data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In Theresa's first $8$ basketball games, she scored $7, 4, 3, 6, 8, 3, 1$ and $5$ points. In her ninth game, she scored fewer than $10$ points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than $10$ points and h... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 4200,
"solution": "1. **Calculate the total points from the first 8 games:**\n \\[\n 7 + 4 + 3 + 6 + 8 + 3 + 1 + 5 = 37\n \\]\n\n2. **Determine the points needed in the ninth game for an integer average:**\n - The total points after 9 games must be a multiple of 9 for the aver... |
deepscaler | [
{
"role": "user",
"content": "Given $$∫_{ 0 }^{ 2 }(\\cos \\frac {π}{4}x+ \\sqrt {4-x^{2}})dx$$, evaluate the definite integral. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi+\\frac{4}{\\pi}"
} | {
"split": "train",
"index": 4201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The four zeros of the polynomial \\(x^4 + px^2 + qx - 144\\) are distinct real numbers in arithmetic progression. Compute the value of \\(p.\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-40"
} | {
"split": "train",
"index": 4202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{P}$ be the matrix for projecting onto the vector $\\begin{pmatrix} -3 \\\\ -2 \\end{pmatrix}.$ Find $\\mathbf{P}^{-1}.$\n\nIf the inverse does not exist, then enter the zero matrix. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 0 & 0 \\\\ 0 & 0 \\end{pmatrix}"
} | {
"split": "train",
"index": 4203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Equilateral $\\triangle ABC$ is inscribed in a circle of radius $2$. Extend $\\overline{AB}$ through $B$ to point $D$ so that $AD=13,$ and extend $\\overline{AC}$ through $C$ to point $E$ so that $AE = 11.$ Through $D,$ draw a line $l_1$ parallel to $\\overline{AE},$ and throug... | math | {
"style": "rule",
"ground_truth": "865"
} | {
"split": "train",
"index": 4204,
"solution": "Note that $AB=2\\sqrt3$, $DF=11$, and $EF=13$. If we take a homothety of the parallelogram with respect to $A$, such that $F$ maps to $G$, we see that $\\frac{[ABG]}{[ACG]}=\\frac{11}{13}$. Since $\\angle AGB=\\angle AGC=60^{\\circ}$, from the sine area formula we h... |
deepscaler | [
{
"role": "user",
"content": "In the equation $\\frac{1}{j} + \\frac{1}{k} = \\frac{1}{4}$, where $j$ and $k$ are positive integers, find the sum of all possible values for $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 4205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\mathbf{7 3 8 , 8 2 6}$. This can be arrived at by stepping down, starting with finding how many combinations are there that begin with a letter other than V or W , and so forth. The answer is $\\frac{8 \\cdot 9!}{2 \\cdot 2}+\\frac{4 \\cdot 7!}{2}+4 \\cdot 6!+4 \\cdot 4!+3!+... | math | {
"style": "rule",
"ground_truth": "738826"
} | {
"split": "train",
"index": 4206,
"solution": "The number of combinations is 738826."
