data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The number of sets $A$ that satisfy $\\{1, 2\\} \\subset A \\subseteq \\{1, 2, 3, 4, 5, 6\\}$ must be determined. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 34949,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $k$ such that $1^2 + 2^2 + 3^2 + \\ldots + k^2$ is a multiple of $360$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "175"
} | {
"split": "train",
"index": 34950,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region? Let's think s... | math | {
"style": "rule",
"ground_truth": "54\\sqrt{3}-18\\pi"
} | {
"split": "train",
"index": 34951,
"solution": "1. **Calculate the area of the hexagon**: \n A regular hexagon can be divided into 6 equilateral triangles. Each side of the hexagon is given as 6. The area \\( A \\) of an equilateral triangle with side length \\( s \\) is given by the formula:\n \\[\n A = \... |
deepscaler | [
{
"role": "user",
"content": "Natural numbers \\( A \\) and \\( B \\) are divisible by all natural numbers from 1 to 65. What is the smallest natural number that \\( A + B \\) might not be divisible by? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 34952,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The total price of 4 floppy disks and 5 CDs is not less than 20 yuan, and the total price of 6 floppy disks and 3 CDs is not more than 24 yuan. What is the minimum amount of money needed to buy 3 floppy disks and 9 CDs? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 34953,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $2022$ users on a social network called Mathbook, and some of them are Mathbook-friends. (On Mathbook, friendship is always mutual and permanent.) \n\nStarting now, Mathbook will only allow a new friendship to be formed between two users if they have [i]at least two[/... | math | {
"style": "rule",
"ground_truth": "3031"
} | {
"split": "train",
"index": 34954,
"solution": "\nLet the number of users on Mathbook be \\( n = 2022 \\). We are tasked with finding the minimum number of friendships that must exist initially so that eventually every user can become friends with every other user, given the condition that a new friendship can o... |
deepscaler | [
{
"role": "user",
"content": "The circumference of the axial cross-section of a cylinder is $90 \\text{ cm}$. What is the maximum possible volume of the cylinder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3375\\pi"
} | {
"split": "train",
"index": 34955,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, if $|\\overrightarrow{AB}|=2$, $|\\overrightarrow{AC}|=3$, $|\\overrightarrow{BC}|=4$, and $O$ is the incenter of $\\triangle ABC$, and $\\overrightarrow{AO}=\\lambda \\overrightarrow{AB}+\\mu \\overrightarrow{BC}$, calculate the value of $\\lambda+\\mu$. L... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"split": "train",
"index": 34956,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The parabola \\(C_{1}: x^{2}=2 p y\\) has a focus at \\(F\\). The hyperbola \\(C_{2}: \\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1\\) has foci \\(F_{1}\\) and \\(F_{2}\\). Point \\(P\\) is a common point of the two curves in the first quadrant. If the points \\(P\\), \\(F\\), an... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 34957,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all the integers between -20.5 and -1.5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-210"
} | {
"split": "train",
"index": 34958,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A survey of participants was conducted at the Olympiad. $50\\%$ of the participants liked the first round, $60\\%$ of the participants liked the second round, $70\\%$ of the participants liked the opening of the Olympiad. It is known that each participant liked either one... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 34959,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mila and Zhenya each came up with a number and wrote down all the natural divisors of their numbers on the board. Mila wrote down 10 numbers, Zhenya wrote down 9 numbers, and the number 6 appeared twice. How many distinct numbers are on the board in total? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 34960,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integers $a$ with $|a| \\leq 2005$ , does the system\r $x^2=y+a$ \r $y^2=x+a$ \r\nhave integer solutions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 34961,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many two-digit positive integers are congruent to 1 (mod 3)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 34962,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB=125$, $AC=117$ and $BC=120$. The angle bisector of angle $A$ intersects $\\overline{BC}$ at point $L$, and the angle bisector of angle $B$ intersects $\\overline{AC}$ at point $K$. Let $M$ and $N$ be the feet of the perpendiculars from $C$ to $\\overline{... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 34963,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $AC=8$, $BC=7$, $\\cos B=-\\frac{1}{7}$.\n(1) Find the measure of angle $A$;\n(2) Find the area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{3}"
