leq 2n, \\binom{a}{k} + \\binom{2n-a}{k} \\geq 2 \\binom{n}{k} \\). (By convention, we set \\( \\binom{a}{k} = 0 \\) when \\( a < k \\).) Proof: We may assume \\( a \\geq n \\), since otherwise we can replace \... |
deepscaler | [
{
"role": "user",
"content": "Solve the system\n\n$$\n\\left\\{\\begin{array}{l}\nx^{3}+3 y^{3}=11 \\\\\nx^{2} y+x y^{2}=6\n\\end{array}\\right.\n$$\n\nCalculate the values of the expression $\\frac{x_{k}}{y_{k}}$ for each solution $\\left(x_{k}, y_{k}\\right)$ of the system and find the smallest among them... | math | {
"style": "rule",
"ground_truth": "-1.31"
} | {
"split": "train",
"index": 20551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 365 cards, in which distinct numbers are written. We may ask for any three cards, the order of numbers written in them. Is it always possible to find out the order of all 365 cards by 2000 such questions? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "1691"
} | {
"split": "train",
"index": 20552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of a parallelogram if one of its sides is 51 and the diagonals are 40 and 74. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1224"
} | {
"split": "train",
"index": 20553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the three-digit positive integer $\\underline{a}\\,\\underline{b}\\,\\underline{c}$ whose representation in base nine is $\\underline{b}\\,\\underline{c}\\,\\underline{a}_{\\,\\text{nine}},$ where $a,$ $b,$ and $c$ are (not necessarily distinct) digits. Let's think step by... | math | {
"style": "rule",
"ground_truth": "227"
} | {
"split": "train",
"index": 20554,
"solution": "As shown in Solution 1, we get $99a = 71b+8c$.\n We can see that $99$ is $28$ larger than $71$, and we have an $8c$. We can clearly see that $56$ is a multiple of $8$, and any larger than $56$ would result in $c$ being larger than $9$. Therefore, our only solution ... |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle has sides of length 2 units. A second equilateral triangle is formed having sides that are $150\\%$ of the length of the sides of the first triangle. A third equilateral triangle is formed having sides that are $150\\%$ of the length of the sides of the ... | math | {
"style": "rule",
"ground_truth": "237.5\\%"
} | {
"split": "train",
"index": 20555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the figure, equilateral hexagon $ABCDEF$ has three nonadjacent acute interior angles that each measure $30^\\circ$. The enclosed area of the hexagon is $6\\sqrt{3}$. What is the perimeter of the hexagon? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "12\\sqrt{3}"
} | {
"split": "train",
"index": 20556,
"solution": "1. **Divide the Hexagon into Triangles**:\n The equilateral hexagon $ABCDEF$ is divided into three isosceles triangles $ABF$, $CBD$, and $EDF$, and one equilateral triangle $BDF$. The three nonadjacent acute interior angles of $30^\\circ$ are at vertices $B$, $D$... |
deepscaler | [
{
"role": "user",
"content": "Given that $F\\_1$ is the left focus of the hyperbola $C$: $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, point $B$ has coordinates $(0, b)$, and the line $F\\_1B$ intersects with the two asymptotes of hyperbola $C$ at points $P$ and $Q$. If $\\overrightarrow{QP}... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 20557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lighting power increased by \n\\[ \\Delta N = N_{\\text {after}} - N_{\\text {before}} = 300\\, \\text{BT} - 240\\, \\text{BT} = 60\\, \\text{BT} \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 20558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer divisible by $10$, $11$, and $12$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "660"
} | {
"split": "train",
"index": 20559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Celine has 240 feet of fencing. She needs to enclose a rectangular area such that the area is eight times the perimeter of the rectangle. If she uses up all her fencing material, how many feet is the largest side of the enclosure? Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 20560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the quadratic equation \\( ax^2 + bx + c \\) and the table of values \\( 6300, 6481, 6664, 6851, 7040, 7231, 7424, 7619, 7816 \\) for a sequence of equally spaced increasing values of \\( x \\), determine the function value that does not belong to the table. Let's think s... | math | {
"style": "rule",
"ground_truth": "6851"