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest common divisor (GCD) and the sum of the numbers 1729 and 867? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2596"
} | {
"split": "train",
"index": 4207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a geometric sequence where the first term is $\\frac{5}{8}$, and the second term is $25$. What is the smallest $n$ for which the $n$th term of the sequence, multiplied by $n!$, is divisible by one billion (i.e., $10^9$)? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 4208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A stalker, to detect a gravitational anomaly (an area where the acceleration due to gravity changes sharply in magnitude), throws a small nut from the surface of the Earth at an angle \\(\\alpha = 30^\\circ\\) to the horizontal with a speed \\(v_0 = 20 \\, \\text{m/s}\\). The n... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 4209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mary told John her score on the American High School Mathematics Examination (AHSME), which was over $80$. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over $80$, John could not have determined... | math | {
"style": "rule",
"ground_truth": "119"
} | {
"split": "train",
"index": 4210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x_{1}, x_{2}, \\ldots, x_{100} \\) be natural numbers greater than 1 (not necessarily distinct). In a \\( 100 \\times 100 \\) table, numbers are placed as follows: at the intersection of the \\( i \\)-th row and the \\( k \\)-th column, the number \\( \\log _{x_{k}} \\f... | math | {
"style": "rule",
"ground_truth": "-10000"
} | {
"split": "train",
"index": 4211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a sequence of positive real numbers where \\( a_1, a_2, \\dots \\) satisfy\n\\[ a_n = 9a_{n-1} - n \\]\nfor all \\( n > 1 \\). Find the smallest possible value of \\( a_1 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{64}"
} | {
"split": "train",
"index": 4212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains 20 candies: 4 chocolate, 6 mint, and 10 butterscotch. Candies are removed randomly from the bag and eaten. What is the minimum number of candies that must be removed to be certain that at least two candies of each flavor have been eaten? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 4213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a natural number \\( N \\), if at least five of the natural numbers from 1 to 9 can divide \\( N \\) evenly, then \\( N \\) is called a \"Five-Divisible Number.\" Find the smallest \"Five-Divisible Number\" that is greater than 2000. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"split": "train",
"index": 4214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A confectionery factory received 5 spools of ribbon, each 60 meters long, for packaging cakes. How many cuts are needed to obtain pieces of ribbon, each 1 meter 50 centimeters long? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "195"
} | {
"split": "train",
"index": 4215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In order to compute the area of a particular circle, Juan first measures the length of its diameter. The actual diameter is 20 cm, but Juan's measurement has an error of up to $20\\%$. What is the largest possible percent error, in percent, in Juan's computed area of the circle... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 4216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of degrees in the smallest positive angle $x$ such that\n\\[8 \\sin x \\cos^5 x - 8 \\sin^5 x \\cos x = 1.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.5^\\circ"
} | {
"split": "train",
"index": 4217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be a set, $|S|=35$. A set $F$ of mappings from $S$ to itself is called to be satisfying property $P(k)$, if for any $x,y\\in S$, there exist $f_1, \\cdots, f_k \\in F$ (not necessarily different), such that $f_k(f_{k-1}(\\cdots (f_1(x))))=f_k(f_{k-1}(\\cdots (f_1(y))))$... | math | {
"style": "rule",
"ground_truth": "595"
} | {
"split": "train",
"index": 4218,
"solution": "\nLet \\( S \\) be a set with \\( |S| = 35 \\). A set \\( F \\) of mappings from \\( S \\) to itself is said to satisfy property \\( P(k) \\) if for any \\( x, y \\in S \\), there exist \\( f_1, f_2, \\ldots, f_k \\in F \\) (not necessarily different) such that \\( ... |
deepscaler | [
{
"role": "user",
"content": "Square $PQRS$ has side length $2$ units. Points $T$ and $U$ are on sides $PQ$ and $SQ$, respectively, with $PT = SU$. When the square is folded along the lines $RT$ and $RU$, sides $PR$ and $SR$ coincide and lie on diagonal $RQ$. Find the length of segment $PT$ which can be exp... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 4219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $(1_8 + 2_8 + 3_8 + \\cdots + 30_8) \\times 3_{10}$, expressed in base $8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1604_8"
} | {
"split": "train",
