data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "For any set \\( S \\), let \\( |S| \\) represent the number of elements in set \\( S \\) and let \\( n(S) \\) represent the number of subsets of set \\( S \\). If \\( A \\), \\( B \\), and \\( C \\) are three finite sets such that:\n\n(1) \\( |A|=|B|=2016 \\);\n(2) \\( n(A) + n... | math | {
"style": "rule",
"ground_truth": "2015"
} | {
"split": "train",
"index": 17549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cubic polynomial $p(x)$ satisfies\n\\[p(n) = \\frac{1}{n^2}\\]for $n = 1, 2, 3,$ and $4.$ Find $p(5).$ 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": 17550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Farmer James wishes to cover a circle with circumference $10 \\pi$ with six different types of colored arcs. Each type of arc has radius 5, has length either $\\pi$ or $2 \\pi$, and is colored either red, green, or blue. He has an unlimited number of each of the six arc types. ... | math | {
"style": "rule",
"ground_truth": "93"
} | {
"split": "train",
"index": 17551,
"solution": "Fix an orientation of the circle, and observe that the problem is equivalent to finding the number of ways to color ten equal arcs of the circle such that each arc is one of three different colors, and any two arcs which are separated by exactly one arc are of diff... |
deepscaler | [
{
"role": "user",
"content": "A polynomial $P$ with integer coefficients is called tricky if it has 4 as a root. A polynomial is called $k$-tiny if it has degree at most 7 and integer coefficients between $-k$ and $k$, inclusive. A polynomial is called nearly tricky if it is the sum of a tricky polynomial a... | math | {
"style": "rule",
"ground_truth": "64912347"
} | {
"split": "train",
"index": 17552,
"solution": "A tricky 7-tiny polynomial takes the form $$\\left(c_{6} x^{6}+\\ldots+c_{1} x+c_{0}\\right)(x-4)$$ For each fixed value of $k, c_{k}-4 c_{k+1}$ should lie in $[-7,7]$, so if we fix $c_{k}$, there are around $15 / 4$ ways of choosing $c_{k+1}$. Therefore if we pick... |
deepscaler | [
{
"role": "user",
"content": "Calculate the nearest integer to $(3+\\sqrt{5})^6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20608"
} | {
"split": "train",
"index": 17553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Dima calculated the factorials of all natural numbers from 80 to 99, found the numbers that are reciprocals to them, and printed the resulting decimal fractions on 20 infinite ribbons (for example, the last ribbon had printed the number $\\frac{1}{99!}=0, \\underbrace{00 \\ldot... | math | {
"style": "rule",
"ground_truth": "155"
} | {
"split": "train",
"index": 17554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f(x)=1+x- \\frac {x^{2}}{2}+ \\frac {x^{3}}{3}$, $g(x)=1-x+ \\frac {x^{2}}{2}- \\frac {x^{3}}{3}$, if the function $F(x)=f(x+3)g(x-4)$, and the zeros of the function $F(x)$ are all within $[a,b]$ $(a < b$, $a$, $b\\in\\mathbb{Z})$, then the minimum value of $b-a$ ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 17555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For point M on the ellipse $\\frac {x^{2}}{9}+ \\frac {y^{2}}{4}=1$, find the minimum distance from M to the line $x+2y-10=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 17556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a recent survey conducted by Mary, she found that $72.4\\%$ of participants believed that rats are typically blind. Among those who held this belief, $38.5\\%$ mistakenly thought that all rats are albino, which is not generally true. Mary noted that 25 people had this specif... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 17557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,...,\\dfrac{n}{2},n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right o... | math | {
"style": "rule",
"ground_truth": "340"
} | {
"split": "train",
"index": 17558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $A B C$ satisfies $\\angle B>\\angle C$. Let $M$ be the midpoint of $B C$, and let the perpendicular bisector of $B C$ meet the circumcircle of $\\triangle A B C$ at a point $D$ such that points $A, D, C$, and $B$ appear on the circle in that order. Given that $\\angle... | math | {
"style": "rule",
"ground_truth": "86^{\\circ}"
} | {
"split": "train",
"index": 17559,