} | {
"split": "train",
"index": 34964,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three balls are randomly and independantly tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $i$ is $2^{-i}$ for $i=1,2,3,....$ More than one ball is allowed in each bin. The probability that the balls end up ... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 34965,
"solution": "To solve this problem, we need to calculate the probability that three balls tossed into bins end up evenly spaced in distinct bins. We start by considering the possible configurations for the balls and their corresponding probabilities.\n\n#### Step 1: Define the ... |
deepscaler | [
{
"role": "user",
"content": "Given a solid $\\Omega$ which is the larger part obtained by cutting a sphere $O$ with radius $4$ by a plane $\\alpha$, and $\\triangle ABC$ is an inscribed triangle of the circular section $O'$ with $\\angle A=90^{\\circ}$. Point $P$ is a moving point on the solid $\\Omega$, a... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 34966,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the integer $k > 3$ for which\n\\[\\log_{10} (k - 3)! + \\log_{10} (k - 2)! + 3 = 2 \\log_{10} k!.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 34967,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers whose sum is $6$ and the absolute value of whose difference is $8$ are roots of the equation: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^2-6x-7=0"
} | {
"split": "train",
"index": 34968,
"solution": "Let the roots of the equation be $x$ and $y$. Given the conditions:\n1. \\(x + y = 6\\)\n2. \\(|x - y| = 8\\)\n\nFrom the first equation, we can express \\(y\\) in terms of \\(x\\):\n\\[ y = 6 - x \\]\n\nSubstituting \\(y\\) into the second equation:\n\\[ |x - (6 -... |
deepscaler | [
{
"role": "user",
"content": "A number is called flippy if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15?$ Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 34969,
"solution": "1. **Identify the conditions for a number to be flippy and divisible by 15**:\n - A flippy number alternates between two distinct digits.\n - A number is divisible by 15 if it is divisible by both 3 and 5.\n\n2. **Condition for divisibility by 5**:\n - The la... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{A} =\\begin{pmatrix} -1 & 2 \\\\ 3 & 4 \\end{pmatrix}.$ Then there exist scalars $p$ and $q$ such that\n\\[\\mathbf{A}^6 = p \\mathbf{A} + q \\mathbf{I}.\\]Enter the ordered pair $(p,q).$ Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "(2223,4510)"
} | {
"split": "train",
"index": 34970,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The vertical drop of a roller coaster is the largest difference in height between any high point and the next low point. The vertical drops of five roller coasters at Mandelbrot Amusement Park are shown in the table. \\begin{tabular}{|l|c|} \\hline\nThe Parabola & 165 feet \\\\... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 34971,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the positive divisors of \\( 2014^{2} \\) be \\( d_{1}, d_{2}, \\cdots, d_{k} \\). Then,\n$$\n\\frac{1}{d_{1}+2014}+\\frac{1}{d_{2}+2014}+\\cdots+\\frac{1}{d_{k}+2014} \\quad = \\quad ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{27}{2014}"
} | {
"split": "train",
"index": 34972,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a\\_n\\}$, the sum of its first $n$ terms is $S\\_n=1-5+9-13+17-21+…+(-1)^{n+1}(4n-3)$. Find the value of $S\\_{15}+S\\_{22}-S\\_{31}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-76"
} | {
"split": "train",
"index": 34973,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Write $2017$ following numbers on the blackboard: $-\\frac{1008}{1008}, -\\frac{1007}{1008}, ..., -\\frac{1}{1008}, 0,\\frac{1}{1008},\\frac{2}{1008}, ... ,\\frac{1007}{1008},\\frac{1008}{1008}$ .\nOne processes some steps as: erase two arbitrary numbers $x, y$ on the bla... | math | {
"style": "rule",
"ground_truth": "-\\frac{144}{1008}"
} | {
"split": "train",
"index": 34974,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twenty-seven increased by twice a number is 39. What is the number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 34975,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the complex number $z$ that satisfies $z= \\frac {1-i}{i+1}$, find the value of $|1+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": 34976,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\frac{1}{330} + \\frac{19}{30}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{11}"
} | {
"split": "train",