"index": 4220,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $63$ houses at the distance of $1, 2, 3, . . . , 63 \\text{ km}$ from the north pole, respectively. Santa Clause wants to distribute vaccine to each house. To do so, he will let his assistants, $63$ elfs named $E_1, E_2, . . . , E_{63}$ , deliever the vaccine ... | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"split": "train",
"index": 4221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $E, I, L, V$ are (not necessarily distinct) nonzero digits in base ten for which the four-digit number $\\underline{E} \\underline{V} \\underline{I} \\underline{L}$ is divisible by 73 , and the four-digit number $\\underline{V} \\underline{I} \\underline{L} \\underline{... | math | {
"style": "rule",
"ground_truth": "9954"
} | {
"split": "train",
"index": 4222,
"solution": "Let $\\underline{E}=2 k$ and $\\underline{V} \\underline{I} \\underline{L}=n$. Then $n \\equiv-2000 k(\\bmod 73)$ and $n \\equiv-k / 5(\\bmod 37)$, so $n \\equiv 1650 k(\\bmod 2701)$. We can now exhaustively list the possible cases for $k$ : - if $k=1$, then $n \\eq... |
deepscaler | [
{
"role": "user",
"content": "What is the product of the [real](https://artofproblemsolving.com/wiki/index.php/Real) [roots](https://artofproblemsolving.com/wiki/index.php/Root) of the [equation](https://artofproblemsolving.com/wiki/index.php/Equation) $x^2 + 18x + 30 = 2 \\sqrt{x^2 + 18x + 45}$? Let's thin... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 4223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $t = 1, 2, 3, 4$, define $S_t = \\sum_{i = 1}^{350}a_i^t$, where $a_i \\in \\{1,2,3,4\\}$. If $S_1 = 513$ and $S_4 = 4745$, find the minimum possible value for $S_2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "905"
} | {
"split": "train",
"index": 4224,
"solution": "Because the order of the $a$'s doesn't matter, we simply need to find the number of $1$s $2$s $3$s and $4$s that minimize $S_2$. So let $w, x, y,$ and $z$ represent the number of $1$s, $2$s, $3$s, and $4$s respectively. Then we can write three equations based on the... |
deepscaler | [
{
"role": "user",
"content": "Given real numbers \\( x, y, z, w \\) satisfying \\( x + y + z + w = 1 \\), find the maximum value of \\( M = xw + 2yw + 3xy + 3zw + 4xz + 5yz \\). 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": 4225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circular sheet of iron with a radius of 6 has a sector removed, which is $\\frac{1}{6}$ of the original area. The remaining part is rolled into the lateral surface of a cone. The volume of the cone is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "\\frac{25\\sqrt{11}}{3}\\pi"
} | {
"split": "train",
"index": 4226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f: \\mathbb{R}^+ \\to \\mathbb{R}^+$ such that for each $x \\in \\mathbb{R}^+$, there is exactly one $y \\in \\mathbb{R}^+$ satisfying $$xf(y)+yf(x) \\leq 2$$ Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "f(x) = \\frac{1}{x}"
} | {
"split": "train",
"index": 4227,
"solution": "\nTo solve the given functional equation problem, we must find all functions \\( f: \\mathbb{R}^+ \\to \\mathbb{R}^+ \\) such that for each \\( x \\in \\mathbb{R}^+ \\), there is exactly one \\( y \\in \\mathbb{R}^+ \\) satisfying\n\n\\[\nxf(y) + yf(x) \\leq 2.\n\\]... |
deepscaler | [
{
"role": "user",
"content": "For all real numbers $x$ and $y$, define the mathematical operation $\\star$ such that the following conditions apply: $x\\ \\star\\ 0 = x+1, x\\ \\star\\ y = y\\ \\star\\ x$, and $(x + 2)\\ \\star\\ y = (x\\ \\star\\ y) + y + 2$. What is the value of $7\\ \\star\\ 3$? Let's th... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 4228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Of the numbers $\\frac{8}{12}, \\frac{5}{6}$, and $\\frac{9}{12}$, which number is the arithmetic mean of the other two? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 4229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with radius \\(5\\) has its center on the \\(x\\)-axis, and the \\(x\\)-coordinate of the center is an integer. The circle is tangent to the line \\(4x+3y-29=0\\).\n(Ⅰ) Find the equation of the circle;\n(Ⅱ) Let the line \\(ax-y+5=0\\) (\\(a > 0\\)) intersect the circle... | math | {
"style": "rule",
"ground_truth": "a = \\dfrac {3}{4}"
} | {
"split": "train",
"index": 4230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Meteorological observations. At the weather station, it was noticed that during a certain period of time, if it rained in the morning, then the evening was clear, and if it rained in the evening, then the morning was clear. There were a total of 9 rainy days: 6 times there were... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 4231,
"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": 4232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $c\\sin\\frac{A+C}{2}=b\\sin C$. \n$(1)$ Find angle $B$; \n$(2)$ Let $BD$ be the altitude from $B$ to side $AC$, and $BD=1$, $b=\\sqrt{3}$. Find the perimeter ... | math | {
"style": "rule",
"ground_truth": "3 + \\sqrt{3}"
} | {
"split": "train",
"index": 4233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve in $ \\mathbb{Z}^2 $ the equation: $ x^2\\left( 1+x^2 \\right) =-1+21^y. $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0, 0), (2, 1), (-2, 1)"
} | {
"split": "train",
"index": 4234,