} | {
"split": "train",
"index": 20561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $\\frac{{m^2-4m+4}}{{m-1}}÷(\\frac{3}{{m-1}}-m-1)$, where $m=\\sqrt{3}-2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-3+4\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 20562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Subtract $555.55$ from $888.88.$ Express the result as a decimal to the nearest hundredth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "333.33"
} | {
"split": "train",
"index": 20563,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The flea Kuzya can make a jump on the plane in any direction for a distance of exactly 17 mm. Its task is to get from point \\( A \\) to point \\( B \\) on the plane, the distance between which is 1947 cm. What is the minimum number of jumps it must make? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "1146"
} | {
"split": "train",
"index": 20564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lunasa, Merlin, and Lyrica each have a distinct hat. Every day, two of these three people, selected randomly, switch their hats. What is the probability that, after 2017 days, every person has their own hat back? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 20565,
"solution": "Imagine that the three hats are the vertices of an equilateral triangle. Then each day the exchange is equivalent to reflecting the triangle along one of its three symmetry axes, which changes the orientation of the triangle (from clockwise to counterclockwise or v... |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S,$ the probability that it is divisible by $9$ is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$ Let's th... | math | {
"style": "rule",
"ground_truth": "913"
} | {
"split": "train",
"index": 20566,
"solution": "A positive integer $n$ has exactly two 1s in its binary representation exactly when $n = 2^j + 2^k$ for $j \\neq k$ nonnegative integers. Thus, the set $S$ is equal to the set $\\{n \\in \\mathbb{Z} \\mid n = 2^j + 2^k \\,\\mathrm{ and }\\, 0 \\leq j < k \\leq 39\\... |
deepscaler | [
{
"role": "user",
"content": "Evaluate the determinant of the given matrix:\n\\[\n\\begin{vmatrix} \\sin \\theta \\sin \\phi & \\sin \\theta \\cos \\phi & \\cos \\theta \\\\ \\cos \\phi & -\\sin \\phi & 0 \\\\ -\\cos \\theta \\sin \\phi & -\\cos \\theta \\cos \\phi & \\sin \\theta \\end{vmatrix}\n\\] Let's ... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 20567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The cost prices of three types of clothing, A, B, and C, are respectively 20 yuan, 30 yuan, and 40 yuan. Their selling prices are 24 yuan, m yuan, and 52 yuan, respectively. After calculation, the total profit from the three types of clothing is the same, and the sum of the sal... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 20568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Remove all perfect squares and perfect cubes from the set\n$$\nA=\\left\\{n \\mid n \\leqslant 10000, n \\in \\mathbf{Z}_{+}\\right\\}\n$$\nand arrange the remaining elements in ascending order. What is the 2014th element of this sequence? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "2068"
} | {
"split": "train",
"index": 20569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\binom{23}{3}=1771$, $\\binom{23}{4}=8855$, and $\\binom{23}{5}=33649$, find $\\binom{25}{5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "53130"
} | {
"split": "train",
"index": 20570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The circular base of a hemisphere of radius $2$ rests on the base of a square pyramid of height $6$. The hemisphere is tangent to the other four faces of the pyramid. What is the edge-length of the base of the pyramid? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 20571,
"solution": "1. **Identify the Geometry and Setup**: Let $ABCDE$ be the pyramid with $ABCD$ as the square base. Let $O$ be the center of square $ABCD$ and $M$ be the midpoint of side $AB$. The hemisphere with radius $2$ is tangent to the triangular face $ABE$ at point $P$.\n\n2... |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle PQR$ be a right triangle with $Q$ as a right angle. A circle with diameter $QR$ intersects side $PR$ at point $S$. If $PS = 3$ and $QS = 9$, find the length of $RS$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 20572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the greatest integer multiple of $36$ all of whose digits are even and no two of whose digits are the same. Find the remainder when $N$ is divided by $1000$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "640"
