data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "\nFind the set of values for parameter \\(a\\) for which the sum of the cubes of the roots of the equation \\(x^{2} + ax + a + 1 = 0\\) is equal to 1. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 33049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of the function $y=\\log_a(x+3)-1$ ($a > 0$ and $a \\neq 1$) always passes through a fixed point $A$. If the point $A$ lies on the line $mx+ny+1=0$, where $mn > 0$, then the minimum value of $\\frac{1}{m}+\\frac{1}{n}$ is Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "3+2\\sqrt{2}"
} | {
"split": "train",
"index": 33050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the infinite series $1 - \\frac{1}{3} - \\frac{1}{9} + \\frac{1}{27} - \\frac{1}{81} - \\frac{1}{243} + \\frac{1}{729} - \\frac{1}{2187} - \\cdots$. Evaluate the sum of this series. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{26}"
} | {
"split": "train",
"index": 33051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains $11$ balls, numbered $1, 2, 3, \\dots 11$. If $6$ balls are drawn simultaneously at random, what is the probability that the sum of the numbers on the balls drawn is odd? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{118}{231}"
} | {
"split": "train",
"index": 33052,
"solution": "To solve this problem, we need to calculate the probability that the sum of the numbers on the six balls drawn is odd. We start by noting that there are 6 odd-numbered balls (1, 3, 5, 7, 9, 11) and 5 even-numbered balls (2, 4, 6, 8, 10) in the box.\n\nThe sum of th... |
deepscaler | [
{
"role": "user",
"content": "Parallelogram $ABCD$ with $A(2,5)$, $B(4,9)$, $C(6,5)$, and $D(4,1)$ is reflected across the $x$-axis to $A'B'C'D'$ and then $A'B'C'D'$ is reflected across the line $y=x+1$ to $A''B''C''D''$. This is done such that $D'$ is the image of $D$, and $D''$ is the image of $D'$. What ... | math | {
"style": "rule",
"ground_truth": "(-2,5)"
} | {
"split": "train",
"index": 33053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\n1. Given non-negative real numbers \\( x, y, z \\) satisfying \\( x^{2} + y^{2} + z^{2} + x + 2y + 3z = \\frac{13}{4} \\), determine the maximum value of \\( x + y + z \\).\n\n2. Given \\( f(x) \\) is an odd function defined on \\( \\mathbb{R} \\) with a period of 3, and when... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 33054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest four-digit number that is divisible by $35$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1200"
} | {
"split": "train",
"index": 33055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two farmers agree that pigs are worth $\\$300$ and that goats are worth $\\$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $\\$390$ debt could be paid with two pi... | math | {
"style": "rule",
"ground_truth": "\\$30"
} | {
"split": "train",
"index": 33056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $n$ is a positive integer such that $n^{3}+2 n^{2}+9 n+8$ is the cube of an integer, find $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 33057,
"solution": "Since $n^{3}<n^{3}+2 n^{2}+9 n+8<(n+2)^{3}$, we must have $n^{3}+2 n^{2}+9 n+8=(n+1)^{3}$. Thus $n^{2}=6 n+7$, so $n=7$."
