sungyub/dapo-math-17k-verl huggingface dataset:
user: Let $\vartriangle ABC$ have $AB = 14$, $BC = 30$, $AC = 40$ and $\vartriangle AB'C'$ with $AB' = 7\sqrt6$, $B'C' = 15\sqrt6$, $AC' = 20\sqrt6$ such that $\angle BAB' = \frac{5\pi}{12}$ . The lines $BB'$ and $CC'$ intersect at point $D$. Let $O$ be the circumcenter of $\vartriangle BCD$, and let $O' $ be the circumcenter of $\vartriangle B'C'D$. Then the length of segment $OO'$ can be expressed as $\frac{a+b \sqrt{c}}{ d}$ , where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$

Answer: 55
user: Let $f(z)=\frac{a z+b}{c z+d}$ for $a, b, c, d \in \mathbb{C}$. Suppose that $f(1)=i, f(2)=i^{2}$, and $f(3)=i^{3}$. If the real part of $f(4)$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m^{2}+n^{2}$.

Answer: 34
user: The sum $$\sum_{m=1}^{2023} \frac{2m}{m^4+m^2+1}$$ can be expressed as $\tfrac{a}{b}$ for relatively prime positive integers $a,b.$ Find the remainder when $a+b$ is divided by $1000.$

Answer: 105
user: Consider the equations $x^2+y^2=16$ and $xy=\tfrac{9}{2}.$ Find the sum, over all ordered pairs $(x,y)$ satisfying these equations, of $|x+y|.$

Answer: 20
user: Let $\triangle A B C$ be a triangle with sidelengths $A B=5, B C=7$, and $C A=6$. Let $D, E, F$ be the feet of the altitudes from $A, B, C$, respectively. Let $L, M, N$ be the midpoints of sides $B C, C A, A B$, respectively. If the area of the convex hexagon with vertices at $D, E, F, L, M, N$ can be written as $\frac{x \sqrt{y}}{z}$ for positive integers $x, y, z$ with $\operatorname{gcd}(x, z)=1$ and $y$ square-free, find $x+y+z$.

Answer: 10043
user: Define a common chord between two intersecting circles to be the line segment connecting their two intersection points. Let $\omega_1,\omega_2,\omega_3$ be three circles of radii $3, 5,$ and $7$, respectively. Suppose they are arranged in such a way that the common chord of $\omega_1$ and $\omega_2$ is a diameter of $\omega_1$, the common chord of $\omega_1$ and $\omega_3$ is a diameter of $\omega_1$, and the common chord of $\omega_2$ and $\omega_3$ is a diameter of $\omega_2$. Compute the square of the area of the triangle formed by the centers of the three circles.

Answer: 96
user: Compute $\left\lfloor \sum_{k=0}^{10}\left(3+2\cos\left(\frac{2k\pi}{11}\right)\right)^{10}\right\rfloor \pmod{100}.$

Answer: 91
user: The integers from $1$ to $25,$ inclusive, are randomly placed into a $5$ by $5$ grid such that in each row, the numbers are increasing from left to right. If the columns from left to right are numbered $1,2,3,4,$ and $5,$ then the expected column number of the entry $23$ can be written as $\tfrac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b.$

Answer: 17
user: Rectangle $ABCD$ has $AB = 24$ and $BC = 7$. Let $d$ be the distance between the centers of the incircles of $\vartriangle ABC$ and $\vartriangle CDA$. Find $d^2$.

Answer: 325
user: Let $S$ denote the set of all positive integers whose prime factors are elements of $\{2,3,5,7,11\}$. (We include 1 in the set $S$.) If

$$
\sum_{q \in S} \frac{\varphi(q)}{q^{2}}
$$

can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, find $a+b$. (Here $\varphi$ denotes Euler's totient function.)

Answer: 1537
user: I have a $2$ by $4$ grid of squares; how many ways can I shade at least one of the squares such that no two shaded squares share an edge?

Answer: 40
user: A quadratic polynomial $f(x)$ is called sparse if its degree is exactly 2 , if it has integer coefficients, and if there exists a nonzero polynomial $g(x)$ with integer coefficients such that $f(x) g(x)$ has degree at most 3 and $f(x) g(x)$ has at most two nonzero coefficients. Find the number of sparse quadratics whose coefficients lie between 0 and 10, inclusive.