"solution": "\nConsider the equation in integers \\( \\mathbb{Z}^2 \\):\n\\[\nx^2 (1 + x^2) = -1 + 21^y.\n\\]\n\nFirst, rewrite the equation as:\n\\[\nx^2 + x^4 = -1 + 21^y.\n\\]\nThus, we have:\n\\[\nx^4 + x^2 + 1 = 21^y.\n\\]\n\nWe're tasked with finding integer solutions \\((... |
deepscaler | [
{
"role": "user",
"content": "9 judges each award 20 competitors a rank from 1 to 20. The competitor's score is the sum of the ranks from the 9 judges, and the winner is the competitor with the lowest score. For each competitor, the difference between the highest and lowest ranking (from different judges) i... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 4235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, with $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. The polar coordinates of point $P$ are $(3, \\frac{\\pi}{4})$. The parametric equation of curve $C$ is $\\r... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}-1}{2}"
} | {
"split": "train",
"index": 4236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest number of acute angles that a convex hexagon can have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 4237,
"solution": "1. **Calculate the sum of interior angles of a hexagon**: \n The sum of the interior angles of a polygon with $n$ sides is given by the formula $(n-2) \\times 180^\\circ$. For a hexagon ($n=6$), this sum is:\n \\[\n (6-2) \\times 180^\\circ = 4 \\times 180^\\c... |
deepscaler | [
{
"role": "user",
"content": "Find all values of $b$ such that $b+\\lceil b\\rceil=17.8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8.8"
} | {
"split": "train",
"index": 4238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle PQR$, points $X$ and $Y$ lie on $\\overline{QR}$ and $\\overline{PR}$, respectively. If $\\overline{PX}$ and $\\overline{QY}$ intersect at $Z$ such that $PZ/ZX = 2$ and $QZ/ZY = 5$, what is $RX/RY$? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 4239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer \\( n \\) is said to be 'good' if \\( n^2 - 1 \\) can be written as the product of three distinct prime numbers. Find the sum of the five smallest 'good' integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "104"
} | {
"split": "train",
"index": 4240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the definite integral:\n\n$$\n\\int_{\\pi / 4}^{\\operatorname{arctg} 3} \\frac{d x}{(3 \\operatorname{tg} x+5) \\sin 2 x}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10} \\ln \\frac{12}{7}"
} | {
"split": "train",
"index": 4241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bug travels from A to B along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there? Let's think... | math | {
"style": "rule",
"ground_truth": "2400"
} | {
"split": "train",
"index": 4242,
"solution": "1. **Identify the number of ways to reach the red arrows:**\n - There is only one way to reach any of the red arrows from point A, as each red arrow has a unique path leading directly from A.\n\n2. **Calculate the number of ways to reach the blue arrows from the r... |
deepscaler | [
{
"role": "user",
"content": "The hyperbola $tx^{2}-y^{2}-1=0$ has asymptotes that are perpendicular to the line $2x+y+1=0$. Find the eccentricity of this hyperbola. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 4243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During a performance, a panda named A Bao is performing on stilts. A Bao's height accounts for $\\frac{1}{4}$ of the total height when he is standing on the stilts. During the performance, A Bao accidentally breaks $20$ dm off each stilt. After the break, A Bao's height account... | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 4244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If for any ${x}_{1},{x}_{2}∈[1,\\frac{π}{2}]$, $x_{1} \\lt x_{2}$, $\\frac{{x}_{2}sin{x}_{1}-{x}_{1}sin{x}_{2}}{{x}_{1}-{x}_{2}}＞a$ always holds, then the maximum value of the real number $a$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 4245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the set \\( M = \\{1, 2, \\cdots, 1000\\} \\). For any non-empty subset \\( X \\) of \\( M \\), let \\( a_X \\) represent the sum of the maximum and minimum numbers in \\( X \\). What is the arithmetic mean of all such \\( a_X \\)? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "1001"
} | {
"split": "train",
"index": 4246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given set \\( S \\) such that \\( |S| = 10 \\). Let \\( A_1, A_2, \\cdots, A_k \\) be non-empty subsets of \\( S \\), and the intersection of any two subsets contains at most two elements. Find the maximum value of \\( k \\). Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "175"
} | {
"split": "train",
"index": 4247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $A^2$, where $A$ is the sum of the absolute values of all roots of the following equation:\n\\[x = \\sqrt{19} + \\frac{91}{{\\sqrt{19}+\\frac{91}{{\\sqrt{19}+\\frac{91}{{\\sqrt{19}+\\frac{91}{{\\sqrt{19}+\\frac{91}{x}}}}}}}}}.\\] Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "383"
} | {
"split": "train",