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given vectors $\\overrightarrow{m}=(b,a-2c)$, $\\overrightarrow{n}=(\\cos A-2\\cos C,\\cos B)$, and $\\overrightarrow{m} \\perp \\overrightarrow{n}$.\n1. Find the value of $\\frac{\\... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{15}}{4}"
} | {
"split": "train",
"index": 33058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the tenth number in the row of Pascal's triangle that has 100 numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\binom{99}{9}"
} | {
"split": "train",
"index": 33059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S_1 = \\{(x, y)|\\log_{10}(1 + x^2 + y^2) \\le 1 + \\log_{10}(x+y)\\}$ and $S_2 = \\{(x, y)|\\log_{10}(2 + x^2 + y^2) \\le 2 + \\log_{10}(x+y)\\}$. What is the ratio of the area of $S_2$ to the area of $S_1$? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "102"
} | {
"split": "train",
"index": 33060,
"solution": "1. **Analyzing the constraint for $S_1$:**\n - Given: $\\log_{10}(1+x^2+y^2)\\le 1+\\log_{10}(x+y)$\n - Since $\\log_{10}(ab) = \\log_{10}(a) + \\log_{10}(b)$, we rewrite the inequality:\n \\[\n \\log_{10}(1+x^2+y^2) \\le \\log_{10}(10) + \\log_{10}(x+y... |
deepscaler | [
{
"role": "user",
"content": "Find the value of $a$ that satisfies the equation $293_{a}+468_{a}=73B_{a}$, where $B_{a}=11_{10}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 33061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To express 20 as a sum of different powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 400 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers?... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 33062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)= \\begin{cases} 2a-(x+ \\frac {4}{x}),x < a\\\\x- \\frac {4}{x},x\\geqslant a\\\\end{cases}$.\n(1) When $a=1$, if $f(x)=3$, then $x=$ \\_\\_\\_\\_\\_\\_;\n(2) When $a\\leqslant -1$, if $f(x)=3$ has three distinct real roots that form an arithmetic sequence, then $a=... | math | {
"style": "rule",
"ground_truth": "- \\frac {11}{6}"
} | {
"split": "train",
"index": 33063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p,$ $q,$ $r,$ and $s$ be the roots of \\[x^4 + 10x^3 + 20x^2 + 15x + 6 = 0.\\] Find the value of \\[\\frac{1}{pq} + \\frac{1}{pr} + \\frac{1}{ps} + \\frac{1}{qr} + \\frac{1}{qs} + \\frac{1}{rs}.\\] 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": 33064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the planar vectors $\\overrightarrow {a}$ and $\\overrightarrow {b}$, with $|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=2$, and $\\overrightarrow {a} \\cdot \\overrightarrow {b} = 1$, find the angle between vectors $\\overrightarrow {a}$ and $\\overrightarrow {b}$.... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 33065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers \\(a, b, c, d\\) belong to the interval \\([-11.5, 11.5]\\). Find the maximum value of the expression \\(a + 2b + c + 2d - ab - bc - cd - da\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "552"
} | {
"split": "train",
"index": 33066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Petya and Vasya are playing the following game. Petya chooses a non-negative random value $\\xi$ with expectation $\\mathbb{E} [\\xi ] = 1$ , after which Vasya chooses his own value $\\eta$ with expectation $\\mathbb{E} [\\eta ] = 1$ without reference to the value of $\... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 33067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( [x] \\) denote the greatest integer less than or equal to the real number \\( x \\). Consider a sequence \\( \\{a_n\\} \\) defined by \\( a_1 = 1 \\) and \\( a_n = \\left[\\sqrt{n a_{n-1}}\\right] \\). Find the value of \\( a_{2017} \\). Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "2015"
} | {
"split": "train",
"index": 33068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\frac {1}{2}x^{2}-2\\ln x+a(a\\in\\mathbb{R})$, $g(x)=-x^{2}+3x-4$. \n$(1)$ Find the intervals of monotonicity for $f(x)$; \n$(2)$ Let $a=0$, the line $x=t$ intersects the graphs of $f(x)$ and $g(x)$ at points $M$ and $N$ respectively. When $|MN|$ r... | math | {
"style": "rule",
"ground_truth": "\\frac {3+ \\sqrt {33}}{6}"
} | {
"split": "train",
"index": 33069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the definite integral:\n\n$$\n\\int_{\\frac{\\pi}{2}}^{2 \\operatorname{arctg} 2} \\frac{d x}{\\sin ^{2} x(1+\\cos x)}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29/24"
} | {
"split": "train",