"index": 34977,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The congruent sides of an isosceles triangle are each 8 cm long, and the perimeter is 26 cm. In centimeters, what is the length of the base? Also, find the area of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5\\sqrt{39}"
} | {
"split": "train",
"index": 34978,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coefficient of the $x^2$ term in the expansion of $\\sum\\limits_{k=1}^{n}{(x+1)}^{k}$ is equal to the coefficient of the $x^{10}$ term. Determine the positive integer value of $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 34979,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A suitcase lock has 3 dials with the digits $0, 1, 2,..., 9$ on each. How many different settings are possible if all three digits have to be different? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 34980,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability that exactly one person gets their hat back when 6 people randomly pick hats? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{30}"
} | {
"split": "train",
"index": 34981,
"solution": "There are 6 people that could get their hat back, so we must multiply 6 by the number of ways that the other 5 people can arrange their hats such that no one gets his/her hat back. So, the number of ways this will happen is ( $6 \\cdot$ derangement of 5 ), or $6 * ... |
deepscaler | [
{
"role": "user",
"content": "Given the hexadecimal system, determine the product of $A$ and $B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6E"
} | {
"split": "train",
"index": 34982,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first $n$ terms of an arithmetic sequence $\\{a\\_n\\}$ is $S\\_n$, where the first term $a\\_1 > 0$ and the common difference $d < 0$. For any $n \\in \\mathbb{N}^*$, there exists $k \\in \\mathbb{N}^*$ such that $a\\_k = S\\_n$. Find the minimum value of $k - 2... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 34983,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A market survey shows that the price $f(t)$ and sales volume $g(t)$ of a particular product in Oriental Department Store over the past month (calculated based on 30 days) approximately satisfy the functions $f(t)=100(1+ \\frac {1}{t})$ and $g(t)= \\begin{cases} 100+t & 1\\leqsl... | math | {
"style": "rule",
"ground_truth": "12100"
} | {
"split": "train",
"index": 34984,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the multiplication of the two numbers $1,002,000,000,000,000,000$ and $999,999,999,999,999,999$. Calculate the number of digits in the product of these two numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 34985,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ is the cube of a positive integer and $d$ is the number of positive integers that are divisors of $x$, then $d$ could be Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "202"
} | {
"split": "train",
"index": 34986,
"solution": "To solve this problem, we need to understand the relationship between the divisors of a cube of an integer and the divisors of the integer itself.\n\n1. **Expression of $x$ as a cube**: Let $x = n^3$ where $n$ is a positive integer.\n\n2. **Prime factorization of $... |
deepscaler | [
{
"role": "user",
"content": "Two isosceles triangles each have at least one angle that measures $70^{\\circ}$. In the first triangle, the measure in degrees of each of the remaining two angles is even. In the second triangle, the measure in degrees of each of the remaining two angles is odd. Let $S$ be the... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 34987,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We have a $100\\times100$ garden and we’ve plant $10000$ trees in the $1\\times1$ squares (exactly one in each.). Find the maximum number of trees that we can cut such that on the segment between each two cut trees, there exists at least one uncut tree. Let's think step b... | math | {
"style": "rule",
"ground_truth": "2500"
} | {
"split": "train",
"index": 34988,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of all $x$ such that the expression $\\frac{x^2+2x+1}{x^2+2x-3}$ is undefined. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 34989,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum of the squares of the roots of the equation \\[x\\sqrt{x} - 8x + 9\\sqrt{x} - 2 = 0,\\] given that all of the roots are real and nonnegative. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 34990,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular prism \\(A B C D - A_{1} B_{1} C_{1} D_{1}\\), \\(A B = A A_{1} = 2\\), \\(A D = 2 \\sqrt{3}\\). Point \\(M\\) lies within plane \\(B A_{1} C_{1}\\). Find the minimum value of \\(\\overrightarrow{M A} \\cdot \\overrightarrow{M C}\\). Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "-\\frac{16}{7}"
} | {
"split": "train",
"index": 34991,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of lengt... | math | {
"style": "rule",
"ground_truth": "\\frac{37}{35}"
} | {
"split": "train",
"index": 34992,