"solution": "Extend $D M$ to hit the circumcircle at $E$. Then, note that since $A D E B$ is a cyclic quadrilateral, $\\angle A B E=180^{\\circ}-\\angle A D E=180^{\\circ}-\\angle A D M=180^{\\circ}-68^{\\circ}=112^{\\circ}$. We also have that $\\angle M E C=\\angle D E C=\\ang... |
deepscaler | [
{
"role": "user",
"content": "Determine the real value of $t$ that minimizes the expression\n\\[\n \\sqrt{t^2 + (t^2 - 1)^2} + \\sqrt{(t-14)^2 + (t^2 - 46)^2}.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7/2"
} | {
"split": "train",
"index": 17560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An easel in a corner hosts three $30 \\text{ cm} \\times 40 \\text{ cm}$ shelves, with equal distances between neighboring shelves. Three spiders resided where the two walls and the middle shelf meet. One spider climbed diagonally up to the corner of the top shelf on one wall, ... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 17561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A magician has $5432_{9}$ tricks for his magical performances. How many tricks are there in base 10? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3998"
} | {
"split": "train",
"index": 17562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) such that every \\( n \\)-element subset of \\( S = \\{1, 2, \\ldots, 150\\} \\) contains 4 numbers that are pairwise coprime (it is known that there are 35 prime numbers in \\( S \\)). Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "111"
} | {
"split": "train",
"index": 17563,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number 123456789 is written on the board. Two adjacent digits are selected from the number, if neither of them is 0, 1 is subtracted from each digit, and the selected digits are swapped (for example, from 123456789, one operation can result in 123436789). What is the smalle... | math | {
"style": "rule",
"ground_truth": "101010101"
} | {
"split": "train",
"index": 17564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $y^{2}=4x$, let $AB$ and $CD$ be two chords perpendicular to each other and passing through its focus. Find the value of $\\frac{1}{|AB|}+\\frac{1}{|CD|}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 17565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You have four textbooks for mandatory courses numbered 1 to 4. They are randomly placed on the same shelf.\n\n1. Calculate the probability that Textbook 2 is to the left of Textbook 4.\n2. Calculate the probability that Textbook 2 is to the left of Textbook 3, and Textbook 3 is... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 17566,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $989 \\cdot 1001 \\cdot 1007+320$ can be written as the product of three distinct primes $p, q, r$ with $p<q<r$. Find $(p, q, r)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(991,997,1009)"
} | {
"split": "train",
"index": 17567,
"solution": "Let $f(x)=x(x-12)(x+6)+320=x^{3}-6 x^{2}-72 x+320$, so that $f(1001)=989 \\cdot 1001 \\cdot 1007+320$. But $f(4)=4(-8)(10)+320=0$, so $f(x)=(x-4)\\left(x^{2}-2 x-80\\right)=(x-4)(x-10)(x+8)$ Thus $f(1001)=991 \\cdot 997 \\cdot 1009$, as desired."
} |
deepscaler | [
{
"role": "user",
"content": "Let $A, B, C$ be points in that order along a line, such that $A B=20$ and $B C=18$. Let $\\omega$ be a circle of nonzero radius centered at $B$, and let $\\ell_{1}$ and $\\ell_{2}$ be tangents to $\\omega$ through $A$ and $C$, respectively. Let $K$ be the intersection of $\\el... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 17568,
"solution": "Note that $B$ is the $K$-excenter of $K X Y$, so $X B$ is the angle bisector of $\\angle A K Y$. As $A B$ and $X Y$ are parallel, $\\angle X A B+2 \\angle A X B=180^{\\circ}$, so $\\angle X B A=180^{\\circ}-\\angle A X B-\\angle X A B$. This means that $A X B$ is i... |
deepscaler | [
{
"role": "user",
"content": "Find all 4-digit numbers $n$ , such that $n=pqr$ , where $p<q<r$ are distinct primes, such that $p+q=r-q$ and $p+q+r=s^2$ , where $s$ is a prime number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2015"
} | {
"split": "train",
"index": 17569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $abc$ represents a three-digit number, if it satisfies $a \\lt b$ and $b \\gt c$, then we call this three-digit number a \"convex number\". The number of three-digit \"convex\" numbers without repeated digits is ______. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "204"
} | {
"split": "train",