"index": 33070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a, b \\in R^{+}$, and $a + b = 1$, find the maximum value of $- \\frac{1}{2a} - \\frac{2}{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{9}{2}"
} | {
"split": "train",
"index": 33071,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each integer $n\\ge 2$ , determine, with proof, which of the two positive real numbers $a$ and $b$ satisfying \\[a^n=a+1,\\qquad b^{2n}=b+3a\\] is larger. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "a > b"
} | {
"split": "train",
"index": 33072,
"solution": "Solution 1\nSquare and rearrange the first equation and also rearrange the second. \\begin{align} a^{2n}-a&=a^2+a+1\\\\ b^{2n}-b&=3a \\end{align} It is trivial that \\begin{align*} (a-1)^2 > 0 \\tag{3} \\end{align*} since $a-1$ clearly cannot equal $0$ (Otherwise $... |
deepscaler | [
{
"role": "user",
"content": "Grandma's garden has three types of apples: Antonovka, Grushovka, and White Naliv. If the amount of Antonovka apples were tripled, the total number of apples would increase by 70%. If the amount of Grushovka apples were tripled, the total number of apples would increase by 50%.... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 33073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $6x^4-14x^3-4x^2+2x-26$ is divided by $2x - 6.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 33074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1(a＞b＞0)$$ with eccentricity $$e= \\frac { \\sqrt {3}}{2}$$, A and B are the left and right vertices of the ellipse, respectively, and P is a point on the ellipse different from A and B. The angles of inclination ... | math | {
"style": "rule",
"ground_truth": "\\frac {3}{5}"
} | {
"split": "train",
"index": 33075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $X+Y$ (in base 10), given the following addition problem in base 7:\n\\[ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c}\n& & & 5 & X & Y_{7}\\\\\n&+& & & 5 & 2_{7}\\\\\n\\cline{2-6}\n& & & 6 & 4 & X_{7}\\\\\n\\end{array} \\] Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 33076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Externally tangent circles with centers at points $A$ and $B$ have radii of lengths $5$ and $3$, respectively. A line externally tangent to both circles intersects ray $AB$ at point $C$. What is $BC$? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 33077,
"solution": "1. **Identify the Configuration**: \n Let $A$ and $B$ be the centers of two externally tangent circles with radii $5$ and $3$, respectively. The line tangent to both circles intersects ray $AB$ at point $C$. Let $D$ and $E$ be the points of tangency on circles ce... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{T}_{n}$ be the set of strings with only 0's or 1's of length $n$ such that any 3 adjacent place numbers sum to at least 1 and no four consecutive place numbers are all zeroes. Find the number of elements in $\\mathcal{T}_{12}$. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "1705"
} | {
"split": "train",
"index": 33078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ is $S_n$, and the sequence $\\{b_n\\}$ is a geometric sequence, satisfying $a_1=3$, $b_1=1$, $b_2+S_2=10$, and $a_5-2b_2=a_3$. The sum of the first $n$ terms of the sequence $\\left\\{ \\frac{a_n}{b_n} \\right... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 33079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a trip from the United States to Canada, Isabella took $d$ U.S. dollars. At the border she exchanged them all, receiving $10$ Canadian dollars for every $7$ U.S. dollars. After spending $60$ Canadian dollars, she had $d$ Canadian dollars left. What is the sum of the digits o... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 33080,
"solution": "1. **Understanding the exchange rate and the amount exchanged**: Isabella exchanges her U.S. dollars to Canadian dollars at a rate where $7$ U.S. dollars yield $10$ Canadian dollars. This means for every $7$ U.S. dollars, she receives $10$ Canadian dollars.\n\n2. *... |
deepscaler | [
{
"role": "user",
"content": "In a 14 team baseball league, each team played each of the other teams 10 times. At the end of the season, the number of games won by each team differed from those won by the team that immediately followed it by the same amount. Determine the greatest number of games the last p... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 33081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_1$ , $a_2$ , $\\ldots$ be an infinite sequence of (positive) integers such that $k$ divides $\\gcd(a_{k-1},a_k)$ for all $k\\geq 2$ . Compute the smallest possible value of $a_1+a_2+\\cdots+a_{10}$ . Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "440"