"index": 4248,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin \\left( \\omega x- \\frac{\\pi }{6} \\right)+\\sin \\left( \\omega x- \\frac{\\pi }{2} \\right)$, where $0 < \\omega < 3$. It is known that $f\\left( \\frac{\\pi }{6} \\right)=0$.\n\n(1) Find $\\omega$;\n\n(2) Stretch the horizontal coordinates of... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 4249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the circle $C$: $x^{2}+y^{2}-2x=0$, find the coordinates of the circle center $C$ and the length of the chord intercepted by the line $y=x$ on the circle $C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 4250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the cyclist encounters red lights at each of 4 intersections with probability $\\frac{1}{3}$ and the events of encountering red lights are independent, calculate the probability that the cyclist does not encounter red lights at the first two intersections and encounters t... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{27}"
} | {
"split": "train",
"index": 4251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at $z$ and delivers milk to houses located at $z^{3}, z^{5}, z^{7}, \\ldots, z^{2013}$, in that order; on Sunday, he begins at 1 and delivers milk to houses located at $z^{2}, z^{4}, z^{6}, \\ld... | math | {
"style": "rule",
"ground_truth": "\\frac{1005}{1006}"
} | {
"split": "train",
"index": 4252,
"solution": "Note that the distance between two points in the complex plane, $m$ and $n$, is $|m-n|$. We have that $$\\sum_{k=1}^{1006}\\left|z^{2 k+1}-z^{2 k-1}\\right|=\\sum_{k=1}^{1006}\\left|z^{2 k}-z^{2 k-2}\\right|=\\sqrt{2012}$$ However, noting that $$|z| \\cdot \\sum_{k=... |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$, if $d < 0$, $T_n$ is the sum of the first $n$ terms of $\\{a_n\\}$, and $T_3=15$, also $a_1+1, a_2+3, a_3+9$ form a geometric sequence with common ratio $q$, calculate the value of $q$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 4253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the sequence created by intermixing the following sets of numbers: the first $1000$ odd numbers, and the squares of the first $100$ integers. What is the median of the new list of $1100$ numbers?\n- $1, 3, 5, \\ldots, 1999$\n- $1^2, 2^2, \\ldots, 100^2$\nA) $1089$\nB) ... | math | {
"style": "rule",
"ground_truth": "1100"
} | {
"split": "train",
"index": 4254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A robot colors natural numbers starting from 1 in ascending order according to the following rule: any natural number that can be expressed as the sum of two composite numbers is colored red; those that do not meet this requirement are colored yellow. For example, 23 can be exp... | math | {
"style": "rule",
"ground_truth": "2001"
} | {
"split": "train",
"index": 4255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S = \\{1, 2, \\ldots, 2005\\}$. If any set of $n$ pairwise coprime numbers from $S$ always contains at least one prime number, find the smallest value of $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 4256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are all equal. ... | math | {
"style": "rule",
"ground_truth": "1152"
} | {
"split": "train",
"index": 4257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten football teams played each other exactly once. As a result, each team ended up with exactly $x$ points.\n\nWhat is the largest possible value of $x$? (A win earns 3 points, a draw earns 1 point, and a loss earns 0 points.) Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 4258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gretchen has eight socks, two of each color: magenta, cyan, black, and white. She randomly draws four socks. What is the probability that she has exactly one pair of socks with the same color? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{24}{35}"
} | {
"split": "train",
"index": 4259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ a_1, a_2,\\ldots ,a_8$ be $8$ distinct points on the circumference of a circle such that no three chords, each joining a pair of the points, are concurrent. Every $4$ of the $8$ points form a quadrilateral which is called a *quad*. If two chords, each joining a p... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 4260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $m$ and $n$ are odd integers, how many terms in the expansion of $(m+n)^6$ are odd? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 4261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a regular hexagon $A B C D E F$, the diagonals $A C$ and $C E$ are divided by points $M$ and $N$ respectively in the following ratios: $\\frac{A M}{A C} = \\frac{C N}{C E} = r$. If points $B$, $M$, and $N$ are collinear, determine the ratio $r$. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 4262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Some people like to write with larger pencils than others. Ed, for instance, likes to write with the longest pencils he can find. However, the halls of MIT are of limited height $L$ and width $L$. What is the longest pencil Ed can bring through the halls so that he can negotiat... | math | {
"style": "rule",
"ground_truth": "3 L"
} | {
"split": "train",
"index": 4263,
"solution": "$3 L$."