"index": 17570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bin has 5 green balls and $k$ purple balls in it, where $k$ is an unknown positive integer. A ball is drawn at random from the bin. If a green ball is drawn, the player wins 2 dollars, but if a purple ball is drawn, the player loses 2 dollars. If the expected amount won fo... | math | {
"style": "rule",
"ground_truth": "k=3"
} | {
"split": "train",
"index": 17571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the length of the diagonal and the area of a rectangle whose one corner is at (1, 1) and the opposite corner is at (9, 7). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 17572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a \\in \\mathbb{R}$, the function $f(x) = ax^3 - 3x^2$, and $x = 2$ is an extreme point of the function $y = f(x)$.\n\n1. Find the value of $a$.\n2. Find the extreme values of the function $f(x)$ in the interval $[-1, 5]$. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 17573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A portion of the graph of $y = f(x)$ is shown in red below, where $f(x)$ is a quadratic function. The distance between grid lines is $1$ unit.\n\nWhat is the sum of all distinct numbers $x$ such that $f(f(f(x)))=-3$ ?\n\n[asy]\n\nsize(150);\n\nreal ticklen=3;\n\nreal tickspace... | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 17574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a multiple-choice math question has incorrect options that cannot be adjacent, calculate the total number of arrangements of $4$ different options that meet the requirements. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 17575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A company purchases 400 tons of a certain type of goods annually. Each purchase is of $x$ tons, and the freight cost is 40,000 yuan per shipment. The annual total storage cost is 4$x$ million yuan. To minimize the sum of the annual freight cost and the total storage cost, find ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 17576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two types of products, A and B, with profits of $p$ ten-thousand yuan and $q$ ten-thousand yuan, respectively. Their relationship with the invested capital $x$ ten-thousand yuan is: $p= \\frac{1}{5}x$, $q= \\frac{3}{5} \\sqrt{x}$. Now, with an investment of 3 ten-thou... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{20}"
} | {
"split": "train",
"index": 17577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be a positive real number. Define\n\\[\n A = \\sum_{k=0}^{\\infty} \\frac{x^{3k}}{(3k)!}, \\quad\n B = \\sum_{k=0}^{\\infty} \\frac{x^{3k+1}}{(3k+1)!}, \\quad\\text{and}\\quad\n C = \\sum_{k=0}^{\\infty} \\frac{x^{3k+2}}{(3k+2)!}.\n\\] Given that $A^3+B^3+C^3 + 8A... | math | {
"style": "rule",
"ground_truth": "183"
} | {
"split": "train",
"index": 17578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence ${{a_{n}}}$ where all terms are non-zero, the sum of the first $n$ terms is ${{S_{n}}}$, and it satisfies ${{a_{1}}=a,}$ $2{{S_{n}}={{a_{n}}{{a_{n+1}}}}}$.\n\n(I) Find the value of ${{a_{2}}}$;\n(II) Find the general formula for the $n^{th}$ term of the sequenc... | math | {
"style": "rule",
"ground_truth": "-15"
} | {
"split": "train",
"index": 17579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), it is known that \\(AB = 3\\), \\(AC = 3\\sqrt{7}\\), and \\(\\angle ABC = 60^\\circ\\). The bisector of angle \\(ABC\\) is extended to intersect at point \\(D\\) with the circle circumscribed around the triangle. Find \\(BD\\). Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 17580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Distribute 7 students into two dormitories, A and B, with each dormitory having at least 2 students. How many different distribution plans are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 17581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\\begin{cases} x=1+\\cos \\alpha \\\\ y=\\sin \\alpha\\end{cases}$ ($\\alpha$ is the parameter), and in the polar coordinate system with the origin as the pole and the positive $x$-axis as the po... | math | {
"style": "rule",
"ground_truth": "\\sqrt {37}-1"
} | {
"split": "train",