} | {
"split": "train",
"index": 20573,
"solution": "If an integer is divisible by $36$, it must also be divisible by $9$ since $9$ is a factor of $36$. It is a well-known fact that, if $N$ is divisible by $9$, the sum of the digits of $N$ is a multiple of $9$. Hence, if $N$ contains all the even digits, the sum of t... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\cos ^{2} \\frac{x}{2}- \\sqrt {3}\\sin x$.\n(I) Find the smallest positive period and the range of the function;\n(II) If $a$ is an angle in the second quadrant and $f(a- \\frac {π}{3})= \\frac {1}{3}$, find the value of $\\frac {\\cos 2a}{1+\\cos 2a... | math | {
"style": "rule",
"ground_truth": "\\frac{1-2\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 20574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Write any natural number on a piece of paper, and rotate the paper 180 degrees. If the value remains the same, such as $0$, $11$, $96$, $888$, etc., we call such numbers \"神马数\" (magical numbers). Among all five-digit numbers, how many different \"magical numbers\" are there? L... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 20575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer with exactly 12 positive integer divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 20576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lily has $3$ gallons of milk and gives $\\frac{12}{5}$ gallons to James. What fraction of a gallon does she have left? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 20577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles of radius $3$ and $6$ are externally tangent to each other and are internally tangent to a circle of radius $9$ . The circle of radius $9$ has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord. Let's think st... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 20578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the curve defined by the equation\n\\[\\theta = \\frac{\\pi}{3}.\\](A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{(A)}"
} | {
"split": "train",
"index": 20579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagonal lengths of a rhombus are 24 units and 10 units. What is the area of the rhombus, in square units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 20580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$ with sides $AB=9$, $AC=3$, and $BC=8$, the angle bisector $AK$ is drawn. A point $M$ is marked on side $AC$ such that $AM : CM=3 : 1$. Point $N$ is the intersection point of $AK$ and $BM$. Find $KN$. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}}{5}"
} | {
"split": "train",
"index": 20581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square flag features a green cross of uniform width, and a yellow square in the center, against a white background. The cross is symmetric with respect to each of the diagonals of the square. Suppose the entire cross (including the green arms and the yellow center) occupies 4... | math | {
"style": "rule",
"ground_truth": "45\\%"
} | {
"split": "train",
"index": 20582,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The smallest 9-digit integer that can be divided by 11 is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100000010"
} | {
"split": "train",
"index": 20583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $63 \\times 57$ in your head. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3591"
} | {
"split": "train",
"index": 20584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An up-right path from $(a, b) \\in \\mathbb{R}^{2}$ to $(c, d) \\in \\mathbb{R}^{2}$ is a finite sequence $\\left(x_{1}, y_{1}\\right), \\ldots,\\left(x_{k}, y_{k}\\right)$ of points in $\\mathbb{R}^{2}$ such that $(a, b)=\\left(x_{1}, y_{1}\\right),(c, d)=\\left(x_{k}, y_{k}\\... | math | {
"style": "rule",
"ground_truth": "1750"
} | {
"split": "train",
"index": 20585,
"solution": "The number of up-right paths from $(0,0)$ to $(4,4)$ is $\\binom{8}{4}$ because any such upright path is identical to a sequence of 4 U's and 4 R's, where $U$ corresponds to a step upwards and R corresponds to a step rightwards. Therefore, the total number of pairs... |
deepscaler | [
{
"role": "user",
"content": "Express $0.3\\overline{45}$ as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{83}{110}"
} | {
"split": "train",