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be the numbers obtained by rolling a pair of dice twice. The probability that the equation $x^{2}-ax+2b=0$ has two distinct real roots is $\\_\\_\\_\\_\\_\\_$. 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": 4264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the equation \n$$\n\\frac{x^{2}+p}{x}=-\\frac{1}{4},\n$$\nwith roots \\(x_{1}\\) and \\(x_{2}\\), determine \\(p\\) such that:\n\na) \\(\\frac{x_{1}}{x_{2}}+\\frac{x_{2}}{x_{1}}=-\\frac{9}{4}\\),\n\nb) one root is 1 less than the square of the other root. Let's think step by... | math | {
"style": "rule",
"ground_truth": "-\\frac{15}{8}"
} | {
"split": "train",
"index": 4265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "581"
} | {
"split": "train",
"index": 4266,
"solution": "Starting with small cases, we see that four chairs give $4 + 1 = 5$, five chairs give $5 + 5 + 1 = 11$, and six chairs give $6 + 6 + 6 + 6 + 1 = 25.$ Thus, n chairs should give $n 2^{n-4} + 1$, as confirmed above. This claim can be verified by the principle of inclu... |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, where $A B = 16$, $B C = 5 \\sqrt{5}$, and $C A = 9$. What is the area of the plane region covered by the set of points outside $\\triangle ABC$ such that the distance to points $B$ and $C$ is less than 6? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "54\\pi + \\frac{5\\sqrt{95}}{4}"
} | {
"split": "train",
"index": 4267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the coefficients of all terms in the expanded form of $(C_4^1x + C_4^2x^2 + C_4^3x^3 + C_4^4x^4)^2$ is 256. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 4268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,b,c,d$ be real numbers such that $b-d \\ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 4269,
"solution": "Using the hint we turn the equation into $\\prod_{k=1} ^4 (x_k-i)(x_k+i) \\implies P(i)P(-i) \\implies (b-d-1)^2 + (a-c)^2 \\implies \\boxed{16}$ . This minimum is achieved when all the $x_i$ are equal to $1$ ."