Answer: 228
user: Given a positive integer $m,$ define the polynomial $$P_m(z) = z^4-\frac{2m^2}{m^2+1} z^3+\frac{3m^2-2}{m^2+1}z^2-\frac{2m^2}{m^2+1}z+1.$$ Let $S$ be the set of roots of the polynomial $P_5(z)\cdot P_7(z)\cdot P_8(z) \cdot P_{18}(z).$ Let $w$ be the point in the complex plane which minimizes $\sum_{z \in S} |z-w|.$ The value of $\sum_{z \in S} |z-w|^2$ equals $\tfrac{a}{b}$ for relatively prime positive integers $a$ and $b.$ Compute $a+b.$

Answer: 171
user: A polynomial $p(x) = \sum_{j=1}^{2n-1} a_j x^j$ with real coefficients is called [i]mountainous[/i] if $n \ge 2$ and there exists a real number such that the polynomial's coefficients satisfy $a_1=1, a_{j+1}-a_j=k$ for $1 \le j \le n-1,$ and $a_{j+1}-a_j=-k$ for $n \le j \le 2n-2;$ we call $k$ the [i]step size[/i] of $p(x).$ A real number $k$ is called [i]good[/i] if there exists a mountainous polynomial $p(x)$ with step size $k$ such that $p(-3)=0.$ Let $S$ be the sum of all good numbers $k$ satisfying $k \ge 5$ or $k \le 3.$ If $S=\tfrac{b}{c}$ for relatively prime positive integers $b,c,$ find $b+c.$

Answer: 101
user: The real quartic $P x^{4}+U x^{3}+M x^{2}+A x+C$ has four different positive real roots. Find the square of the smallest real number $z$ for which the expression $M^{2}-2 U A+z P C$ is always positive, regardless of what the roots of the quartic are.

Answer: 16
user: Find the largest integer $x<1000$ such that $\left(\begin{array}{c}1515 \\ x\end{array}\right)$ and $\left(\begin{array}{c}1975 \\ x\end{array}\right)$ are both odd.

Answer: 419
user: If $\theta$ is the unique solution in $(0,\pi)$ to the equation $2\sin(x)+3\sin(\tfrac{3x}{2})+\sin(2x)+3\sin(\tfrac{5x}{2})=0,$ then $\cos(\theta)=\tfrac{a-\sqrt{b}}{c}$ for positive integers $a,b,c$ such that $a$ and $c$ are relatively prime. Find $a+b+c.$

Answer: 110
user: Kelvin the frog is hopping on the coordinate plane $\mathbb{R}^{2}$. He starts at the origin, and every second, he hops one unit to the right, left, up, or down, such that he always remains in the first quadrant $\{(x, y): x \geq 0, y \geq 0\}$. In how many ways can Kelvin make his first 14 jumps such that his 14 th jump lands at the origin?

Answer: 613470
user: Let $\oplus$ denote the xor binary operation. Define $x \star y=(x+y)-(x\oplus y).$ Compute $$\sum_{k=1}^{63} (k \star 45).$$([i]Remark:[/i] The xor operation works as follows: when considered in binary, the $k$th binary digit of $a \oplus b$ is $1$ exactly when the $k$th binary digits of $a$ and $b$ are different. For example, $5 \oplus 12 = 0101_2 \oplus 1100_2=1001_2=9.$)

Answer: 2880
user: Alien Connor starts at $(0,0)$ and walks around on the integer lattice. Specifically, he takes one step of length one in a uniformly random cardinal direction every minute, unless his previous four steps were all in the same directionin which case he randomly picks a new direction to step in. Every time he takes a step, he leaves toxic air on the lattice point he just left, and the toxic cloud remains there for $150$ seconds. After taking $5$ steps total, the probability that he has not encountered his own toxic waste canb be written as $\tfrac{a}{b}$ for relatively prime positive integers $a,b.$ Find $a+b.$

Answer: 505
user: Let $a,b,c,d,e,f$ be real numbers such that $a^2+b^2+c^2=14, d^2+e^2+f^2=77,$ and $ad+be+cf=32.$ Find $(bf-ce)^2+(cd-af)^2+(ae-bd)^2.$