} | {
"split": "train",
"index": 33082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, square $PQRS$ has side length 2. Points $M$ and $N$ are the midpoints of $SR$ and $RQ$, respectively. The value of $\\cos (\\angle MPN)$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 33083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A soccer ball rolls at $4 \\mathrm{~m} / \\mathrm{s}$ towards Marcos in a direct line from Michael. The ball is $15 \\mathrm{~m}$ ahead of Michael who is chasing it at $9 \\mathrm{~m} / \\mathrm{s}$. Marcos is $30 \\mathrm{~m}$ away from the ball and is running towards it at $8... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 33084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the positive integers \\( a, b, c \\) satisfy \\( 2017 \\geqslant 10a \\geqslant 100b \\geqslant 1000c \\), find the number of possible triples \\( (a, b, c) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "574"
} | {
"split": "train",
"index": 33085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four identical isosceles triangles $A W B, B X C, C Y D$, and $D Z E$ are arranged with points $A, B, C, D$, and $E$ lying on the same straight line. A new triangle is formed with sides the same lengths as $A X, A Y,$ and $A Z$. If $A Z = A E$, find the largest integer value of... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 33086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( a, b, \\) and \\( c \\) are all positive integers, and the parabola \\( y = ax^2 + bx + c \\) intersects the x-axis at two distinct points \\( A \\) and \\( B \\). If both \\( A \\) and \\( B \\) are less than 1 unit away from the origin, what is the minimum valu... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 33087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n$$\n\\int_{1}^{2} \\frac{9x+4}{x^{5}+3x^{2}+x} \\, dx. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\ln \\frac{80}{23}"
} | {
"split": "train",
"index": 33088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nWe traveled by train from Anglchester to Klinkerton. But an hour after the train started, a locomotive malfunction was discovered. We had to continue the journey at a speed that was $\\frac{3}{5}$ of the original speed. As a result, we arrived in Klinkerton with a delay of 2 ... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 33089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $y = \\frac{p(x)}{q(x)}$ where $p(x)$ is quadratic and $q(x)$ is quadratic is given conceptually (imagine a graph with necessary features). The function has vertical asymptotes at $x = -4$ and $x = 1$. The graph passes through the point $(0,0)$ and $(2,-1)$. Determ... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 33090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the region described by $x \\ge 0,$ $y \\ge 0,$ and\n\\[50 \\{x\\} \\ge \\lfloor x \\rfloor - \\lfloor y \\rfloor.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25.5"
} | {
"split": "train",
"index": 33091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the integers 334 and 419 has digits whose product is 36. How many 3-digit positive integers have digits whose product is 36? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 33092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the area of the circle described by the equation \\(3x^2 + 3y^2 - 15x + 9y + 27 = 0\\) in terms of \\(\\pi\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 33093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \\( A, B, C \\) are three angles such that \\( A \\geq B \\geq C \\geq \\frac{\\pi}{8} \\) and \\( A + B + C = \\frac{\\pi}{2} \\). Find the largest possible value of the product \\( 720 \\times (\\sin A) \\times (\\cos B) \\times (\\sin C) \\). Let's think step by step... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 33094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a senior high school class, there are two study groups, Group A and Group B, each with 10 students. Group A has 4 female students and 6 male students; Group B has 6 female students and 4 male students. Now, stratified sampling is used to randomly select 2 students from each ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{31}{75}"
} | {
"split": "train",