"index": 20586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nChicks hatch on the night from Sunday to Monday. For two weeks, a chick sits with its beak open, during the third week it silently grows feathers, and during the fourth week it flies out of the nest. Last week, there were 20 chicks in the nest sitting with their beaks open, a... | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 20587,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $a$ such that $x^4 + a^2$ is not prime for any integer $x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 20588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, . . . , 2020$ . The students are initially divided up into $20$ groups of $101$ , with each division into groups equally likely. In each of the groups, the contestant with the l... | math | {
"style": "rule",
"ground_truth": "2123"
} | {
"split": "train",
"index": 20589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An $8$ by $2\\sqrt{2}$ rectangle has the same center as a circle of radius $2$. The area of the region common to both the rectangle and the circle is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\pi+4"
} | {
"split": "train",
"index": 20590,
"solution": "1. **Visualize and Analyze the Geometry**: Consider a rectangle with dimensions $8$ by $2\\sqrt{2}$ and a circle with radius $2$, both sharing the same center. We need to find the area of the region common to both the rectangle and the circle.\n\n2. **Identify Key ... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, suppose a curve $C_1$ is given by $x^2+y^2=1$. All points on curve $C_1$ have their $x$ and $y$ coordinates stretched by a factor of $\\sqrt{2}$ and $\\sqrt{3}$, respectively, resulting in a new curve $C_2$.\n$(1)$ Write down the parametric e... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{10}}{2}"
} | {
"split": "train",
"index": 20591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the equation of the line that passes through the intersection of the lines $2x+3y+5=0$ and $2x+5y+7=0$, and is parallel to the line $x+3y=0$. Also, calculate the distance between these two parallel lines. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{10}}{5}"
} | {
"split": "train",
"index": 20592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Billy wants to determine the average number of cookies in a package of cookies. He purchases 7 packages of cookies, opens them, and counts the number of cookies in each one. He finds that the packages have 8, 10, 12, 15, 16, 17, and 20 cookies. Based on these observations, what... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 20593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average of a set of data 1, 3, 2, 5, $x$ is 3. What is the standard deviation of this set of data? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 20594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a rectangle $ABCD$ containing three squares. Two smaller squares each occupy a part of rectangle $ABCD$, and each smaller square has an area of 1 square inch. A larger square, also inside rectangle $ABCD$ and not overlapping with the smaller squares, has a side length ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 20595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Erin the ant starts at a given corner of a hypercube (4-dimensional cube) and crawls along exactly 15 edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point, determine the number o... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 20596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the complex number $z$ that satisfies $$z= \\frac {1-i}{i}$$ (where $i$ is the imaginary unit), find $z^2$ and $|z|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}"
} | {
"split": "train",
"index": 20597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f: N \\to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \\in\\{0,1\\}$ for all $m,n \\in N$ . Determine $f (2014)$ . $N_0=\\{0,1,2,...\\}$ Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "671"
} | {
"split": "train",
"index": 20598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1, \\sqrt{2k})$. Evaluate\n\\[\n\\sum_{k=1}^\\infty (-1)^{k-1} \\frac{A(k)}{k}.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi^2}{16}"
} | {
"split": "train",
"index": 20599,
"solution": "We will prove that the sum converges to $\\pi^2/16$.\nNote first that the sum does not converge absolutely, so we are not free to rearrange it arbitrarily. For that matter, the standard alternating sum test does not apply because the absolute values of the terms do... |