Answer: 54
user: Let $\vartriangle ABC$ be a triangle with $AB = 4$ and $AC = \frac72$ . Let $\omega$ denote the $A$-excircle of $\vartriangle ABC$. Let $\omega$ touch lines $AB$, $AC$ at the points $D$, $E$, respectively. Let $\Omega$ denote the circumcircle of $\vartriangle ADE$. Consider the line $\ell$ parallel to $BC$ such that $\ell$ is tangent to $\omega$ at a point $F$ and such that $\ell$ does not intersect $\Omega$. Let $\ell$ intersect lines $AB$, $AC$ at the points $X$, $Y$ , respectively, with $XY = 18$ and $AX = 16$. Let the perpendicular bisector of $XY$ meet the circumcircle of $\vartriangle AXY$ at $P$, $Q$, where the distance from $P$ to $F$ is smaller than the distance from $Q$ to$ F$. Let ray $\overrightarrow {PF}$ meet $\Omega$ for the first time at the point $Z$. If $PZ^2 = \frac{m}{n}$ for relatively prime positive integers $m$, $n$, find $m + n$.

Answer: 1159
user: What is the sum of all possible $\left(\begin{array}{l}i \\ j\end{array}\right)$ subject to the restrictions that $i \geq 10, j \geq 0$, and $i+j \leq 20$ ? Count different $i, j$ that yield the same value separately - for example, count both $\left(\begin{array}{c}10 \\ 1\end{array}\right)$ and $\left(\begin{array}{c}10 \\ 9\end{array}\right)$.

Answer: 27633
user: Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers $a, b, c$, respectively. They then compute the polynomial $f(x)=x^{3}+p x^{2}+q x+r$ with roots $a, b, c$. If the expected value of the sum of the squares of the coefficients of $f(x)$ is $\frac{m}{n}$ for relatively prime positive integers $m, n$, find the remainder when $m+n$ is divided by 1000 .

Answer: 551
user: Let $a_n = \sqrt{1 + (1 - \tfrac{1}{n})^2} + \sqrt{1 + (1 + \tfrac{1}{n})^2}, n \ge 1$. Evaluate $\tfrac{1}{a_1} + \tfrac{1}{a_2} + \ldots + \tfrac{1}{a_{20}}$.

Answer: 7
user: Find the largest real number $x$ such that \[\left(\dfrac{x}{x-1}\right)^2+\left(\dfrac{x}{x+1}\right)^2=\dfrac{325}{144}.\]

Answer: 5
user: If \[\frac{1}{1+2} + \frac{1}{1+2+3} + \ldots + \frac{1}{1+2 + \ldots + 20} = \frac{m}{n}\]
where $m$ and $n$ are positive integers with no common divisor, find $m + n$.

Answer: 40
user: A circle radius $320$ is tangent to the inside of a circle radius $1000$. The smaller circle is tangent to a diameter of the larger circle at a point $P$. How far is the point $P$ from the outside of the larger circle?

Answer: 400
user: Let $r$ be a real number such that $\sqrt[3]{r} - \frac{1}{\sqrt[3]{r}}=2$. Find $r^3 - \frac{1}{r^3}$.

Answer: 2786
user: Let $x_1$ and $x_2$ be the roots of the equation $x^2 + 3x + 1 = 0$. Compute \[\left(\frac{x_1}{x_2 + 1}\right)^2 + \left(\frac{x_2}{x_1 + 1}\right)^2\]

Answer: 18
user: What is the smallest number that could be the date of the first Saturday after the second Monday following the second Thursday of a month?

Answer: 1
user: For each positive integer $m$ and $n$ define function $f(m, n)$ by $f(1, 1) = 1$, $f(m+ 1, n) = f(m, n) +m$ and $f(m, n + 1) = f(m, n) - n$. Find the sum of all the values of $p$ such that $f(p, q) = 2004$ for some $q$.

Answer: 3007
user: In $\triangle ABC$, $\angle A = 30^{\circ}$ and $AB = AC = 16$ in. Let $D$ lie on segment $BC$ such that $\frac{DB}{DC} = \frac23$ . Let $E$ and $F$ be the orthogonal projections of $D$ onto $AB$ and $AC$, respectively. Find $DE + DF$ in inches.