"index": 33095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that the area of $\\triangle ABC$ is $3\\sin A$, the perimeter is $4(\\sqrt{2}+1)$, and $\\sin B + \\sin C = \\sqrt{2}\\sin A$.\n\n1. Find the values of $a$ and $\\cos... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{6} - 7}{18}"
} | {
"split": "train",
"index": 33096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "According to the graph, what is the average monthly balance, in dollars, of David's savings account during the four-month period shown? [asy]\ndraw((0,0)--(13,0)--(13,8)--(0,8)--cycle,linewidth(1));\ndraw((0,2)--(13,2),linewidth(1));\ndraw((0,4)--(13,4),linewidth(1));\ndraw((0,... | math | {
"style": "rule",
"ground_truth": "\\$150"
} | {
"split": "train",
"index": 33097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a class of 50 students, each student must be paired with another student for a project. Due to prior groupings in other classes, 10 students already have assigned partners. If the pairing for the remaining students is done randomly, what is the probability that Alex is paire... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{29}"
} | {
"split": "train",
"index": 33098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the arc length of the curve defined by the equation in the rectangular coordinate system.\n\n\\[ y = \\ln 7 - \\ln x, \\sqrt{3} \\leq x \\leq \\sqrt{8} \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1 + \\frac{1}{2} \\ln \\frac{3}{2}"
} | {
"split": "train",
"index": 33099,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six students sign up for three different intellectual competition events. How many different registration methods are there under the following conditions? (Not all six students must participate)\n\n(1) Each person participates in exactly one event, with no limit on the number ... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 33100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $G$ be the centroid of triangle $ABC$ with $AB=13,BC=14,CA=15$ . Calculate the sum of the distances from $G$ to the three sides of the triangle.\n\nNote: The *centroid* of a triangle is the point that lies on each of the three line segments between a vertex and the m... | math | {
"style": "rule",
"ground_truth": "\\frac{2348}{195}"
} | {
"split": "train",
"index": 33101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the digits 0 to 9, how many three-digit even numbers can be formed without repeating any digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 33102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the greatest common divisor of $4^{8}-1$ and $8^{12}-1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 33103,
"solution": "Let $d=\\operatorname{gcd}(a, b)$ for some $a, b \\in \\mathbb{Z}^{+}$. Then, we can write $d=a x-b y$, where $x, y \\in \\mathbb{Z}^{+}$, and $$\\begin{align*} & 2^{a}-1 \\mid 2^{a x}-1 \\tag{1}\\\\ & 2^{b}-1 \\mid 2^{b y}-1 \\tag{2} \\end{align*}$$ Multiplying t... |
deepscaler | [
{
"role": "user",
"content": "In the trapezoid \\(ABCD\\), the lengths of the bases \\(AD = 24 \\text{ cm}\\), \\(BC = 8 \\text{ cm}\\), and the diagonals \\(AC = 13 \\text{ cm}\\), \\(BD = 5\\sqrt{17} \\text{ cm}\\) are known. Calculate the area of the trapezoid. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 33104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Pine Ridge Elementary School, one third of the students ride the school bus home. One fifth of the students are picked up by car. One eighth of the students go home on their skateboards. Another one tenth of the students share rides with classmates. The rest of the students ... | math | {
"style": "rule",
"ground_truth": "\\frac{29}{120}"
} | {
"split": "train",
"index": 33105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which of the following numbers is not an integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2014}{4}"
} | {
"split": "train",
"index": 33106,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find \\( b \\) if \\( b \\) is the remainder when \\( 1998^{10} \\) is divided by \\( 10^{4} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"split": "train",
"index": 33107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the curve defined by the equation\n\\[r = \\frac{1}{1 - \\cos \\theta}.\\](A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{(C)}"
} | {
"split": "train",
"index": 33108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has $AC = 450$ and $BC = 300$. Points $K$ and $L$ are located on $\\overline{AC}$ and $\\overline{AB}$ respectively so that $AK = CK$, and $\\overline{CL}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\\overline{BK}$ and $\\overlin... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 33109,
