data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let $n$ and $k$ be integers satisfying $\\binom{2k}{2} + n = 60$ . It is known that $n$ days before Evan's 16th birthday, something happened. Compute $60-n$ .\n\n*Proposed by Evan Chen* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 19649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The solutions to the equation \\( x^3 - 4 \\lfloor x \\rfloor = 5 \\), where \\( x \\) is a real number, are denoted by \\( x_1, x_2, x_3, \\ldots, x_k \\) for some positive integer \\( k \\). Find \\( \\sum_{i=1}^{k} x_{i}^{3} \\). Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 19650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the median of the following list of $4040$ numbers?\n\\[1, 2, 3, \\ldots, 2020, 1^2, 2^2, 3^2, \\ldots, 2020^2\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1976.5"
} | {
"split": "train",
"index": 19651,
"solution": "1. **Identify the total number of terms and the position of the median**: \n The list consists of $2020$ integers from $1$ to $2020$ and $2020$ squares from $1^2$ to $2020^2$. Thus, the total number of terms is $2020 + 2020 = 4040$. The median of an even number o... |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), the left and right foci coincide with the symmetric points about the two asymptotes, respectively. Then, the eccentricity of the hyperbola is __________. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 19652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the integer \\( m \\), \\( -180 \\leq m \\leq 180 \\), such that \\(\\sin m^\\circ = \\sin 945^\\circ.\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-135"
} | {
"split": "train",
"index": 19653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), angle \\( \\angle C \\) is a right angle, \\( AC = 3 \\) and \\( BC = 4 \\). In triangle \\( \\triangle ABD \\), angle \\( \\angle A \\) is a right angle, and \\( AD = 12 \\). Points \\( C \\) and \\( D \\) are on opposite sides of \\( AB \\)... | math | {
"style": "rule",
"ground_truth": "128"
} | {
"split": "train",
"index": 19654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p=2^{24036583}-1$, the largest prime currently known. For how many positive integers $c$ do the quadratics \\pm x^{2} \\pm p x \\pm c all have rational roots? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 19655,
"solution": "This is equivalent to both discriminants $p^{2} \\pm 4 c$ being squares. In other words, $p^{2}$ must be the average of two squares $a^{2}$ and $b^{2}$. Note that $a$ and $b$ must have the same parity, and that \\left(\\frac{a+b}{2}\\right)^{2}+\\left(\\frac{a-b}{2... |
deepscaler | [
{
"role": "user",
"content": "Determine the maximum integer value of the expression \n\\[\\frac{3x^2 + 9x + 28}{3x^2 + 9x + 7}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 19656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nProfessor Severus Snape brewed three potions, each in a volume of 600 ml. The first potion makes the drinker intelligent, the second makes them beautiful, and the third makes them strong. To have the effect of the potion, it is sufficient to drink at least 30 ml of each potio... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 19657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Oleg drew an empty $50 \\times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 being irrational. Then, in each cell of the table, he wrote t... | math | {
"style": "rule",
"ground_truth": "1250"
} | {
"split": "train",
"index": 19658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the minimum possible value of the sum\n\\[\\frac{a}{3b} + \\frac{b}{5c} + \\frac{c}{7a},\\]\nwhere $a,$ $b,$ and $c$ are positive real numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{\\sqrt[3]{105}}"
} | {
"split": "train",
"index": 19659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ be a function such that $f(0) = 1$ and\n\\[f(xy) = f \\left( \\frac{x^2 + y^2}{2} \\right) + (x - y)^2\\]for all real numbers $x$ and $y.$ Find $f(x).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 - 2x"
} | {
"split": "train",
"index": 19660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right triangle with integer leg lengths is called \"cool'' if the number of square units in its area is equal to twice the number of units in the sum of the lengths of its legs. What is the sum of all the different possible areas of cool right triangles? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "118"