"index": 17582,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the integer solutions to the system of inequalities\n\\[\n\\begin{cases}\n9x - a \\geq 0, \\\\\n8x - b < 0\n\\end{cases}\n\\]\nare only 1, 2, and 3, find the number of ordered pairs \\((a, b)\\) of integers that satisfy this system. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 17583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When three positive integers are divided by $12$, the remainders are $7,$ $9,$ and $10,$ respectively.\n\nWhen the sum of the three integers is divided by $12$, what is the remainder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 17584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the m... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{8}"
} | {
"split": "train",
"index": 17585,
"solution": "To solve this problem, we need to understand how the iterative average process works and how the order of numbers affects the final result. The iterative average process can be thought of as a weighted average where numbers added later in the sequence have a greate... |
deepscaler | [
{
"role": "user",
"content": "Find the units digit of the following within the indicated number base: $52_7 + 62_7$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 17586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(x)=x^3+ax^2+bx+c$ be a polynomial where $a,b,c$ are integers and $c$ is odd. Let $p_{i}$ be the value of $P(x)$ at $x=i$ . Given that $p_{1}^3+p_{2}^{3}+p_{3}^{3}=3p_{1}p_{2}p_{3}$ , find the value of $p_{2}+2p_{1}-3p_{0}.$ Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 17587,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a psychiatric hospital, there is a chief doctor and many madmen. During the week, each madman bit someone once a day (possibly themselves). At the end of the week, it was found that each patient has two bites, and the chief doctor has one hundred bites. How many madmen are t... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 17588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ordered triples \\((x, y, z)\\) satisfy the following conditions:\n\\[ x^2 + y^2 + z^2 = 9, \\]\n\\[ x^4 + y^4 + z^4 = 33, \\]\n\\[ xyz = -4? \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 17589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the solution set of the inequality $x^{2}-2x+1-m^{2} \\lt 0$ is $A$; \n$(1)$ Find $A$; \n$(2)$ If $0 \\lt m \\lt 1$, and $A=\\{x\\left|\\right.a \\lt x \\lt b\\}$, find the minimum value of $\\frac{1}{{8a+2b}}-\\frac{1}{{3a-3b}}$. Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 17590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $a$, $b$, and $c$ are positive integers such that $a \\geq b \\geq c$ and $a+b+c=2010$. Furthermore, $a!b!c! = m \\cdot 10^n$, where $m$ and $n$ are integers, and $m$ is not divisible by $10$. What is the smallest possible value of $n$?\nA) 499\nB) 500\nC) 502\nD) 504\n... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 17591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 5 female students and 2 male students in a class. Find the number of different distribution schemes in which they can be divided into two groups, with each group having both female and male students. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 17592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many triangles with integer sides have a perimeter equal to 27? (Triangles that differ only in the order of sides, such as 7, 10, 10 and 10, 10, 7, are considered the same triangle.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 17593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One line is parameterized by\n\\[\\begin{pmatrix} -1 + s \\\\ 3 - ks \\\\ 1 + ks \\end{pmatrix}.\\]Another line is parameterized by\n\\[\\begin{pmatrix} t/2 \\\\ 1 + t \\\\ 2 - t \\end{pmatrix}.\\]If the lines are coplanar (i.e. there is a plane that contains both lines), then ... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 17594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A, B, C, D$ be points chosen on a circle, in that order. Line $B D$ is reflected over lines $A B$ and $D A$ to obtain lines $\\ell_{1}$ and $\\ell_{2}$ respectively. If lines $\\ell_{1}, \\ell_{2}$, and $A C$ meet at a common point and if $A B=4, B C=3, C D=2$, compute the... | math | {
"style": "rule",
"ground_truth": "\\sqrt{21}"
} | {
"split": "train",
"index": 17595,
"solution": "Let the common point be $E$. Then since lines $B E$ and $B D$ are symmetric about line $B A, B A$ is an exterior bisector of $\\angle D B E$, and similarly $D A$ is also an exterior bisector of $\\angle B D E$. Therefore $A$ is the $E$-excenter of triangle $B D E$ ... |