} |
deepscaler | [
{
"role": "user",
"content": "A chocolate bar originally weighed 400 grams and cost 150 rubles. Recently, to save money, the manufacturer reduced the weight of the bar to 300 grams and increased its price to 180 rubles. By what percentage did the manufacturer's revenue increase? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 4270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the terminal side of angle θ passes through point P(-x, -6) and $$cosθ=- \\frac {5}{13}$$, find the value of $$tan(θ+ \\frac {π}{4})$$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac {17}{7}"
} | {
"split": "train",
"index": 4271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfy the equation $x^{2}+y^{2}-4x+1=0$.\n$(1)$ Find the maximum and minimum values of $\\dfrac {y}{x}$;\n$(2)$ Find the maximum and minimum values of $y-x$;\n$(3)$ Find the maximum and minimum values of $x^{2}+y^{2}$. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "7-4 \\sqrt {3}"
} | {
"split": "train",
"index": 4272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Central Middle School the $108$ students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of $15$ cookies, lists these ite... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 4273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x$, and that $f(x) = 1 - |x - 2|$ for $1\\leq x \\leq 3$. Find the smallest $x$ for which $f(x) = f(2001)$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "429"
} | {
"split": "train",
"index": 4274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 4275,
"solution": "1. **Understanding the Circle and Arcs**: The circle is divided into 12 equal arcs, and each arc corresponds to a central angle at the center $O$. Since the circle's total degrees is $360^\\circ$, each central angle measures:\n \\[\n \\frac{360^\\circ}{12} = 30^... |
deepscaler | [
{
"role": "user",
"content": "A triangle has sides of length $48$ , $55$ , and $73$ . A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If $c$ ... | math | {
"style": "rule",
"ground_truth": "200689"
} | {
"split": "train",
"index": 4276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four teams play in a tournament in which each team plays exactly one game against each of the other three teams. At the end of each game, either the two teams tie or one team wins and the other team loses. A team is awarded 3 points for a win, 0 points for a loss, and 1 point f... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 4277,
"solution": "Suppose that the four teams in the league are called W, X, Y, and Z. Then there is a total of 6 games played: W against X, W against Y, W against Z, X against Y, X against Z, Y against Z. In each game that is played, either one team is awarded 3 points for a win and... |
deepscaler | [
{
"role": "user",
"content": "Container A holds 4 red balls and 6 green balls; containers B and C each hold 6 red balls and 4 green balls. A container is selected at random and then a ball is randomly selected from that container. What is the probability that the ball selected is green? Express your answer ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{15}"
} | {
"split": "train",
"index": 4278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One end of a bus route is at Station $A$ and the other end is at Station $B$. The bus company has the following rules:\n(1) Each bus must complete a one-way trip within 50 minutes (including the stopping time at intermediate stations), and it stops for 10 minutes when reaching ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 4279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area in the plane contained by the graph of\n\\[\n|x + 2y| + |2x - y| \\le 6.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5.76"
} | {
"split": "train",
"index": 4280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert the following radians to degrees and degrees to radians: \n(1) $$\\frac {\\pi}{12}$$ = \\_\\_\\_\\_\\_\\_ ; (2) $$\\frac {13\\pi}{6}$$ = \\_\\_\\_\\_\\_\\_ ; (3) -$$\\frac {5}{12}$$π = \\_\\_\\_\\_\\_\\_ . \n(4) 36° = \\_\\_\\_\\_\\_\\_ rad; (5) -105° = \\_\\_\\_\\_\\... | math | {
"style": "rule",
"ground_truth": "- \\frac {7}{12}\\pi"
} | {
"split": "train",
"index": 4281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nWhen I saw Eleonora, I found her very pretty. After a brief trivial conversation, I told her my age and asked how old she was. She answered:\n\n- When you were as old as I am now, you were three times older than me. And when I will be three times older than I am now, together... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 4282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Call a number feared if it contains the digits 13 as a contiguous substring and fearless otherwise. (For example, 132 is feared, while 123 is fearless.) Compute the smallest positive integer $n$ such that there exists a positive integer $a<100$ such that $n$ and $n+10 a$ are fe... | math | {
"style": "rule",
"ground_truth": "1287"
} | {
"split": "train",
"index": 4283,
"solution": "First of all, note that we cannot have $n, n+a, \\ldots, n+10 a$ be less than 1000, since we cannot have fearless numbers have 13 as their last two digits since $a<100$, and $129,130,131, \\ldots, 139$ doesn't work as 139 is feared. Thus, we must utilize numbers of ... |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\sum_{k = 1}^\\infty \\frac{6^k}{(3^k - 2^k)(3^{k + 1} - 2^{k + 1})}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 4284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ and $q$ be the two distinct solutions to the equation $$(x-5)(2x+9) = x^2-13x+40.$$What is $(p + 3)(q + 3)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-112"