"solution": "#### Analyzing the first right triangle:\nConsider a right triangle $ABC$ with sides $3$, $4$, and $5$, where $5$ is the hypotenuse. Let a square be inscribed such that one vertex of the square coincides with the right-angle vertex $C$ of the triangle. Let the side... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\arctan ( \\tan 65^\\circ - 2 \\tan 40^\\circ )$. (Express your answer in degrees as an angle between $0^\\circ$ and $180^\\circ$.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25^\\circ"
} | {
"split": "train",
"index": 34993,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $DEF$, $DE=130$, $DF=110$, and $EF=140$. The angle bisector of angle $D$ intersects $\\overline{EF}$ at point $T$, and the angle bisector of angle $E$ intersects $\\overline{DF}$ at point $S$. Let $R$ and $U$ be the feet of the perpendiculars from $F$ to $\\overline... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 34994,
"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$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25\\pi"
} | {
"split": "train",
"index": 34995,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A large $5 \\times 5 \\times 5$ cube is formed using 125 small $1 \\times 1 \\times 1$ cubes. There are three central columns, each passing through the small cube at the very centre of the large cube: one from top to bottom, one from front to back, and one from left to right. A... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 34996,
"solution": "The original $5 \\times 5 \\times 5$ cube has 6 faces, each of which is $5 \\times 5$. When the three central columns of cubes is removed, one of the '1 \\times 1$ squares' on each face is removed. This means that the surface area of each face is reduced by 1 to $5... |
deepscaler | [
{
"role": "user",
"content": "What is the probability that in a random sequence of 8 ones and 2 zeros, there are exactly three ones between the two zeros? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2/15"
} | {
"split": "train",
"index": 34997,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{40}"
} | {
"split": "train",
"index": 34998,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, square PQRS has side length 40. Points J, K, L, and M are on the sides of PQRS, so that JQ = KR = LS = MP = 10. Line segments JZ, KW, LX, and MY are drawn parallel to the diagonals of the square so that W is on JZ, X is on KW, Y is on LX, and Z is on MY. Find th... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 34999,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCDE$ be a convex pentagon such that: $\\angle ABC=90,\\angle BCD=135,\\angle DEA=60$ and $AB=BC=CD=DE$ . Find angle $\\angle DAE$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 35000,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point \\(P\\) is inside an equilateral \\(\\triangle ABC\\) such that the measures of \\(\\angle APB, \\angle BPC, \\angle CPA\\) are in the ratio 5:6:7. Determine the ratio of the measures of the angles of the triangle formed by \\(PA, PB, PC\\) (in increasing order). Let's th... | math | {
"style": "rule",
"ground_truth": "2: 3: 4"
} | {
"split": "train",
"index": 35001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $3$ times their perimeters? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 35002,
"solution": "1. **Define the variables and the relationship between area and perimeter:**\n Let $a$ and $b$ be the two legs of the right triangle, and $c$ be the hypotenuse. The area of the triangle is given by $\\frac{ab}{2}$, and the perimeter is $a + b + c$. The problem st... |
deepscaler | [
{
"role": "user",
"content": "In circle $O$ with radius 10 units, chords $AC$ and $BD$ intersect at right angles at point $P$. If $BD$ is a diameter of the circle, and the length of $PC$ is 3 units, calculate the product $AP \\cdot PB$. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 35003,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a right circular cone with a base radius of \\(1 \\, \\text{cm}\\) and a slant height of \\(3 \\, \\text{cm}\\), point \\(P\\) is on the circumference of the base. Determine the shortest distance from the vertex \\(V\\) of the cone to the shortest path from \\(P\\) back t... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 35004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To investigate the growth inhibitory effect of a certain drug on mice, $40$ mice were divided into two groups, a control group (without the drug) and an experimental group (with the drug).<br/>$(1)$ Suppose the number of mice in the control group among two mice is $X$, find the... | math | {
"style": "rule",
"ground_truth": "95\\%"
} | {
"split": "train",