} | {
"split": "train",
"index": 19661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all prime numbers $p$ for which there exists a unique $a \\in\\{1,2, \\ldots, p\\}$ such that $a^{3}-3 a+1$ is divisible by $p$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 19662,
"solution": "We show that $p=3$ is the only prime that satisfies the condition. Let $f(x)=x^{3}-3 x+1$. As preparation, let's compute the roots of $f(x)$. By Cardano's formula, it can be seen that the roots are $2 \\operatorname{Re} \\sqrt[3]{\\frac{-1}{2}+\\sqrt{\\left(\\frac{... |
deepscaler | [
{
"role": "user",
"content": "From the natural numbers 1 to 2008, the maximum number of numbers that can be selected such that the sum of any two selected numbers is not divisible by 3 is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "671"
} | {
"split": "train",
"index": 19663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let positive integers \\( a, b, c, d \\) satisfy \\( a > b > c > d \\) and \\( a+b+c+d=2004 \\), \\( a^2 - b^2 + c^2 - d^2 = 2004 \\). Find the minimum value of \\( a \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "503"
} | {
"split": "train",
"index": 19664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Side $AB$ of triangle $ABC$ has length 8 inches. Line $DEF$ is drawn parallel to $AB$ so that $D$ is on segment $AC$, and $E$ is on segment $BC$. Line $AE$ extended bisects angle $FEC$. If $DE$ has length $5$ inches, then the length of $CE$, in inches, is: Let's think step by s... | math | {
"style": "rule",
"ground_truth": "\\frac{40}{3}"
} | {
"split": "train",
"index": 19665,
"solution": "1. **Identify Given Information and Draw Auxiliary Lines**: \n - Triangle $ABC$ has side $AB = 8$ inches.\n - Line $DEF$ is parallel to $AB$, with $D$ on $AC$, $E$ on $BC$, and $DE = 5$ inches.\n - Line $AE$ extended bisects angle $FEC$.\n\n2. **Use of Parall... |
deepscaler | [
{
"role": "user",
"content": "Form a five-digit number without repeating digits using the numbers 0, 1, 2, 3, 4, where exactly one even number is sandwiched between two odd numbers. How many such five-digit numbers are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 19666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among all the five-digit numbers formed without repeating any of the digits 0, 1, 2, 3, 4, if they are arranged in ascending order, determine the position of the number 12340. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 19667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "John surveyed a group of people about their knowledge of rats. To the nearest tenth of a percent, he found that $86.8\\%$ of the people surveyed thought rats carried diseases. Of the people who thought rats carried diseases, $45.7\\%$ said that rats frequently carried rabies. S... | math | {
"style": "rule",
"ground_truth": "53"
} | {
"split": "train",
"index": 19668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convex pentagon $ABCDE$ has side lengths $AB=5$, $BC=CD=DE=6$, and $EA=7$. Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of $ABCDE$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 19669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three people, John, Macky, and Rik, play a game of passing a basketball from one to another. Find the number of ways of passing the ball starting with Macky and reaching Macky again at the end of the seventh pass. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 19670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers from 1 to 200, inclusive, are placed in a bag. A number is randomly selected from the bag. What is the probability that it is neither a perfect square, a perfect cube, nor a multiple of 7? Express your answer as a common fraction. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\frac{39}{50}"
} | {
"split": "train",
"index": 19671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can 4 purple balls and 4 green balls be placed into a $4 \\times 4$ grid such that every row and column contains one purple ball and one green ball? Only one ball may be placed in each box, and rotations and reflections of a single configuration are considered ... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 19672,