Answer: 8
user: Let $f$ be a real-valued function of real and positive argument such that $f(x) + 3xf(\tfrac1x) = 2(x + 1)$ for all real numbers $x > 0$. Find $f(2003)$.

Answer: 1002
user: In $\triangle ABC$, max $\{\angle A, \angle B \} = \angle C + 30^{\circ}$ and $\frac{R}{r} = \sqrt{3} + 1$, where $R$ is the radius of the circumcircle and $r$ is the radius of the incircle. Find $\angle C$ in degrees.

Answer: 60
user: A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer layer of unit cubes are removed from the block, more than half the original unit cubes will still remain?

Answer: 20
user: How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\]

Answer: 5
user: A week ago, Sandy’s seasonal Little League batting average was $360$. After five more at bats this week, Sandy’s batting average is up to $400$. What is the smallest number of hits that Sandy could have had this season?

Answer: 12
user: Let $a$ and $b$ be real numbers satisfying \[a^4 + 8b = 4(a^3 - 1) - 16 \sqrt{3}\] and \[b^4 + 8a = 4(b^3 - 1) + 16 \sqrt{3}.\] Find $a^4 + b^4$.

Answer: 56
user: Jerry purchased some stock for $ \$14,400$ at the same time that Susan purchased a bond for $ \$6,250$. Jerry’s investment went up $20$ percent the first year, fell $10$ percent the second year, and rose another $20$ percent the third year. Susan’s investment grew at a constant rate of compound interest for three years. If both investments are worth the same after three years, what was the annual percentage increase of Susan’s investment?

Answer: 44
user: Evaluate the product \[\left(1 + \frac{2}{3}\right)\left(1 + \frac{2}{4}\right)\left(1 + \frac{2}{5}\right) \cdots \left(1 + \frac{2}{98}\right).\]

Answer: 825
user: Find the number of different quadruples $(a, b, c, d)$ of positive integers such that $ab =cd = a + b + c + d - 3$.

Answer: 10
user: Let $M$ and $m$ be the largest and the smallest values of $x$, respectively, which satisfy $4x(x - 5) \le 375$. Find $M - m$.

Answer: 9
user: The number $2.5081081081081 \ldots$ can be written as $m/n$ where $m$ and $n$ are natural numbers with no common factors. Find $m + n$.

Answer: 649
user: A $70$ foot pole stands vertically in a plane supported by three $490$ foot wires, all attached to the top of the pole, pulled taut, and anchored to three equally spaced points in the plane. How many feet apart are any two of those anchor points?

Answer: 840
user: One rainy afternoon you write the number $1$ once, the number $2$ twice, the number $3$ three times, and so forth until you have written the number $99$ ninety-nine times. What is the $2005$ th digit that you write?

Answer: 5
user: How far is it from the point $(9, 17)$ to its reflection across the line \[3x + 4y = 15?\]

Answer: 32
user: We want to paint some identically-sized cubes so that each face of each cube is painted a solid color and each cube is painted with six different colors. If we have seven different colors to choose from, how many distinguishable cubes can we produce?

Answer: 210
user: There are three bags. One bag contains three green candies and one red candy. One bag contains two green candies and two red candies. One bag contains one green candy and three red candies. A child randomly selects one of the bags, randomly chooses a first candy from that bag, and eats the candy. If the first candy had been green, the child randomly chooses one of the other two bags and randomly selects a second candy from that bag. If the first candy had been red, the child randomly selects a second candy from the same bag as the first candy. If the probability that the second candy is green is given by the fraction $m/n$ in lowest terms, find $m + n$.

Answer: 217
user: How many non-overlapping $2$ by $2$ squares will fit into a circle with radius $8$?

Answer: 40
user: Find the number of addition problems in which a two digit number is added to a second two digit number to give a two digit answer, such as in the three examples: \[\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}23\\42\\\hline 65\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}36\\36\\\hline 72\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}42\\23\\\hline 65\end{tabular}\,.\]

Answer: 3240
user: How many positive integers less that $200$ are relatively prime to either $15$ or $24$?

Answer: 120
user: Write the number $2004_{(5)}$ [ $2004$ base $5$ ] as a number in base $6$.

Answer: 1102
user: A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer two layers of unit cubes are removed from the block, more than half the original unit cubes will still remain?