"index": 35005,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A market survey shows that the price $f(x)$ (in yuan) and the sales volume $g(x)$ (in units) of a certain product in the past $20$ days are both functions of time $x$ (in days), and the price satisfies $f(x)=20-\\frac{1}{2}|x-10|$, and the sales volume satisfies $g(x)=80-2x$, w... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 35006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all values of $z$ such that $z^4 - 4z^2 + 3 = 0$. Enter all the solutions, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{3},-1,1,\\sqrt{3}"
} | {
"split": "train",
"index": 35007,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Expanding $(1+0.2)^{1000}_{}$ by the binomial theorem and doing no further manipulation gives\n${1000 \\choose 0}(0.2)^0+{1000 \\choose 1}(0.2)^1+{1000 \\choose 2}(0.2)^2+\\cdots+{1000 \\choose 1000}(0.2)^{1000}$ $= A_0 + A_1 + A_2 + \\cdots + A_{1000},$ where $A_k = {1000 \\ch... | math | {
"style": "rule",
"ground_truth": "166"
} | {
"split": "train",
"index": 35008,
"solution": "We know that $A_k$ will increase as $k$ increases until certain $k=m$, where $A_0 < A_1 < A_2 < \\dots < A_{m-2} < A_{m-1} < A_m$ and\n$A_m > A_{m+1} > A_{m+2} > \\dots > A_{1000}.$\n Next, to change $A_{k-1}$ to $A_k$, we multiply $A_{k-1}$ by $\\frac{1000-k+1}{5k... |
deepscaler | [
{
"role": "user",
"content": "An up-right path between two lattice points $P$ and $Q$ is a path from $P$ to $Q$ that takes steps of 1 unit either up or to the right. A lattice point $(x, y)$ with $0 \\leq x, y \\leq 5$ is chosen uniformly at random. Compute the expected number of up-right paths from $(0,0)$... | math | {
"style": "rule",
"ground_truth": "175"
} | {
"split": "train",
"index": 35009,
"solution": "For a lattice point $(x, y)$, let $F(x, y)$ denote the number of up-right paths from $(0,0)$ to $(5,5)$ that don't pass through $(x, y)$, and let $$S=\\sum_{0 \\leq x \\leq 5} \\sum_{0 \\leq y \\leq 5} F(x, y)$$ Our answer is $\\frac{S}{36}$, as there are 36 lattic... |
deepscaler | [
{
"role": "user",
"content": "Given a regular hexagon \\( A_6 \\) where each of its 6 vertices is colored with either red or blue, determine the number of type II colorings of the vertices of the hexagon. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 35010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can George choose two out of seven colors to paint his room? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 35011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$, and $(0,1)$. The probability that the slope of the line determined by $P$ and the point $\\left(\\frac58, \\frac38 \\right)$ is greater than or equal to $\\f... | math | {
"style": "rule",
"ground_truth": "171"
} | {
"split": "train",
"index": 35012,
"solution": "The areas bounded by the unit square and alternately bounded by the lines through $\\left(\\frac{5}{8},\\frac{3}{8}\\right)$ that are vertical or have a slope of $1/2$ show where $P$ can be placed to satisfy the condition. One of the areas is a trapezoid with bases... |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of the function \\( f(x) = \\tan^2 x - 4 \\tan x - 8 \\cot x + 4 \\cot^2 x + 5 \\) on the interval \\( \\left( \\frac{\\pi}{2}, \\pi \\right) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9 - 8\\sqrt{2}"
} | {
"split": "train",
"index": 35013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest solution to \\[\\lfloor x \\rfloor = 5 + 100 \\{ x \\},\\]where $\\{x\\} = x - \\lfloor x \\rfloor.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "104.99"
} | {
"split": "train",
"index": 35014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$ is a constant such that $4x^2 + 14x + a$ is the square of a binomial, then what is $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{49}{4}"
} | {
"split": "train",
"index": 35015,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "[asy] draw(circle((4,1),1),black+linewidth(.75)); draw((0,0)--(8,0)--(8,6)--cycle,black+linewidth(.75)); MP(\"A\",(0,0),SW);MP(\"B\",(8,0),SE);MP(\"C\",(8,6),NE);MP(\"P\",(4,1),NW); MP(\"8\",(4,0),S);MP(\"6\",(8,3),E);MP(\"10\",(4,3),NW); MP(\"->\",(5,1),E); dot((4,1)); [/asy]T... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 35016,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the slope of a line perpendicular to the line containing the points $(4,-7)$ and $(-5,-1)$? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 35017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The fenced area of a yard is a 15-foot by 12-foot rectangular region with a 3-foot by 3-foot square cut out, as shown. What is the area of the region within the fence, in square feet?\n\n[asy]draw((0,0)--(16,0)--(16,12)--(28,12)--(28,0)--(60,0)--(60,48)--(0,48)--cycle);\nlabel(... | math | {
"style": "rule",
"ground_truth": "171"
} | {
"split": "train",
"index": 35018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given positive integers \\(a\\) and \\(b\\) such that \\(15a + 16b\\) and \\(16a - 15b\\) are both perfect squares, find the smallest possible value of these two perfect squares. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "231361"
} | {
"split": "train",