"solution": "There are $4!=24$ ways to place the four purple balls into the grid. Choose any purple ball, and place two green balls, one in its row and the other in its column. There are four boxes that do not yet lie in the same row or column as a green ball, and at least one ... |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C$ with its left and right foci at $F_{1}(-\\sqrt{3},0)$ and $F_{2}(\\sqrt{3},0)$, respectively, and the ellipse passes through the point $(-1, \\frac{\\sqrt{3}}{2})$.\n\n(Ⅰ) Find the equation of the ellipse $C$; \n(Ⅱ) Given a fixed point $A(1, \\frac{1}{2})$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 19673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$. Let's think ... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 19674,
"solution": "Let $AB=10x+y$ and $CD=10y+x$. It follows that $CO=\\frac{AB}{2}=\\frac{10x+y}{2}$ and $CH=\\frac{CD}{2}=\\frac{10y+x}{2}$. Scale up this triangle by 2 to ease the arithmetic. Applying the Pythagorean Theorem on $2CO$, $2OH$ and $2CH$, we deduce \\[(2OH)^2=\\left(1... |
deepscaler | [
{
"role": "user",
"content": "Mary's top book shelf holds five books with the following widths, in centimeters: $6$, $\\dfrac{1}{2}$, $1$, $2.5$, and $10$. What is the average book width, in centimeters? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 19675,
"solution": "1. **List the widths of the books**: The widths of the books are given as $6$, $\\dfrac{1}{2}$, $1$, $2.5$, and $10$ centimeters.\n\n2. **Calculate the total sum of the widths**:\n \\[\n 6 + \\dfrac{1}{2} + 1 + 2.5 + 10 = 6 + 0.5 + 1 + 2.5 + 10\n \\]\n \\[\... |
deepscaler | [
{
"role": "user",
"content": "Circles $\\mathcal{C}_1, \\mathcal{C}_2,$ and $\\mathcal{C}_3$ have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line $t_1$ is a common internal tangent to $\\mathcal{C}_1$ and $\\mathcal{C}_2$ and has a positive slope, and line $t_2$ is... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 19676,
"solution": "Call the centers $O_1, O_2, O_3$, the points of tangency $r_1, r_2, s_1, s_2$ (with $r$ on $t_1$ and $s$ on $t_2$, and $s_2$ on $\\mathcal{C}_2$), and the intersection of each common internal tangent to the X-axis $r, s$. $\\triangle O_1r_1r \\sim \\triangle O_2r_2... |
deepscaler | [
{
"role": "user",
"content": "Compute the remainder when\n${2007 \\choose 0} + {2007 \\choose 3} + \\cdots + {2007 \\choose 2007}$\nis divided by 1000.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 19677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lengths of the three sides of a triangle are $7$, $x+4$ and $2x+1$. The perimeter of the triangle is 36. What is the length of the longest side of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 19678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\frac{2+4+6}{1+3+5} - \\frac{1+3+5}{2+4+6}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{12}"
} | {
"split": "train",
"index": 19679,
"solution": "1. **Calculate the sums in the numerators and denominators:**\n - The sum of the numerators in the first fraction: $2 + 4 + 6 = 12$.\n - The sum of the denominators in the first fraction: $1 + 3 + 5 = 9$.\n - The sum of the numerators in the second fraction: ... |
deepscaler | [
{
"role": "user",
"content": "Using systematic sampling method to select 32 people from 960 for a questionnaire survey, they are randomly numbered from 1 to 960. After grouping, the number drawn by simple random sampling in the first group is 9. Among the 32 people drawn, those with numbers in the interval ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 19680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find \\(\\sin \\alpha\\) if \\(\\cos \\alpha = \\operatorname{tg} \\beta\\), \\(\\cos \\beta = \\operatorname{tg} \\gamma\\), \\(\\cos \\gamma = \\operatorname{tg} \\alpha\\) \\((0 < \\alpha < \\frac{\\pi}{2}, 0 < \\beta < \\frac{\\pi}{2}, 0 < \\gamma < \\frac{\\pi}{2})\\). Let... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 19681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the graph of a certain quadratic $y = ax^2 + bx + c$, the vertex of the parabola is $(3,7)$ and one of the $x$-intercepts is $(-2,0)$. What is the $x$-coordinate of the other $x$-intercept? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 19682,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Max bought a new dirt bike and paid $10\\%$ of the cost upfront, which was $\\$150$. What was the price of the bike? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\$ 1500"