Answer: 10
user: In the addition problem 
\[ \setlength{\tabcolsep}{1mm}\begin{tabular}{cccccc}& W & H & I & T & E\\ + & W & A & T & E & R \\\hline P & I & C & N & I & C\end{tabular} \] each distinct letter represents a diﬀerent digit. Find the number represented by the answer PICNIC.

Answer: 169069
user: Find $x$ so that $2^{2^{3^{2^{2}}}} = 4^{4^{x}}$.

Answer: 40
user: Find $x$ such that \[\frac{\frac{5}{x-50}+ \frac{7}{x+25}}{\frac{2}{x-50}- \frac{3}{x+25}} = 17.\]

Answer: 125
user: What is the $1000$ th digit to the right of the decimal point in the decimal representation of $\tfrac{37}{5500}$?

Answer: 7
user: Find the number of nonnegative integers $n$ for which $(n^2 - 3n + 1)^2 + 1$ is a prime number

Answer: 4
user: We glue together $990$ one inch cubes into a $9$ by $10$ by $11$ inch rectangular solid. Then we paint the outside of the solid. How many of the original $990$ cubes have just one of their sides painted?

Answer: 382
user: The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $150$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?

Answer: 30
user: Eight identical cubes with of size $1 \times 1 \times 1$ each have the numbers $1$ through $6$ written on their faces with the number $1$ written on the face opposite number $2$, number $3$ written on the face opposite number $5$, and number $4$ written on the face opposite number $6$. The eight cubes are stacked into a single $2 \times 2 \times 2$ cube. Add all of the numbers appearing on the outer surface of the new cube. Let $M$ be the maximum possible value for this sum, and $N$ be the minimum possible value for this sum. Find $M - N$.

Answer: 40
user: The side lengths of a trapezoid are $\sqrt[4]{3}, \sqrt[4]{3}, \sqrt[4]{3}$, and $2 \cdot \sqrt[4]{3}$. Its area is the ratio of two relatively prime positive integers, $m$ and $n$. Find $m + n$.

Answer: 13
user: And it came to pass that Jeb owned over a thousand chickens. So Jeb counted his chickens. And Jeb reported the count to Hannah. And Hannah reported the count to Joshua. And Joshua reported the count to Caleb. And Caleb reported the count to Rachel. But as fate would have it, Jeb had over-counted his chickens by nine chickens. Then Hannah interchanged the last two digits of the count before reporting it to Joshua. And Joshua interchanged the first and the third digits of the number reported to him before reporting it to Caleb. Then Caleb doubled the number reported to him before reporting it to Rachel. Now it so happens that the count reported to Rachel was the correct number of chickens that Jeb owned. How many chickens was that?

Answer: 2052
user: The number $1$ is special. The number $2$ is special because it is relatively prime to $1$. The number $3$ is not special because it is not relatively prime to the sum of the special numbers less than it, $1 + 2$. The number $4$ is special because it is relatively prime to the sum of the special numbers less than it. So, a number bigger than $1$ is special only if it is relatively prime to the sum of the special numbers less than it. Find the twentieth special number.

Answer: 29
user: Jerry is mowing a rectangular lawn which is $77$ feet north to south by $83$ feet east to west. His lawn mower cuts a path $18$ inches wide. Jerry mows the grass by cutting a path from west to east across the north side of the lawn and then making a right turn cutting a path along the east side of the lawn. When he completes mowing each side of the lawn, he continues by making right turns to mow a path along the next side. How many right turns will he make?

Answer: 102
user: Let $d_k$ be the greatest odd divisor of $k$ for $k = 1, 2, 3, \ldots$. Find $d_1 + d_2 + d_3 + \ldots + d_{1024}$.

Answer: 349526
user: Find the number of nonnegative integers $n$ for which $(n^2 - 3n +1)^2 + 1$ is a prime number.

Answer: 4
user: The cost of producing each item is inversely proportional to the square root of the number of items produced. The cost of producing ten items is $ \$2100$. If items sell for $ \$30$ each, how many items need to be sold so that the producers break even?

Answer: 490
user: A palindrome is a number that reads the same forwards and backwards such as $3773$ or $42924$. What is the smallest $9$ digit palindrome which is a multiple of $3$ and has at least two digits which are $5$'s and two digits which are $7$'s?