deepscaler | [
{
"role": "user",
"content": "The perpendicular bisectors of the sides of triangle $PQR$ meet its circumcircle at points $P',$ $Q',$ and $R',$ respectively. If the perimeter of triangle $PQR$ is 30 and the radius of the circumcircle is 7, then find the area of hexagon $PQ'RP'QR'.$ Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 17596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We are given 5771 weights weighing 1,2,3,...,5770,5771. We partition the weights into $n$ sets of equal weight. What is the maximal $n$ for which this is possible? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2886"
} | {
"split": "train",
"index": 17597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) such that for any given \\( n \\) rectangles with side lengths not exceeding 100, there always exist 3 rectangles \\( R_{1}, R_{2}, R_{3} \\) such that \\( R_{1} \\) can be nested inside \\( R_{2} \\) and \\( R_{2} \\) can be nested ... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 17598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{v}$ and $\\mathbf{w}$ be the vectors such that $\\mathbf{v} \\cdot \\mathbf{w} = -3$ and $\\|\\mathbf{w}\\| = 5.$ Find the magnitude of $\\operatorname{proj}_{\\mathbf{w}} \\mathbf{v}.$ 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": 17599,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For finite sets $A$ and $B$ , call a function $f: A \\rightarrow B$ an \\emph{antibijection} if there does not exist a set $S \\subseteq A \\cap B$ such that $S$ has at least two elements and, for all $s \\in S$ , there exists exactly one element $s'$ of $S$ such ... | math | {
"style": "rule",
"ground_truth": "1363641"
} | {
"split": "train",
"index": 17600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of the length to the width of a rectangle is $4$ : $3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12}{25}"
} | {
"split": "train",
"index": 17601,
"solution": "1. **Assign Variables to Dimensions:**\n Let the length of the rectangle be $4x$ and the width be $3x$. This assignment is based on the given ratio of length to width, which is $4:3$.\n\n2. **Use the Pythagorean Theorem:**\n The diagonal $d$ of the rectangle fo... |
deepscaler | [
{
"role": "user",
"content": "The number of lattice points (points with integer coordinates) inside the region bounded by the right branch of the hyperbola \\( x^{2}-y^{2}=1 \\) and the line \\( x=100 \\), excluding the boundary, is \\(\\qquad\\) . Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "9800"
} | {
"split": "train",
"index": 17602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a_n\\}$ satisfies $\\{a_1=2, a_2=1,\\}$ and $\\frac{a_n \\cdot a_{n-1}}{a_{n-1}-a_n}=\\frac{a_n \\cdot a_{n+1}}{a_n-a_{n+1}}(n\\geqslant 2)$, determine the $100^{\\text{th}}$ term of the sequence $\\{a_n\\}$. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{50}"
} | {
"split": "train",
"index": 17603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), point \\(P\\) is located one-third of the way along segment \\(AB\\) closer to point \\(A\\). Point \\(R\\) is one-third of the way along segment \\(PB\\) closer to point \\(P\\), and point \\(Q\\) lies on segment \\(BC\\) such that angles \\(PCB\\) and \... | math | {
"style": "rule",
"ground_truth": "9:2"
} | {
"split": "train",
"index": 17604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given Angela has 4 files that are each 1.2 MB, 8 files that are 0.9 MB each, and 10 files that are 0.6 MB each, calculate the minimum number of disks Angela will need to store all her files. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 17605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the ten volumes of the collected works of Theodore Sturgeon is available in paperback for $\\$$15 or in hardcover for $\\$$25. Theresa buys a copy of each of the ten volumes for a total of $\\$$220. How many hardcover volumes did she buy? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 17606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive value of $x$ which satisfies\n\\[\\log_5 (x + 2) + \\log_{\\sqrt{5}} (x^2 + 2) + \\log_{\\frac{1}{5}} (x + 2) = 3.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\sqrt{125} - 2}"
} | {
"split": "train",