} | {
"split": "train",
"index": 19683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain ellipse, the center is at $(-3,1),$ one focus is at $(-3,0),$ and one endpoint of a semi-major axis is at $(-3,3).$ Find the semi-minor axis of the ellipse. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 19684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the coordinates of a point on the terminal side of angle $\\alpha$ are $(\\frac{\\sqrt{3}}{2},-\\frac{1}{2})$, determine the smallest positive value of angle $\\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11\\pi}{6}"
} | {
"split": "train",
"index": 19685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a spherical surface with an area of $60\\pi$, there are four points $S$, $A$, $B$, and $C$, and $\\triangle ABC$ is an equilateral triangle. The distance from the center $O$ of the sphere to the plane $ABC$ is $\\sqrt{3}$. If the plane $SAB$ is perpendicular to the plane $AB... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 19686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that when 81849, 106392, and 124374 are divided by an integer \\( n \\), the remainders are equal. If \\( a \\) is the maximum value of \\( n \\), find \\( a \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "243"
} | {
"split": "train",
"index": 19687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle ABC, $a-b=4$, $a+c=2b$, and the largest angle is $120^\\circ$. Find the perimeter of this triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 19688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the six positive integer factors of 30, each multiplied by 2? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 19689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\(0 \\le x_0 < 1\\), let \n\\[x_n = \\left\\{ \\begin{array}{ll}\n3x_{n-1} & \\text{if } 3x_{n-1} < 1 \\\\\n3x_{n-1} - 1 & \\text{if } 1 \\le 3x_{n-1} < 2 \\\\\n3x_{n-1} - 2 & \\text{if } 3x_{n-1} \\ge 2 \n\\end{array}\\right.\\]\nfor all integers \\(n > 0\\), determine ... | math | {
"style": "rule",
"ground_truth": "729"
} | {
"split": "train",
"index": 19690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of the greatest common divisor and the least common multiple of $100$ and $120.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12000"
} | {
"split": "train",
"index": 19691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the last two digits of $8^{25} + 12^{25}?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 19692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\\sin^4 x + \\frac{3}{2} \\cos^4 x,\\]as $x$ varies over all real numbers. 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": 19693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of ways to select 99 cells of a $19 \\times 19$ square grid such that no two selected cells share an edge or vertex. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 19694,
"solution": "We claim the number of ways to select $n^{2}-1$ such cells from a $(2 n-1) \\times(2 n-1)$ grid is exactly $n^{3}$, which implies the answer to this question is 1000 . Partition the board into $n^{2}$ regions, as pictured. Also, shade red every cell in an odd row a... |
deepscaler | [
{
"role": "user",
"content": "Simplify\n\\[\\frac{\\cos x}{1 + \\sin x} + \\frac{1 + \\sin x}{\\cos x}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sec x"
} | {
"split": "train",
"index": 19695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Barney Schwinn noted that his bike's odometer showed a reading of $2332$, a palindrome. After riding for $5$ hours one day and $4$ hours the next day, he observed that the odometer displayed another palindrome, $2552$. Calculate Barney's average riding speed during this period.... | math | {
"style": "rule",
"ground_truth": "\\frac{220}{9}"
} | {
"split": "train",
"index": 19696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rounded to 3 decimal places, what is $\\frac{8}{11}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.727"
} | {
"split": "train",
"index": 19697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest five-digit number whose digits add up to 20? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99200"
} | {
"split": "train",