"index": 35019,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point $P$ is on the line $y=2x+1$, and point $Q$ is on the curve $y=x+\\ln x$, determine the minimum distance between points $P$ and $Q$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 35020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a regular octagon, find the ratio of the length of the shortest diagonal to the longest diagonal. Express your answer as a common fraction in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 35021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence $a_1$, $a_2$, $\\ldots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\\geq1$. If $a_1=999$, $a_2<999$, and $a_{2006}=1$, how many different values of $a_2$ are possible? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "324"
} | {
"split": "train",
"index": 35022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle, one leg is 24 inches, and the other leg is 10 inches more than twice the shorter leg. Calculate the area of the triangle and find the length of the hypotenuse. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3940}"
} | {
"split": "train",
"index": 35023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Given $\\cos \\alpha =\\frac{\\sqrt{5}}{3}, \\alpha \\in \\left(-\\frac{\\pi }{2},0\\right)$, find $\\sin (\\pi -\\alpha)$;\n\n(2) Given $\\cos \\left(\\theta+ \\frac{\\pi}{4}\\right)= \\frac{4}{5}, \\theta \\in \\left(0, \\frac{\\pi}{2}\\right)$, find $\\cos \\left(\\frac{... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 35024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A piece of string fits exactly once around the perimeter of a rectangle whose area is 180. The ratio of the length to the width of the rectangle is 3:2. Rounded to the nearest whole number, what is the area of the largest circle that can be formed from this piece of string? Let... | math | {
"style": "rule",
"ground_truth": "239"
} | {
"split": "train",
"index": 35025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $e = \\frac{\\sqrt{6}}{3}$, the distance from the origin to the line passing through points $A(0, -b)$ and $B(a, 0)$ is $\\frac{\\sqrt{3}}{2}$.\n\n$(1)$ Find the equation of the ellipse;\n\... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{6}"
} | {
"split": "train",
"index": 35026,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Vila Par, all the truth coins weigh an even quantity of grams and the false coins weigh an odd quantity of grams. The eletronic device only gives the parity of the weight of a set of coins. If there are $2020$ truth coins and $2$ false coins, determine the least $k$, such th... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 35027,
"solution": "\nIn the given problem, we have 2020 true coins, each weighing an even number of grams, and 2 false coins, each weighing an odd number of grams. The electronic device available can detect the parity (even or odd) of the total weight of a set of coins. We need to de... |
deepscaler | [
{
"role": "user",
"content": "Elliot and Emily run a 12 km race. They start at the same point, run 6 km up a hill, and return to the starting point by the same route. Elliot has a 8 minute head start and runs at the rate of 12 km/hr uphill and 18 km/hr downhill. Emily runs 14 km/hr uphill and 20 km/hr downh... | math | {
"style": "rule",
"ground_truth": "\\frac{169}{48}"
} | {
"split": "train",
"index": 35028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\dbinom{9}{8}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 35029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be the set $\\{k^{19}-k: 1<k<20, k\\in N\\}$ . Let $G$ be the GCD of all elements of $A$ .\nThen the value of $G$ is? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "798"
} | {
"split": "train",
"index": 35030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six standard fair six-sided dice are rolled and arranged in a row at random. Compute the expected number of dice showing the same number as the sixth die in the row. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{6}"
} | {
"split": "train",
"index": 35031,
"solution": "For each $i=1,2, \\ldots, 6$, let $X_{i}$ denote the indicator variable of whether the $i$-th die shows the same number as the sixth die. Clearly, $X_{6}=1$ always. For all other $i, X_{i}$ is 1 with probability $\\frac{1}{6}$ and 0 otherwise, so $\\mathbb{E}\\left... |
deepscaler | [
{
"role": "user",
"content": "Let the sequence $x, 3x+3, 5x+5, \\dots$ be in geometric progression. What is the fourth term of this sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{125}{12}"
} | {
"split": "train",
"index": 35032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of 36 consecutive integers is $6^4$. What is their median? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 35033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g(x) = ax^7 + bx^3 + dx^2 + cx - 8$. If $g(-7) = 3$, then find $g(7)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-19"