deepscaler | [
{
"role": "user",
"content": "A rectangular piece of paper $ABCD$ has sides of lengths $AB = 1$ , $BC = 2$ . The rectangle is folded in half such that $AD$ coincides with $BC$ and $EF$ is the folding line. Then fold the paper along a line $BM$ such that the corner $A$ falls on line $EF$ . How... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 20600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Ming arrives at the departure station between 7:50 and 8:30 to catch the high-speed train departing at 7:00, 8:00, or 8:30. Calculate the probability that his waiting time does not exceed 10 minutes. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "\\frac {2}{3}"
} | {
"split": "train",
"index": 20601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the cells of an $8\\times 8$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least three cells which are still free. Find the greatest possible number of marbles that c... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 20602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\#$ be the relation defined by $A \\# B = A^2 + B^2$. If $A \\# 5 = 169$, what is the positive value of $A$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 20603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of\n\\[(\\sin x + \\csc x)^2 + (\\cos x + \\sec x)^2\\]for $0 < x < \\frac{\\pi}{2}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 20604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\tan\\alpha=3$, calculate the following:\n(1) $\\frac{\\sin\\alpha+\\cos\\alpha}{2\\sin\\alpha-\\cos\\alpha}$\n(2) $\\sin^2\\alpha+\\sin\\alpha\\cos\\alpha+3\\cos^2\\alpha$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 20605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $n$ between 1 and 1000 inclusive does the decimal representation of $\\frac{n}{1400}$ terminate? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "142"
} | {
"split": "train",
"index": 20606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $F$ fractions $\\frac{m}{n}$ with the properties: $m$ and $n$ are positive integers with $m<n$, $\\frac{m}{n}$ is in lowest terms, $n$ is not divisible by the square of any integer larger than 1, and the shortest sequence of consecutive digits that repeats consecutive... | math | {
"style": "rule",
"ground_truth": "181"
} | {
"split": "train",
"index": 20607,
"solution": "We start with 1111109 fractions, as above, and want to remove all of the fractions in $U, V$ and $W$. Since each fraction in $W$ is in $U$ and $V$, it is enough to remove those $U$ and $V$ only. The total number of fractions in $U$ and $V$ (that is, in $U \\cup V$)... |
deepscaler | [
{
"role": "user",
"content": "Two different natural numbers end with 7 zeros and have exactly 72 divisors. Find their sum. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "70000000"
} | {
"split": "train",
"index": 20608,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Last year, 10% of the net income from our school's ball was allocated to clubs for purchases, and the remaining part covered the rental cost of the sports field. This year, we cannot sell more tickets, and the rental cost remains the same, so increasing the share for the clubs ... | math | {
"style": "rule",
"ground_truth": "12.5"
} | {
"split": "train",
"index": 20609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all the positive perfect cubes that are not divisible by $10$ such that the number obtained by erasing the last three digits is also a perfect cube. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1331 \\text{ and } 1728"
} | {
"split": "train",
"index": 20610,
"solution": "\nTo solve the given problem, we need to find all positive perfect cubes that are not divisible by \\(10\\) and have the property that when the last three digits are erased, the resulting number is also a perfect cube.\n\n1. **Understanding the Cube Condition**: Le... |
deepscaler | [
{
"role": "user",