"index": 17607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the derivative of the $(n-1)$th derivative as the $n$th derivative $(n \\in N^{*}, n \\geqslant 2)$, that is, $f^{(n)}(x)=[f^{(n-1)}(x)]'$. They are denoted as $f''(x)$, $f'''(x)$, $f^{(4)}(x)$, ..., $f^{(n)}(x)$. If $f(x) = xe^{x}$, then the $2023$rd derivative of the f... | math | {
"style": "rule",
"ground_truth": "-\\frac{2023}{2024}"
} | {
"split": "train",
"index": 17608,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The side lengths of a cyclic quadrilateral are 25, 39, 52, and 60. What is the diameter of the circle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 17609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $x$ is $x^2 < 7x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 17610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle of radius $2$ is centered at $O$. Square $OABC$ has side length $1$. Sides $AB$ and $CB$ are extended past $B$ to meet the circle at $D$ and $E$, respectively. What is the area of the shaded region in the figure, which is bounded by $BD$, $BE$, and the minor arc connec... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}+1-\\sqrt{3}"
} | {
"split": "train",
"index": 17611,
"solution": "1. **Identify the Geometry and Key Points**: We have a circle centered at $O$ with radius $2$, and a square $OABC$ with side length $1$. The sides $AB$ and $CB$ are extended to meet the circle at points $D$ and $E$ respectively.\n\n2. **Calculate $DA$ and $CE$ Usin... |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[ x^6 - 3x^3 - x^2 - x - 2 = q_1(x) q_2(x) \\dotsm q_m(x), \\]\nwhere each non-constant polynomial $q_i(x)$ is monic with integer coefficients, and cannot be factored further over the integers. Compute $q_1(3) + q_2(3) + \\dots + q_m(3)$. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "634"
} | {
"split": "train",
"index": 17612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, if $\\angle A=60^{\\circ}$, $\\angle C=45^{\\circ}$, and $b=4$, then the smallest side of this triangle is $\\_\\_\\_\\_\\_\\_\\_.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}-4"
} | {
"split": "train",
"index": 17613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, triangle $ABC$ has been reflected over its median $\\overline{AM}$ to produce triangle $AB'C'$. If $AE = 6$, $EC =12$, and $BD = 10$, then find $AB$.\n\n[asy]\nsize(250);\npair A,B,C,D,M,BB,CC,EE;\nB = (0,0);\nD = (10,0);\nM = (15,0);\nC=2*M;\nA = D + (sc... | math | {
"style": "rule",
"ground_truth": "8\\sqrt{3}"
} | {
"split": "train",
"index": 17614,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For real numbers $x$, let \\[f(x) = \\left\\{\n\\begin{array}{cl}\nx+2 &\\text{ if }x>3, \\\\\n2x+a &\\text{ if }x\\le 3.\n\\end{array}\n\\right.\\]What must the value of $a$ be to make the piecewise function continuous (which means that its graph can be drawn without lifting y... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 17615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can $17$ identical red and $10$ identical white balls be distributed into $4$ distinct boxes such that the number of red balls is greater than the number of white balls in each box? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "5720"
} | {
"split": "train",
"index": 17616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive integers $n$ and $k$, let $f(n, k)$ be the remainder when $n$ is divided by $k$, and for $n > 1$ let $F(n) = \\max_{\\substack{1\\le k\\le \\frac{n}{2}}} f(n, k)$. Find the remainder when $\\sum\\limits_{n=20}^{100} F(n)$ is divided by $1000$. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "512"
} | {
"split": "train",
"index": 17617,
"solution": "$Lemma:$ Highest remainder when $n$ is divided by $1\\leq k\\leq n/2$ is obtained for $k_0 = (n + (3 - n \\pmod{3}))/3$ and the remainder thus obtained is $(n - k_0*2) = [(n - 6)/3 + (2/3)*n \\pmod{3}]$.\n$Note:$ This is the second highest remainder when $n$ is div... |
deepscaler | [
{
"role": "user",
"content": "Let $f$ be a function from the nonnegative integers to the positive reals such that $f(x+y)=f(x) \\cdot f(y)$ holds for all nonnegative integers $x$ and $y$. If $f(19)=524288 k$, find $f(4)$ in terms of $k$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "16 k^{4 / 19}"
} | {
"split": "train",
"index": 17618,
"solution": "The given condition implies $f(m n)=f(m)^{n}$, so $$f(4)^{19}=f(4 \\cdot 19)=f(19 \\cdot 4)=f(19)^{4}$$ and it follows that $f(4)=16 k^{4 / 19}$."