} | {
"split": "train",
"index": 35034,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the center of the circle with equation $x^2 - 2x + y^2 - 4y - 28 = 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(1, 2)"
} | {
"split": "train",
"index": 35035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. It is given that $2b\\cos C=a\\cos C+c\\cos A$. \n$(I)$ Find the magnitude of angle $C$; \n$(II)$ If $b=2$ and $c= \\sqrt {7}$, find $a$ and the area of $\\triangle ABC$... | math | {
"style": "rule",
"ground_truth": "\\dfrac {3 \\sqrt {3}}{2}"
} | {
"split": "train",
"index": 35036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can the digits of $47,\\!770$ be arranged to form a 5-digit number, ensuring the number does not begin with 0? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 35037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "[b]Problem Section #1\na) A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189, 320, 287, 264, x$ , and y. Find the greatest possible value of: $x + y$ .\n\n<span style=\"color:red\">NOTE: There is a high chance ... | math | {
"style": "rule",
"ground_truth": "761"
} | {
"split": "train",
"index": 35038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Use the Horner's method to calculate the value of the polynomial $f(x) = 2x^5 + 5x^3 - x^2 + 9x + 1$ when $x = 3$. What is the value of $v_3$ in the third step? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "68"
} | {
"split": "train",
"index": 35039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a rectangle $ABCD, E$ is the midpoint of $AB, F$ is a point on $AC$ such that $BF$ is perpendicular to $AC$ , and $FE$ perpendicular to $BD$ . Suppose $BC = 8\\sqrt3$ . Find $AB$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 35040,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\sqrt{\\frac{1}{9} + \\frac{1}{16}} = $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"split": "train",
"index": 35041,
"solution": "1. **Identify the common denominator** for the fractions inside the square root:\n \\[\n \\frac{1}{9} + \\frac{1}{16}\n \\]\n The least common multiple of 9 and 16 is 144. Therefore, we rewrite the fractions with this common denominator:\n \\[\n \\frac{... |
deepscaler | [
{
"role": "user",
"content": "Find all odd natural numbers greater than 500 but less than 1000, for which the sum of the last digits of all divisors (including 1 and the number itself) is equal to 33. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "729"
} | {
"split": "train",
"index": 35042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers $2 \\leq a \\leq 101$ have the property that there exists a positive integer $N$ for which the last two digits in the decimal representation of $a^{2^{n}}$ is the same for all $n \\geq N$ ? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 35043,
"solution": "Solution 1. It suffices to consider the remainder $\\bmod 100$. We start with the four numbers that have the same last two digits when squared: $0,1,25,76$. We can now go backwards, repeatedly solving equations of the form $x^{2} \\equiv n(\\bmod 100)$ where $n$ is... |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be a set of integers such that for each integer $m$, there exists an integer $a \\in A$ and positive integer $n$ such that $a^{n} \\equiv m(\\bmod 100)$. What is the smallest possible value of $|A|$? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 35044,
"solution": "Work in $R=\\mathbb{Z} / 100 \\mathbb{Z} \\cong \\mathbb{Z} / 4 \\mathbb{Z} \\times \\mathbb{Z} / 25 \\mathbb{Z}$. Call an element $r \\in R$ type $(s, t)$ if $s=\\nu_{2}(r) \\leq 2$ and $t=\\nu_{5}(r) \\leq 2$. Also, define an element $r \\in R$ to be coprime if i... |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to put 4 distinguishable balls into 2 distinguishable boxes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 35045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Observe the sequence: (1), (4, 7), (10, 13, 16), (19, 22, 25, 28), ..., then 2008 is in the $\\boxed{\\text{th}}$ group. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 35046,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of trapezoid \\(ABCD (AD \\| BC)\\) if its bases are in the ratio \\(5:3\\), and the area of triangle \\(ADM\\) is 50, where \\(M\\) is the point of intersection of lines \\(AB\\) and \\(CD\\). Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 35047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a Cartesian coordinate plane, call a rectangle $standard$ if all of its sides are parallel to the $x$ - and $y$ - axes, and call a set of points $nice$ if no two of them have the same $x$ - or $y$ - coordinate. First, Bert chooses a nice set $B$ of $2016$ points ... | math | {
"style": "rule",
"ground_truth": "2015"
} | {
"split": "train",
"index": 35048,
"solution": ""
} |
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