"solution": "Using the diagram above, we can solve this problem by using mass points. By angle bisector theorem: \\[\\frac{BL}{CB}=\\frac{AL}{CA}\\implies\\frac{BL}{300}=\\frac{AL}{450}\\implies 3BL=2AL\\] So, we can weight $A$ as $2$ and $B$ as $3$ and $L$ as $5$. Since $K$ is... |
deepscaler | [
{
"role": "user",
"content": "A loonie is a $\\$ 1$ coin and a dime is a $\\$ 0.10$ coin. One loonie has the same mass as 4 dimes. A bag of dimes has the same mass as a bag of loonies. The coins in the bag of loonies are worth $\\$ 400$ in total. How much are the coins in the bag of dimes worth? Let's think... | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 33110,
"solution": "Since the coins in the bag of loonies are worth $\\$ 400$, then there are 400 coins in the bag. Since 1 loonie has the same mass as 4 dimes, then 400 loonies have the same mass as $4(400)$ or 1600 dimes. Therefore, the bag of dimes contains 1600 dimes, and so the c... |
deepscaler | [
{
"role": "user",
"content": "Simplify and evaluate\n\n(Ⅰ) Evaluate \\\\( \\dfrac{ \\sqrt{3}\\sin (- \\dfrac{20}{3}\\pi)}{\\tan \\dfrac{11}{3}\\pi}-\\cos \\dfrac{13}{4}\\pi\\cdot\\tan (- \\dfrac{35}{4}\\pi) \\).\n\n(Ⅱ) Evaluate: \\\\( \\dfrac{\\sqrt{1-2\\sin {10}^{\\circ }\\cos {10}^{\\circ }}}{\\cos {10}^{... | math | {
"style": "rule",
"ground_truth": "- \\dfrac{ \\sqrt{7}}{2}"
} | {
"split": "train",
"index": 33111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The figure below shows line $\\ell$ with a regular, infinite, recurring pattern of squares and line segments.\n[asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 33112,
"solution": "We analyze each of the four statements to determine which transformations will map the figure onto itself.\n\n1. **Some rotation around a point on line $\\ell$:**\n - Consider a rotation of $180^\\circ$ around a point that is exactly halfway between an up-facing ... |
deepscaler | [
{
"role": "user",
"content": "Divide a 7-meter-long rope into 8 equal parts, each part is meters, and each part is of the whole rope. (Fill in the fraction) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 33113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three balls are randomly placed into three boxes. Let the random variable $\\xi$ denote the maximum number of balls in any one box. Determine the mathematical expectation $E(\\xi)$ of $\\xi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{9}"
} | {
"split": "train",
"index": 33114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 33115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A board game spinner is divided into four regions labeled $A$, $B$, $C$, and $D$. The probability of the arrow stopping on region $A$ is $\\frac{3}{8}$, the probability of it stopping in $B$ is $\\frac{1}{4}$, and the probability of it stopping in region $C$ is equal to the pr... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"split": "train",
"index": 33116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I have a picture with dimensions $x$ and $y$ (in inches), such that $x$ and $y$ are both integers greater than one. I would like to place this picture in an elongated frame of dimensions $(2x + 3)$ and $(y+2)$. If I measured the area of the frame to be $34$ square inches, what... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 33117,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the operations: \n\\( a \\bigcirc b = a^{\\log _{7} b} \\),\n\\( a \\otimes b = a^{\\frac{1}{\\log ^{6} b}} \\), where \\( a, b \\in \\mathbb{R} \\). \n\nA sequence \\(\\{a_{n}\\} (n \\geqslant 4)\\) is given such that:\n\\[ a_{3} = 3 \\otimes 2, \\quad a_{n} = (n \\otim... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 33118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a $6\\times 6\\times h$ rectangular box containing a larger sphere of radius $3$ and four smaller spheres, each with radius $2$. The smaller spheres are placed at each corner of the bottom square face of the box and are tangent to two adjacent sides of the box. The lar... | math | {
"style": "rule",
"ground_truth": "5 + \\sqrt{23}"
} | {
"split": "train",
"index": 33119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the coefficient of the $x^2$ term in the expansion of $(1-ax)(1+2x)^4$ is $4$, then $\\int_{\\frac{e}{2}}^{a}{\\frac{1}{x}}dx =$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\ln(5) - 1"
} | {
"split": "train",