"content": "Fill in the blanks with unique digits in the following equation: \n\\[ \\square \\times(\\square+\\square \\square) \\times(\\square+\\square+\\square+\\square \\square) = 2014 \\]\nThe maximum sum of the five one-digit numbers among the choices is: Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 20611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The slant height of a cone forms an angle $\\alpha$ with the plane of its base, where $\\cos \\alpha = \\frac{1}{4}$. A sphere is inscribed in the cone, and a plane is drawn through the circle of tangency of the sphere and the lateral surface of the cone. The volume of the part... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 20612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lines $l$ and $k$ are parallel to each other. $m\\angle A = 120^\\circ$, and $m\\angle C = 80^\\circ$. What is the number of degrees in $m\\angle B$?\n\n[asy]\nsize(100); real h = 1.2; currentpen = fontsize(10pt);\ndraw(Label(\"$l$\",Relative(1)),(0,0)--(1,0),E);\ndraw(Label(\"... | math | {
"style": "rule",
"ground_truth": "160^\\circ"
} | {
"split": "train",
"index": 20613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $F(z)=\\dfrac{z+i}{z-i}$ for all complex numbers $z\\neq i$, and let $z_n=F(z_{n-1})$ for all positive integers $n$. Given that $z_0=\\dfrac{1}{137}+i$ and $z_{2002}=a+bi$, where $a$ and $b$ are real numbers, find $a+b$. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "275"
} | {
"split": "train",
"index": 20614,
"solution": "Iterating $F$ we get:\n\\begin{align*} F(z) &= \\frac{z+i}{z-i}\\\\ F(F(z)) &= \\frac{\\frac{z+i}{z-i}+i}{\\frac{z+i}{z-i}-i} = \\frac{(z+i)+i(z-i)}{(z+i)-i(z-i)}= \\frac{z+i+zi+1}{z+i-zi-1}= \\frac{(z+1)(i+1)}{(z-1)(1-i)}\\\\ &= \\frac{(z+1)(i+1)^2}{(z-1)(1^2+1^2)... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=(2-a)(x-1)-2\\ln x$ $(a\\in \\mathbb{R})$.\n\n(Ⅰ) If the tangent line at the point $(1,g(1))$ on the curve $g(x)=f(x)+x$ passes through the point $(0,2)$, find the interval where the function $g(x)$ is decreasing;\n\n(Ⅱ) If the function $y=f(x)$ has no ... | math | {
"style": "rule",
"ground_truth": "2-4\\ln 2"
} | {
"split": "train",
"index": 20615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Lucy starts with an initial term of 8 in her sequence, where each subsequent term is generated by either doubling the previous term and subtracting 2 if a coin lands on heads, or halving the previous term and subtracting 2 if a coin lands on tails, determine the probabili... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 20616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One of the roots of $z^2 = -55 + 48i$ is $3 + 8i.$ What is the other root? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3 - 8i"
} | {
"split": "train",
"index": 20617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The probability of an event happening is $\\frac{1}{2}$, find the relation between this probability and the outcome of two repeated experiments. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50\\%"
} | {
"split": "train",
"index": 20618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\arccos (\\cos 7).$ All functions are in radians. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7 - 2 \\pi"
} | {
"split": "train",
"index": 20619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 20620,
"solution": "\nTo solve this problem, we need to determine the probability that all three red chips are drawn before both green chips are drawn. We can approach this problem by considering the order in which the chips are drawn.\n\n#### Step-by-step Analysis:\n\n1. **Total Arra... |
deepscaler | [
{
"role": "user",
"content": "What is the 7th term of an arithmetic sequence of 15 terms where the first term is 3 and the last term is 72? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 20621,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( ABC \\), it is given that \\( AC = 5\\sqrt{2} \\), \\( BC = 5 \\), and \\( \\angle BAC = 30^\\circ \\). What is the largest possible size in degrees of \\( \\angle ABC \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "135"
} | {
"split": "train",
"index": 20622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain country, the airline system is arranged in such a way that any city is connected by airlines with no more than three other cities, and from any city to any other, one can travel with no more than one stopover. What is the maximum number of cities that can be in thi... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 20623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two real numbers $p>1$ and $q>1$ so that $\\frac{1}{p} + \\frac{1}{q} = 1$ and $pq = 4$, what is $q$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 20624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\frac{1000^2}{252^2-248^2}$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 20625,