"index": 19698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a rectangular grid of dots with 5 rows and 6 columns, determine how many different squares can be formed using these dots as vertices. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 19699,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin (ωx+φ)(ω > 0,|φ|\\leqslant \\dfrac {π}{2})$, $y=f(x- \\dfrac {π}{4})$ is an odd function, $x= \\dfrac {π}{4}$ is the symmetric axis of the graph of $y=f(x)$, and $f(x)$ is monotonic in $(\\dfrac {π}{14}, \\dfrac {13π}{84})$, determine the maximum ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 19700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the following number:\n\n\\[\n\\frac{12346 \\cdot 24689 \\cdot 37033 + 12347 \\cdot 37034}{12345^{2}}\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "74072"
} | {
"split": "train",
"index": 19701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(x) = x^2 + ax + b$ be a quadratic polynomial. For how many pairs $(a, b)$ of positive integers where $a, b < 1000$ do the quadratics $P(x+1)$ and $P(x) + 1$ have at least one root in common? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 19702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the polar coordinate system, the polar coordinate equation of the curve $\\Gamma$ is $\\rho= \\frac {4\\cos \\theta}{\\sin ^{2}\\theta}$. Establish a rectangular coordinate system with the pole as the origin, the polar axis as the positive semi-axis of $x$, and the unit leng... | math | {
"style": "rule",
"ground_truth": "4 \\sqrt {2}"
} | {
"split": "train",
"index": 19703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set $\\{a,b,c\\}=\\{1,2,3\\}$, and three relations: $①a\\neq 3$, $②b=3$, $③c\\neq 1$. Only one of these relations is correct. Find the value of $100a+10b+c=\\_\\_\\_\\_\\_\\_.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "312"
} | {
"split": "train",
"index": 19704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\lfloor x \\rfloor$ be the greatest integer less than or equal to $x$. Then the number of real solutions to $4x^2-40\\lfloor x \\rfloor +51=0$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 19705,
"solution": "1. **Rewrite the equation**: Start by rewriting the given equation:\n \\[\n 4x^2 - 40\\lfloor x \\rfloor + 51 = 0\n \\]\n Rearrange this to isolate $4x^2$:\n \\[\n 4x^2 = 40\\lfloor x \\rfloor - 51\n \\]\n Let $n = 40\\lfloor x \\rfloor - 51$, which... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the curve $C$ is given by the parametric equations $\\begin{cases} x= \\sqrt {3}+2\\cos \\alpha \\\\ y=1+2\\sin \\alpha\\end{cases}$ (where $\\alpha$ is the parameter). A polar coordinate system is established with the origin of the Car... | math | {
"style": "rule",
"ground_truth": "3 \\sqrt {3}"
} | {
"split": "train",
"index": 19706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number \\( n \\) such that both \\( n^2 \\) and \\( (n+1)^2 \\) contain the digit 7. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 19707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(n)$ represent the product of all non-zero digits of a positive integer $n$. For example: $P(123) = 1 \\times 2 \\times 3 = 6$ and $P(206) = 2 \\times 6 = 12$. Find the value of $P(1) + P(2) + \\cdots + P(999)$. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "97335"
} | {
"split": "train",
"index": 19708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many numbers are in the list $250, 243, 236, \\ldots, 29, 22?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 19709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $PQRS TUVW$ is a regular octagon. There are 70 ways in which four of its sides can be chosen at random. If four of its sides are chosen at random and each of these sides is extended infinitely in both directions, what is the probability that they will meet to form ... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{35}"
} | {
"split": "train",
"index": 19710,
"solution": "If the four sides that are chosen are adjacent, then when these four sides are extended, they will not form a quadrilateral that encloses the octagon. If the four sides are chosen so that there are exactly three adjacent sides that are not chosen and one other side... |
deepscaler | [
{
"role": "user",
"content": "Let \\(ABCD\\) be a convex quadrilateral such that \\(AB + BC = 2021\\) and \\(AD = CD\\). We are also given that \\(\\angle ABC = \\angle CDA = 90^\\circ\\). Determine the length of the diagonal \\(BD\\). Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\frac{2021}{\\sqrt{2}}"