} | {
"split": "train",
"index": 4285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$, $b$, $c$, $d$, $e$, and $f$ are integers for which $8x^3 + 125 = (ax^2 + bx + c)(d x^2 + ex + f)$ for all $x$, then what is $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "770"
} | {
"split": "train",
"index": 4286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $2017$ distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points? Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 4287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four numbers, 2613, 2243, 1503, and 985, when divided by the same positive integer, yield the same remainder (which is not zero). Find the divisor and the remainder. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 4288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a convex pentagon \\( P Q R S T \\), the angle \\( P R T \\) is half of the angle \\( Q R S \\), and all sides are equal. Find the angle \\( P R T \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 4289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a game of Fish, R2 and R3 are each holding a positive number of cards so that they are collectively holding a total of 24 cards. Each player gives an integer estimate for the number of cards he is holding, such that each estimate is an integer between $80 \\%$ of his actual ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 4290,
"solution": "To minimize the sum, we want each player to say an estimate as small as possible-i.e. an estimate as close to $80 \\%$ of his actual number of cards as possible. We claim that the minimum possible sum is 20. First, this is achievable when R2 has 10 cards and estimat... |
deepscaler | [
{
"role": "user",
"content": "A school club buys 1200 candy bars at a price of four for $3 dollars, and sells all the candy bars at a price of three for $2 dollars, or five for $3 dollars if more than 50 are bought at once. Calculate their total profit in dollars. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "-100"
} | {
"split": "train",
"index": 4291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sizes of circular pizzas are determined by their diameter. If Lana's initial pizza was 14 inches in diameter and she decides to order a larger pizza with a diameter of 18 inches instead, what is the percent increase in the area of her pizza? Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "65.31\\%"
} | {
"split": "train",
"index": 4292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $3 \\sin A + 4 \\cos B = 6$ and $4 \\sin B + 3 \\cos A = 1$. Find all possible values of $\\angle C,$ in degrees. Enter all the possible values, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30^\\circ"
} | {
"split": "train",
"index": 4293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the set of solutions to the system of constraints\n\n$$\n\\left\\{\\begin{array}{l}\n2-2 x_{1}-x_{2} \\geqslant 0 \\\\\n2-x_{1}+x_{2} \\geqslant 0 \\\\\n5-x_{1}-x_{2} \\geqslant 0 \\\\\nx_{1} \\geqslant 0, \\quad x_{2} \\geqslant 0\n\\end{array}\\right.\n$$\n\nfind the minim... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 4294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be positive real numbers such that each of the equations $x^2 + ax + 2b = 0$ and $x^2 + 2bx + a = 0$ has real roots. Find the smallest possible value of $a + b.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 4295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the integer value of $y$ in the arithmetic sequence $2^2, y, 2^4$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 4296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "According to the latest revision of the \"Regulations on the Application and Use of Motor Vehicle Driving Licenses\" by the Ministry of Public Security: each driving license applicant must pass the \"Subject One\" (theoretical subject) and \"Comprehensive Subject\" (combination... | math | {
"style": "rule",
"ground_truth": "0.9988"
} | {
"split": "train",
"index": 4297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ be an equilateral triangle with side length $55.$ Points $D,$ $E,$ and $F$ lie on $\\overline{BC},$ $\\overline{CA},$ and $\\overline{AB},$ respectively, with $BD = 7,$ $CE=30,$ and $AF=40.$ Point $P$ inside $\\triangle ABC$ has the property that \\[\\angle... | math | {
"style": "rule",
"ground_truth": "075"
} | {
"split": "train",
"index": 4298,
"solution": "We begin by using the fact stated in Solution 3 that, for any point in an equilateral triangle, the lengths of the three perpendicular lines dropped to the sides of the triangle add up to the altitude of that triangle. To make things simple, let's assign $\\angle AE... |
deepscaler | [
{
"role": "user",
"content": "Let triangle $PQR$ be a right triangle with $\\angle PRQ = 90^\\circ$. A circle is tangent to the sides $PQ$ and $PR$ at points $S$ and $T$ respectively. The points on the circle diametrically opposite $S$ and $T$ both lie on side $QR$. Given that $PQ = 12$, find the area of th... | math | {
"style": "rule",
"ground_truth": "4\\pi - 8"
} | {
"split": "train",
"index": 4299,
"solution": ""
} |
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