Answer: 105717501
user: Find $x$ if\[\cfrac{1}{\cfrac{1}{\cfrac{1}{\cfrac{1}{x}+\cfrac12}+\cfrac{1}{\cfrac{1}{x}+\cfrac12}}+\cfrac{1}{\cfrac{1}{\cfrac{1}{x}+\cfrac12}+\cfrac{1}{\cfrac{1}{x}+\cfrac12}}}=\frac{x}{36}.\]

Answer: 70
user: A palindrome is a number that reads the same forwards and backwards such as $3773$ or $42924$. Find the sum of the twelve smallest five digit palindromes.

Answer: 126632
user: Find the number of quadruples $(a,b,c,d)$ of integers which satisfy both \begin{align*}\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} &= \frac{1}{2}\qquad\text{and}\\\\2(a+b+c+d) &= ab + cd + (a+b)(c+d) + 1.\end{align*}

Answer: 48
user: A tailor met a tortoise sitting under a tree. When the tortoise was the tailor’s age, the tailor was only a quarter of his current age. When the tree was the tortoise’s age, the tortoise was only a seventh of its current age. If the sum of their ages is now $264$, how old is the tortoise?

Answer: 77
user: The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $165$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?

Answer: 30
user: Functions $f$ and $g$ are defined so that $f(1) = 4$, $g(1) = 9$, and for each integer $n \ge 1$, $f(n+1) = 2f(n) + 3g(n) + 2n $ and $g(n+1) = 2g(n) + 3 f(n) + 5$. Find $f(2005) - g(2005)$.

Answer: 1999
user: Let $T$ be a $30-60-90$ triangle with hypotenuse of length $20$. Three circles, each externally tangent to the other two, have centers at the three vertices of $T$. The area of the union of the circles intersected with $T$ is $(m + n \sqrt{3}) \pi$ for rational numbers $m$ and $n$. Find $m + n$.

Answer: 50
user: We have two positive integers both less than $1000$. The arithmetic mean and the geometric mean of these numbers are consecutive odd integers. Find the maximum possible value of the difference of the two integers.

Answer: 120
user: At the beginning of each hour from $1$ o’clock AM to $12$ NOON and from $1$ o’clock PM to $12$ MIDNIGHT a coo-coo clock’s coo-coo bird coo-coos the number of times equal to the number of the hour. In addition, the coo-coo clock’s coo-coo bird coo-coos a single time at $30$ minutes past each hour. How many times does the coo-coo bird coo-coo from $12:42$ PM on Monday until $3:42$ AM on Wednesday?

Answer: 279
user: Find the sum of all the prime numbers less than $100$ which are one more than a multiple of six.

Answer: 527
user: $12$ students need to form five study groups. They will form three study groups with $2$ students each and two study groups with $3$ students each. In how many ways can these groups be formed?

Answer: 138600
user: Consider all ordered pairs $(m, n)$ of positive integers satisfying $59 m - 68 n = mn$. Find the sum of all the possible values of $n$ in these ordered pairs.

Answer: 237
user: In triangle $ABC$, $AB = 52$, $BC = 56$, $CA = 60$. Let $D$ be the foot of the altitude from $A$ and $E$ be the intersection of the internal angle bisector of $\angle BAC$ with $BC$. Find $DE$.

Answer: 6
user: The rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. Each day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. This continued through the day when they caught more squirrels than rabbits. Up through that day how many rabbits did they catch in all?

Answer: 5491
user: At a movie theater tickets for adults cost $4$ dollars more than tickets for children. One afternoon the theater sold $100$ more child tickets than adult tickets for a total sales amount of $1475$ dollars. How much money would the theater have taken in if the same tickets were sold, but the costs of the child tickets and adult tickets were reversed?

Answer: 1875
user: Consider the polynomials \begin{align*}P(x) &= (x + \sqrt{2})(x^2 - 2x + 2)\\Q(x) &= (x - \sqrt{2})(x^2 + 2x + 2)\\R(x) &= (x^2 + 2)(x^8 + 16).\end{align*} Find the coefficient of $x^4$ in $P(x)\cdot Q(x)\cdot R(x)$.

Answer: 0