} | {
"split": "train",
"index": 19711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 19712,
"solution": "Let the two nonzero real numbers be $x$ and $y$. According to the problem, the sum of these two numbers is 4 times their product. This can be expressed as:\n\\[ x + y = 4xy. \\]\n\nWe are asked to find the sum of the reciprocals of $x$ and $y$. Let's denote the rec... |
deepscaler | [
{
"role": "user",
"content": "Given that $α \\in ( \\frac{5}{4}π, \\frac{3}{2}π)$, and it satisfies $\\tan α + \\frac{1}{\\tan α} = 8$, find the value of $\\sin α \\cos α$ and $\\sin α - \\cos α$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 19713,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An organization initially consists of 10 leaders and a certain number of regular members. Each year, all leaders are replaced, and each regular member recruits two new members. After this, 10 new leaders are elected from outside the organization. Initially, there are 30 people ... | math | {
"style": "rule",
"ground_truth": "1180990"
} | {
"split": "train",
"index": 19714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the unique positive integer $n$ such that $\\frac{n^{3}-1989}{n}$ is a perfect square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 19715,
"solution": "We need $n^{2}-\\frac{1989}{n}$ to be a perfect square, so $n \\mid 1989$. Also, this perfect square would be less than $n^{2}$, so it would be at most $(n-1)^{2}=n^{2}-2 n+1$. Thus, $$\\frac{1989}{n} \\geq 2 n-1 \\Longrightarrow 1989 \\geq 2 n^{2}-n$$ so $n \\leq ... |
deepscaler | [
{
"role": "user",
"content": "The expression\n\\[\\frac{P+Q}{P-Q}-\\frac{P-Q}{P+Q}\\]\nwhere $P=x+y$ and $Q=x-y$, is equivalent to: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{x^2-y^2}{xy}"
} | {
"split": "train",
"index": 19716,
"solution": "1. **Substitute the expressions for \\( P \\) and \\( Q \\):**\n Given \\( P = x + y \\) and \\( Q = x - y \\), substitute these into the original expression:\n \\[\n \\frac{P+Q}{P-Q} - \\frac{P-Q}{P+Q} = \\frac{(x+y)+(x-y)}{(x+y)-(x-y)} - \\frac{(x+y)-(x-y)}... |
deepscaler | [
{
"role": "user",
"content": "Express $3.\\overline{36}$ as a common fraction in lowest terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"split": "train",
"index": 19717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The vertical drops of six roller coasters at Fractal Fun Park are listed in the table below:\n\\begin{tabular}{|l|c|}\n\\hline\nThe Spiral Slide & 173 feet \\\\\n\\hline\nThe Velocity Vortex & 125 feet \\\\\n\\hline\nThe Cyclone & 150 feet \\\\\n\\hline\nThe Apex Jump & 310 fee... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 19718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $5x^4 - 12x^3 + 3x^2 - 5x + 15$ is divided by $3x - 9$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 19719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$A B C$ is a triangle with $A B=15, B C=14$, and $C A=13$. The altitude from $A$ to $B C$ is extended to meet the circumcircle of $A B C$ at $D$. Find $A D$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{63}{4}"
} | {
"split": "train",
"index": 19720,
"solution": "Let the altitude from $A$ to $B C$ meet $B C$ at $E$. The altitude $A E$ has length 12 ; one way to see this is that it splits the triangle $A B C$ into a $9-12-15$ right triangle and a $5-12-13$ right triangle; from this, we also know that $B E=9$ and $C E=5$. Now... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer \\( n \\) such that a cube with side length \\( n \\) can be divided into 1996 smaller cubes, each with side length a positive integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 19721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability that the square root of a randomly selected two-digit whole number is less than eight? 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}{5}"
} | {
"split": "train",
"index": 19722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the first seven prime numbers that have a units digit of 7. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "379"
} | {
"split": "train",