"index": 33120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( n = \\overline{abc} \\) be a three-digit number, where \\( a, b, \\) and \\( c \\) are the digits of the number. If \\( a, b, \\) and \\( c \\) can form an isosceles triangle (including equilateral triangles), how many such three-digit numbers \\( n \\) are there? Let's... | math | {
"style": "rule",
"ground_truth": "165"
} | {
"split": "train",
"index": 33121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the equation, $\\overline{\\mathrm{ABCD}}+\\overline{\\mathrm{EFG}}=2020$, different letters represent different digits. What is $A+B+C+D+E+F+G=$ $\\qquad$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 33122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be an integer greater than 1. If all digits of $97n$ are odd, find the smallest possible value of $n$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 33123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest common divisor of the numbers $2002+2,2002^{2}+2,2002^{3}+2, \\ldots$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 33124,
"solution": "Notice that $2002+2$ divides $2002^{2}-2^{2}$, so any common divisor of $2002+2$ and $2002^{2}+2$ must divide $\\left(2002^{2}+2\\right)-\\left(2002^{2}-2^{2}\\right)=6$. On the other hand, every number in the sequence is even, and the $n$th number is always congru... |
deepscaler | [
{
"role": "user",
"content": "Find the greatest integer value of $b$ for which the expression $\\frac{9x^3+4x^2+11x+7}{x^2+bx+8}$ has a domain of all real numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 33125,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xibing is a local specialty in Haiyang, with a unique flavor, symbolizing joy and reunion. Person A and person B went to the market to purchase the same kind of gift box filled with Xibing at the same price. Person A bought $2400$ yuan worth of Xibing, which was $10$ boxes less... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 33126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the fraction by rationalizing the denominator: $$\\frac{4}{\\sqrt{108}+2\\sqrt{12}+2\\sqrt{27}}.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{12}"
} | {
"split": "train",
"index": 33127,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the area of a triangle with side lengths 7, 7, and 5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{42.75}}{2}"
} | {
"split": "train",
"index": 33128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $0 < a < 1$, and $\\left[a+\\frac{1}{2020}\\right]+\\left[a+\\frac{2}{2020}\\right]+\\cdots+\\left[a+\\frac{2019}{2020}\\right] = 2018$, calculate the value of $[1010 a]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1009"
} | {
"split": "train",
"index": 33129,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circular park 20 feet in diameter with a straight walking path 5 feet wide passing through its center, calculate the area of the park covered by grass after the path is laid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100\\pi - 100"
} | {
"split": "train",
"index": 33130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Without using a calculator, find the largest prime factor of $17^4 + 2 \\times 17^2 + 1 - 16^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 33131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x) = \\begin{cases} x+2 & (x\\leq-1) \\\\ x^{2} & (-1<x<2) \\\\ 2x & (x\\geq2) \\end{cases}$, determine the value of $x$ if $f(x)=3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 33132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a class of 60 students, a sample of size 5 is to be drawn using systematic sampling. The students are randomly assigned numbers from 01 to 60, and then divided into 5 groups in order of their numbers (1-5, 6-10, etc.). If the second number drawn is 16, the number drawn from ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 33133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points $A(-3, -4)$ and $B(6, 3)$ in the xy-plane; point $C(1, m)$ is taken so that $AC + CB$ is a minimum. Find the value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{8}{9}"
} | {
"split": "train",