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equation \\( 2 \\sqrt{2} \\sin ^{3}\\left(\\frac{\\pi x}{4}\\right) = \\sin \\left(\\frac{\\pi}{4}(1+x)\\right) \\).\n\nHow many solutions of this equation satisfy the condition: \\( 2000 \\leq x \\leq 3000 \\)? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 17619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S^{}_{}$ be the set of all rational numbers $r^{}_{}$, $0^{}_{}<r<1$, that have a repeating decimal expansion in the form $0.abcabcabc\\ldots=0.\\overline{abc}$, where the digits $a^{}_{}$, $b^{}_{}$, and $c^{}_{}$ are not necessarily distinct. To write the elements of $S^... | math | {
"style": "rule",
"ground_truth": "660"
} | {
"split": "train",
"index": 17620,
"solution": "We can see that the denominator of the fraction will be 999b, where b is either 1, 1/3, 1/111, 1/9, or 1/333. This means that the numerator stays abc unless abc divides 999. Finding the numbers relatively prime to 999 gives us the numerators when the fraction canno... |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, $sin(A+\\frac{π}{4})sin(B+\\frac{π}{4})=cosAcosB$. Find:<br/>\n$(1)$ the value of angle $C$;<br/>\n$(2)$ if $AB=\\sqrt{2}$, find the minimum value of $\\overrightarrow{CA}•\\overrightarrow{CB}$. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "-\\sqrt{2}+1"
} | {
"split": "train",
"index": 17621,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each successive term in the sequence $2048, 512, 128, x,\ny, 2, \\frac 12, \\frac 18,...$ is obtained by multiplying the previous term by a constant. What is the value of $x + y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 17622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the roots of the equation $(x-a)(x-b)=(x-c)(x-d)$, if you know that $a+d=b+c=2015$ and $a \\ne c$ (numbers $a, b, c, d$ are not given). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2015}{2}"
} | {
"split": "train",
"index": 17623,
"solution": "\nTo find the roots of the equation \\((x-a)(x-b)=(x-c)(x-d)\\), given that \\(a + d = b + c = 2015\\) and \\(a \\neq c\\), we will simplify the equation and determine the solutions.\n\n### Step 1: Expand Both Sides\nExpanding both sides of the equation, we have:\n... |
deepscaler | [
{
"role": "user",
"content": "One and one-half of what number is 30? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 17624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gavrila is in an elevator cabin which is moving downward with a deceleration of 5 m/s². Find the force with which Gavrila presses on the floor. Gavrila's mass is 70 kg, and the acceleration due to gravity is 10 m/s². Give the answer in newtons, rounding to the nearest whole num... | math | {
"style": "rule",
"ground_truth": "350"
} | {
"split": "train",
"index": 17625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a, b, c \\) be the side lengths of a right triangle, with \\( a \\leqslant b < c \\). Determine the maximum constant \\( k \\) such that \\( a^{2}(b+c) + b^{2}(c+a) + c^{2}(a+b) \\geqslant k a b c \\) holds for all right triangles, and identify when equality occurs. Let... | math | {
"style": "rule",
"ground_truth": "2 + 3\\sqrt{2}"
} | {
"split": "train",
"index": 17626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular table measures $12'$ in length and $9'$ in width and is currently placed against one side of a rectangular room. The owners desire to move the table to lay diagonally in the room. Determine the minimum length of the shorter side of the room, denoted as $S$, in fee... | math | {
"style": "rule",
"ground_truth": "15'"
} | {
"split": "train",
"index": 17627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a cube $ABCDEFGH$ where each side has length $2$ units. Find $\\sin \\angle GAC$. (Consider this by extending the calculations needed for finding $\\sin \\angle HAC$) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 17628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five doctors A, B, C, D, and E are assigned to four different service positions located in the Sichuan disaster zone, labeled A, B, C, and D. Each position must be filled by at least one doctor. Calculate the total number of ways doctors A and B can serve independently in diffe... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 17629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two regular tetrahedrons $A$ and $B$ are made with the 8 vertices of a unit cube. (this way is unique)\r\n\r\nWhat's the volume of $A\\cup B$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 17630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, which is equilateral with a side length $s$, lines $\\overline{LM}$, $\\overline{NO}$, and $\\overline{PQ}$ are parallel to $\\overline{YZ}$, and $XL = LN = NP = QY$. Determine the ratio of the area of trapezoid $PQYZ$ to the area of triangle $XYZ$. Let's thi... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"split": "train",
"index": 17631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r$ be a real number, $|r| < 2,$ and let $z$ be a complex number such that\n\\[z + \\frac{1}{z} = r.\\]Find $|z|.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 17632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The digits in a two-digit positive integer are reversed. The new two-digit integer minus the original integer equals 54. What is the positive difference between the two digits of the original integer? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 17633,