"index": 19723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a\\in R$, $b \\gt 0$, $a+b=2$, then the minimum value of $\\frac{1}{2|a|}+\\frac{|a|}{b}$ is ______. 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": 19724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain country, there are 100 cities. The Ministry of Aviation requires that each pair of cities be connected by a bidirectional flight operated by exactly one airline, and that for each airline, it must be possible to travel from any city to any other city (possibly with... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 19725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of the solutions of: $|y|=2(|y|-1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 19726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We colour all the sides and diagonals of a regular polygon $P$ with $43$ vertices either\nred or blue in such a way that every vertex is an endpoint of $20$ red segments and $22$ blue segments.\nA triangle formed by vertices of $P$ is called monochromatic if all of its sides ha... | math | {
"style": "rule",
"ground_truth": "859"
} | {
"split": "train",
"index": 19727,
"solution": "\nGiven a regular polygon \\( P \\) with 43 vertices, each segment (sides and diagonals) of this polygon is colored either red or blue. We know the following conditions:\n- Every vertex is an endpoint of 20 red segments.\n- Every vertex is an endpoint of 22 blue se... |
deepscaler | [
{
"role": "user",
"content": "Given two four-digit numbers \\( M \\) and \\( N \\) which are reverses of each other, and have \\( q^{p}-1 \\) identical positive divisors, \\( M \\) and \\( N \\) can be factorized into prime factors as \\( p q^{q} r \\) and \\( q^{p+q} r \\) respectively, where \\( p \\), \\... | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"split": "train",
"index": 19728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest solution to the equation \\[\\lfloor x^2 \\rfloor - \\lfloor x \\rfloor^2 = 19.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{119}"
} | {
"split": "train",
"index": 19729,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bill can buy jags, jigs and jogs for $\\$1$, $\\$2$ and $\\$7$ each, respectively. What is the largest number of jogs he can purchase if he must buy at least one of each item and will spend exactly $\\$50$? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 19730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\(a > 0\\), \\(b > 0\\), \\(c > 1\\), and \\(a + b = 1\\). Find the minimum value of \\(\\left(\\frac{2a + b}{ab} - 3\\right)c + \\frac{\\sqrt{2}}{c - 1}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 + 2\\sqrt{2}"
} | {
"split": "train",
"index": 19731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, where $b=2$. \n$(1)$ If $A+C=120^{\\circ}$ and $a=2c$, find the length of side $c$. \n$(2)$ If $A-C=15^{\\circ}$ and $a=\\sqrt{2}c\\sin A$, find the are... | math | {
"style": "rule",
"ground_truth": "3 - \\sqrt{3}"
} | {
"split": "train",
"index": 19732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A train leaves station K for station L at 09:30, while another train leaves station L for station K at 10:00. The first train arrives at station L 40 minutes after the trains pass each other. The second train arrives at station K 1 hour and 40 minutes after the trains pass each... | math | {
"style": "rule",
"ground_truth": "10:50"
} | {
"split": "train",
"index": 19733,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. What is the smallest possible value of the length of the side of the original square? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 19734,
"solution": "1. **Understanding the Problem:**\n - We need to find the smallest integer side length of a square that can be divided into 10 smaller squares, where at least 8 of these squares have an area of 1 (i.e., side length of 1).\n\n2. **Eliminating the Smallest Option:*... |
deepscaler | [
{
"role": "user",
"content": "The inscribed circle of triangle $DEF$ is tangent to $\\overline{DE}$ at point $P$ and its radius is $13$. Given that $DP = 17$ and $PE = 31$, and the tangent from vertex $F$ to the circle is $20$, determine the perimeter of triangle $DEF$. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "136"
} | {
"split": "train",
"index": 19735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given right triangle $ABC$, with $AB=4, BC=3$, and $CA=5$. Circle $\\omega$ passes through $A$ and is tangent to $BC$ at $C$. What is the radius of $\\omega$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{8}"