"solution": "1. **Identify the expression and apply the difference of squares formula**: \n The given expression is $\\frac{1000^2}{252^2-248^2}$. We can simplify the denominator using the difference of squares formula, which states that $a^2 - b^2 = (a-b)(a+b)$.\n\n2. **App... |
deepscaler | [
{
"role": "user",
"content": "Jori has 3 gallons of distilled water. She uses 5/4 gallons in the first science experiment and 1/3 gallon in a second experiment. How much distilled water does she have left after both experiments? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{12}"
} | {
"split": "train",
"index": 20626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{v}$ be a vector such that\n\\[\\left\\| \\mathbf{v} + \\begin{pmatrix} 4 \\\\ 2 \\end{pmatrix} \\right\\| = 10.\\]\nFind the smallest possible value of $\\|\\mathbf{v}\\|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10 - 2 \\sqrt{5}"
} | {
"split": "train",
"index": 20627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function $f(x)= \\sqrt{3}\\cos^2\\omega x+\\sin \\omega x\\cos \\omega x+a$ where $\\omega > 0$, $a\\in\\mathbb{R}$, and the graph of $f(x)$ has its first highest point on the right side of the y-axis at the x-coordinate $\\dfrac{\\pi}{6}$.\n\n(Ⅰ) Find the smallest posi... | math | {
"style": "rule",
"ground_truth": "\\dfrac{ \\sqrt{3}+1}{2}"
} | {
"split": "train",
"index": 20628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain school conducted a survey on the daily reading time of students during the summer vacation, as shown in the table below:\n\n| | Mean | Variance | Number of Students |\n|---------|------|----------|--------------------|\n| Grade 10 | $2.7$ | $1$ | $800$ |\n| Gr... | math | {
"style": "rule",
"ground_truth": "1.966"
} | {
"split": "train",
"index": 20629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider that Henry's little brother now has 10 identical stickers and 5 identical sheets of paper. How many ways can he distribute all the stickers on the sheets of paper, if only the number of stickers on each sheet matters and no sheet can remain empty? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 20630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABCDE$ is a regular pentagon. $AP$, $AQ$ and $AR$ are the perpendiculars dropped from $A$ onto $CD$, $CB$ extended and $DE$ extended, respectively. Let $O$ be the center of the pentagon. If $OP = 1$, then find $AO + AQ + AR$.\n\n[asy]\n\nunitsize(2 cm);\n\npair A, B, C, D, ... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 20631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Elisa creates a sequence in a manner similar to Jacob but starts with the first term as 10. Each succeeding term depends on the outcome of flipping a fair coin: If it lands heads, the next term is obtained by doubling the previous term and subtracting 1; if it lands tails, the ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 20632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many groups of integer solutions are there for the equation $xyz = 2009$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 20633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle PQR$ be a right triangle such that $Q$ is a right angle. A circle with diameter $QR$ intersects side $PR$ at $S$. If $PS = 3$ and $QS = 9$, what is $RS$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 20634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Paco uses a spinner to select a number from 1 through 5, each with equal probability. Manu uses a different spinner to select a number from 1 through 10, each with equal probability. What is the probability that the product of Manu's number and Paco's number is less than 30? Ex... | math | {
"style": "rule",
"ground_truth": "\\frac{41}{50}"
} | {
"split": "train",
"index": 20635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parametric equation of line C1 as $$\\begin{cases} x=2+t \\\\ y=t \\end{cases}$$ (where t is the parameter), and the polar coordinate equation of the ellipse C2 as ρ²cos²θ + 9ρ²sin²θ = 9. Establish a rectangular coordinate system with the origin O as the pole and the ... | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{3}}{5}"
} | {
"split": "train",
"index": 20636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For every real number $x$, what is the value of the expression $(x+1)^{2} - x^{2}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2x + 1"
} | {
"split": "train",
"index": 20637,
"solution": "Expanding and simplifying, $(x+1)^{2} - x^{2} = (x^{2} + 2x + 1) - x^{2} = 2x + 1$."