"solution": "Suppose that the original integer has tens digit $a$ and ones (units) digit $b$. This integer is equal to $10 a+b$. When the digits are reversed, the tens digit of the new integer is $b$ and the ones digit is $a$. This new integer is equal to $10 b+a$. Since the ne... |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers that can be represented in decimal form as $\\overline{13 x y 45 z}$ and are divisible by 792, where $x, y, z$ are unknown digits. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1380456"
} | {
"split": "train",
"index": 17634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in △ABC, the sides opposite to the internal angles A, B, and C are a, b, and c respectively, and $b^{2}=c^{2}+a^{2}- \\sqrt {2}ac$.\n(I) Find the value of angle B;\n(II) If $a= \\sqrt {2}$ and $cosA= \\frac {4}{5}$, find the area of △ABC. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac {7}{6}"
} | {
"split": "train",
"index": 17635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}$ be unit vectors such that $\\mathbf{a} \\cdot \\mathbf{b} = \\mathbf{a} \\cdot \\mathbf{c} = 0,$ and the angle between $\\mathbf{b}$ and $\\mathbf{c}$ is $\\frac{\\pi}{4}.$ Then\n\\[\\mathbf{a} = k (\\mathbf{b} \\times \\math... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}, -\\sqrt{2}"
} | {
"split": "train",
"index": 17636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a \\(10 \\times 10\\) grid, there are 11 horizontal grid lines and 11 vertical grid lines. The line segments connecting adjacent nodes on the same line are called \"links.\" What is the minimum number of links that must be removed so that at each node, there are at most 3 re... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 17637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\left(\\dfrac{3}{4}\\right)^5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{243}{1024}"
} | {
"split": "train",
"index": 17638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For some positive integer $n$, the number $120n^3$ has $120$ positive integer divisors, including $1$ and the number $120n^3$. How many positive integer divisors does the number $64n^4$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "375"
} | {
"split": "train",
"index": 17639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$f (x) = x + 3$ and $g(x) = x^2 -6$, what is the value of $f (g(2))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 17640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the complex plane, the parallelogram formed by the points 0, $z,$ $\\frac{1}{z},$ and $z + \\frac{1}{z}$ has an area of $\\frac{12}{13}.$ If the real part of $z$ is positive, compute the smallest possible value of $\\left| z + \\frac{1}{z} \\right|^2.$ Let's think step by st... | math | {
"style": "rule",
"ground_truth": "\\frac{36}{13}"
} | {
"split": "train",
"index": 17641,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers 2, 3, 5, 7, 11, 13 are arranged in a multiplication table, with three along the top and the other three down the left. The multiplication table is completed and the sum of the nine entries is tabulated. What is the largest possible sum of the nine entries?\n\n\\[\... | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 17642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $35_8-74_8?$ Express your answer in base 8. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-37_8"
} | {
"split": "train",
"index": 17643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the $8$ vertices of a cube, select $4$ vertices. The probability that these $4$ vertices lie in the same plane is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{35}"
} | {
"split": "train",
"index": 17644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three positive integers $a,$ $b,$ and $x$ form an O'Hara triple $(a,b,x)$ if $\\sqrt{a}+\\sqrt{b}=x.$ For example, $(1,4,3)$ is an O'Hara triple because $\\sqrt{1}+\\sqrt{4}=3.$\n\nIf $(36,25,x)$ is an O'Hara triple, determine the value of $x.$ Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 17645,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\(a\\) copies of a right-angled isosceles triangle with hypotenuse \\(\\sqrt{2} \\, \\mathrm{cm}\\) can be assembled to form a trapezium with perimeter equal to \\(b \\, \\mathrm{cm}\\), find the least possible value of \\(b\\). (Give the answer in surd form.) Let's think s... | math | {
"style": "rule",
"ground_truth": "4 + 2\\sqrt{2}"
} | {
"split": "train",
"index": 17646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The parametric equation of line $l$ is \n$$\n\\begin{cases} \nx= \\frac { \\sqrt {2}}{2}t \\\\ \ny= \\frac { \\sqrt {2}}{2}t+4 \\sqrt {2}\n\\end{cases}\n$$\n(where $t$ is the parameter), and the polar equation of circle $c$ is $\\rho=2\\cos(\\theta+ \\frac{\\pi}{4})$. Tangent l... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{6}"
} | {
"split": "train",
"index": 17647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The circle centered at point $A$ with radius $19$ and the circle centered at point $B$ with radius $32$ are both internally tangent to a circle centered at point $C$ with radius $100$ such that point $C$ lies on segment $\\overline{AB}$ . Point $M$ is on the ci... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 17648,
"solution": ""
} |
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