} | {
"split": "train",
"index": 19736,
"solution": "Let $O$ be the center of $\\omega$, and let $M$ be the midpoint of $AC$. Since $OA=OC$, $OM \\perp AC$. Also, $\\angle OCM=\\angle BAC$, and so triangles $ABC$ and $CMO$ are similar. Then, $CO/CM=AC/AB$, from which we obtain that the radius of $\\omega$ is $CO=\\fr... |
deepscaler | [
{
"role": "user",
"content": "Thirteen blue and six green hexagonal tiles were used to create the figure below. If a new figure is created by attaching a border of green tiles with the same size and shape as the others, what will be the difference between the total number of green tiles and the total number... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 19737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$ let $I$ be the center of the inscribed circle, and let the bisector of $\\angle ACB$ intersect $AB$ at $L$. The line through $C$ and $L$ intersects the circumscribed circle of $\\triangle ABC$ at the two points $C$ and $D$. If $LI=2$ and $LD=3$, then $IC=\\t... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 19738,
"solution": "Let $AB=c,BC=a,CA=b$, and $x=\\tfrac{a+b}{c}$. Then, notice that $\\tfrac{CI}{IL}=\\tfrac{a+b}{c}=x$, so $CI=IL\\cdot{}x=2x$. Also, by the incenter-excenter lemma, $AD=BD=ID=IL+LD=5$. Therefore, by Ptolemy's Theorem on cyclic quadrilateral $ABCD$, $5a+5b=c(2x+5)$, ... |
deepscaler | [
{
"role": "user",
"content": "Solve \\[\\frac{x+4}{x-2}=3\\]for $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 19739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the distance between the vertices of the hyperbola\n\\[\\frac{x^2}{99} - \\frac{y^2}{36} = 1.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6 \\sqrt{11}"
} | {
"split": "train",
"index": 19740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten identical crates each of dimensions $3\\mathrm{ft}\\times 4\\mathrm{ft}\\times 6\\mathrm{ft}$. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen a... | math | {
"style": "rule",
"ground_truth": "190"
} | {
"split": "train",
"index": 19741,
"solution": "Only the heights matter, and each crate is either 3, 4, or 6 feet tall with equal probability. We have the following:\n\\begin{align*}3a + 4b + 6c &= 41\\\\ a + b + c &= 10\\end{align*}\n Subtracting 3 times the second from the first gives $b + 3c = 11$, or $(b,c) ... |
deepscaler | [
{
"role": "user",
"content": "Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\\sqrt[3]{n}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"split": "train",
"index": 19742,
"solution": "\nGiven the problem, we are tasked to find the largest integer \\( n \\) such that \\( n \\) is divisible by all positive integers less than \\( \\sqrt[3]{n} \\).\n\n### Step-by-Step Solution:\n\n1. **Understanding the condition:**\n\n We know \\( n \\) must be d... |
deepscaler | [
{
"role": "user",
"content": "In the sequence ${a_{n}}$, $a_{1}=1$, $a_{n+2}+(-1)^{n}a_{n}=1$. Let $s_{n}$ be the sum of the first $n$ terms of the sequence ${a_{n}}$. Find $s_{100}$ = \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1300"
} | {
"split": "train",
"index": 19743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\omega$ be a nonreal root of $x^3 = 1.$ Compute\n\\[(1 - \\omega + \\omega^2)^4 + (1 + \\omega - \\omega^2)^4.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"split": "train",
"index": 19744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a $\\it{good\\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is nev... | math | {
"style": "rule",
"ground_truth": "94"
} | {
"split": "train",
"index": 19745,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles as shown. Two large triangles a... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 19746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer $n$ such that $7350$ is a factor of $n!$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 19747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles $\\omega_1$, $\\omega_2$, and $\\omega_3$ each have radius $6$ and are placed in the plane so that each circle is externally tangent to the other two. Points $Q_1$, $Q_2$, and $Q_3$ lie on $\\omega_1$, $\\omega_2$, and $\\omega_3$ respectively such that triangle $\\tria... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 19748,
"solution": ""
} |
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