"index": 33134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose the graph of $y=f(x)$ includes the points $(1,5),$ $(2,3),$ and $(3,1)$.\n\nBased only on this information, there are two points that must be on the graph of $y=f(f(x))$. If we call those points $(a,b)$ and $(c,d),$ what is $ab+cd$? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 33135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of Jim's weight and Bob's weight is 180 pounds. If you subtract Jim's weight from Bob's weight, you get half of Bob's weight. How many pounds does Bob weigh? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 33136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b, c$, and $d$ be positive real numbers such that\n\\[\\begin{array}{c@{\\hspace{3pt}}c@{\\hspace{3pt}}c@{\\hspace{3pt}}c@{\\hspace{3pt}}c}a^2+b^2&=&c^2+d^2&=&2008,\\\\ ac&=&bd&=&1000.\\end{array}\\]\nIf $S=a+b+c+d$, compute the value of $\\lfloor S\\rfloor$.\n Let's th... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 33137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the six positive integer factors of 18? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 33138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the quadrilateral \\(ABCD\\), it is known that \\(AB = BD\\), \\(\\angle ABD = \\angle DBC\\), and \\(\\angle BCD = 90^\\circ\\). On the segment \\(BC\\), there is a point \\(E\\) such that \\(AD = DE\\). What is the length of segment \\(BD\\) if it is known that \\(BE = 7\\... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 33139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We have a triangle $\\triangle ABC$ and a point $K$ on $BC$ such that $AK$ is an altitude of $\\triangle ABC$. If $AC = 10,$ $BK = 7$, and $BC = 13,$ then what is the area of $\\triangle ABC$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 33140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Wendy is playing darts with a circular dartboard of radius 20. Whenever she throws a dart, it lands uniformly at random on the dartboard. At the start of her game, there are 2020 darts placed randomly on the board. Every turn, she takes the dart farthest from the center, and th... | math | {
"style": "rule",
"ground_truth": "6060"
} | {
"split": "train",
"index": 33141,
"solution": "Consider an individual dart. There is a $\\frac{1}{4}$ probability it is already within 10 units of the center. If not, for every throw there is a $\\frac{1}{4}$ probability it is not thrown again. Thus, if $E$ is the expected value of times it is thrown, we find $... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin \\alpha + \\cos \\alpha = \\frac{\\sqrt{2}}{3}$, and $0 < \\alpha < \\pi$, find $\\tan\\left(\\alpha - \\frac{\\pi}{4}\\right) = \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 33142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five brothers equally divided an inheritance from their father. The inheritance included three houses. Since three houses could not be divided into 5 parts, the three older brothers took the houses, and the younger brothers were compensated with money. Each of the three older b... | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 33143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many primes \\( p < 50 \\) is \\( p^{4} + 5p^{3} + 4 \\) divisible by 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 33144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of the expression $2+3 \times 5+2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 33145,
"solution": "Calculating, $2+3 \times 5+2=2+15+2=19$."
} |
deepscaler | [
{
"role": "user",
"content": "There are two colors Jessica can use to color this 2 by 2 grid. If non-overlapping regions that share a side cannot be the same color, how many possible ways are there for Jessica to color the grid?\n\n[asy]\nsize(101);\ndraw(unitsquare);\ndraw((0,.5)--(1,.5));\ndraw((.5,0)--(.... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 33146,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle, the lengths of the three sides are integers \\( l, m, n \\), with \\( l > m > n \\). It is known that \\( \\left\\{ \\frac{3^{l}}{10^{4}} \\right\\} = \\left\\{ \\frac{3^{m}}{10^{4}} \\right\\} = \\left\\{ \\frac{3^{n}}{10^{4}} \\right\\} \\), where \\( \\{x\\} \... | math | {
"style": "rule",
"ground_truth": "3003"
} | {
"split": "train",
"index": 33147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has area $30$, one side of length $10$, and the median to that side of length $9$. Let $\\theta$ be the acute angle formed by that side and the median. What is $\\sin{\\theta}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 33148,
"solution": "1. **Identify the given values and the formula for the area of a triangle:**\n - Given: Area of the triangle $= 30$, one side (let's call it $a$) $= 10$, and the median to that side $= 9$.\n - The formula for the area of a triangle using a side and the median t... |
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