} |
deepscaler | [
{
"role": "user",
"content": "Consider a 5-minute interval. In this period, an average of 5 bites occur on the first fishing rod, and 1 bite on the second fishing rod. Therefore, the total average number of bites on both rods during these 5 minutes is 6. Determine the average waiting time for the first bite... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 20638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $x$ such that $\\log_x 81=\\log_2 16$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 20639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest four-digit number whose digits add up to 23? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9950"
} | {
"split": "train",
"index": 20640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a \"clearance game,\" the rules stipulate that in round \\( n \\), a dice is to be rolled \\( n \\) times. If the sum of the points of these \\( n \\) rolls is greater than \\( 2^{n} \\), the player clears the round. \n\n(1) What is the maximum number of rounds a player can ... | math | {
"style": "rule",
"ground_truth": "\\frac{100}{243}"
} | {
"split": "train",
"index": 20641,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $\\sqrt{104\\sqrt{6}+468\\sqrt{10}+144\\sqrt{15}+2006}$ can be written as $a\\sqrt{2}+b\\sqrt{3}+c\\sqrt{5},$ where $a, b,$ and $c$ are positive integers. Find $abc$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "936"
} | {
"split": "train",
"index": 20642,
"solution": "We begin by equating the two expressions:\n\\[a\\sqrt{2}+b\\sqrt{3}+c\\sqrt{5} = \\sqrt{104\\sqrt{6}+468\\sqrt{10}+144\\sqrt{15}+2006}\\]\n Squaring both sides yields:\n\\[2ab\\sqrt{6} + 2ac\\sqrt{10} + 2bc\\sqrt{15} + 2a^2 + 3b^2 + 5c^2 = 104\\sqrt{6}+468\\sqrt{10... |
deepscaler | [
{
"role": "user",
"content": "Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are $6$ cm in diameter and $12$ cm high. Felicia buys cat food in cylindrical cans that are $12$ cm in diameter and $6$ cm high. What is the ratio of the volume of one of Alex's cans to the vol... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 20643,
"solution": "To find the ratio of the volumes of Alex's and Felicia's cat food cans, we first need to calculate the volume of each can. The volume $V$ of a cylinder is given by the formula:\n\\[ V = \\pi r^2 h \\]\nwhere $r$ is the radius and $h$ is the height of the cylinder.\... |
deepscaler | [
{
"role": "user",
"content": "$M$ is the midpoint of the side $AB$ in an equilateral triangle $\\triangle ABC.$ The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1.$ On the line passing through $C$ and parallel to $MD$ there is a point $T$ inside the triangle $\\triangle ABC$ such that $\\angle CTA... | math | {
"style": "rule",
"ground_truth": "120^\\circ"
} | {
"split": "train",
"index": 20644,
"solution": "\nGiven an equilateral triangle \\(\\triangle ABC\\), where \\(M\\) is the midpoint of side \\(AB\\). We have a point \\(D\\) on side \\(BC\\) such that the segment \\(BD : DC = 3 : 1\\). We need to find \\(\\angle MTD\\) given that there is a point \\(T\\) on a li... |
deepscaler | [
{
"role": "user",
"content": "Find all real values of $x$ that satisfy $\\frac{x(x+1)}{(x-4)^2} \\ge 12.$ (Give your answer in interval notation.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "[3, 4) \\cup \\left(4, \\frac{64}{11}\\right]"
} | {
"split": "train",
"index": 20645,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A school has between 150 and 250 students enrolled. Each day, all the students split into eight different sections for a special workshop. If two students are absent, each section can contain an equal number of students. Find the sum of all possible values of student enrollment... | math | {
"style": "rule",
"ground_truth": "2626"
} | {
"split": "train",
"index": 20646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest possible rational root of the equation $ax^2 + bx + c = 0{}$ where $a, b$ and $c{}$ are positive integers that do not exceed $100{}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{99}"
} | {
"split": "train",
"index": 20647,
"solution": "\nTo determine the largest possible rational root of the quadratic equation \\( ax^2 + bx + c = 0 \\), where \\( a, b, \\) and \\( c \\) are positive integers not exceeding 100, we use the Rational Root Theorem. This theorem states that any rational root, expressed... |
deepscaler | [
{
"role": "user",
"content": "There is more than one integer greater than $1$ which, when divided by any integer $k$ such that $2 \\le k \\le 11$, has a remainder of $1$. \nWhat is the difference between the two smallest such integers? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "27720"
} | {
"split": "train",
"index": 20648,
"solution": "1. **Identify the Problem Requirements:**\n We need to find integers greater than $1$ that, when divided by any integer $k$ such that $2 \\le k \\le 11$, leave a remainder of $1$. We are asked to find the difference between the two smallest such integers.\n\n2. *... |
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