meetforever/GRPO_CL huggingface dataset:
user: A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin \(k\) is \(2^{-k}\) for \(k = 1,2,3,\ldots\). What is the probability that the red ball is tossed into a higher-numbered bin than the green ball? The original answer is in the form \(\frac{k}{m}\), please give the value of \(k + m\).
Answer: 4
user: The base-ten representation for $19!$ is $121,6T5,100,40M,832,H00$, where $T$, $M$, and $H$ denote digits that are not given. Find the value of $T+M+H$.
Answer: 12
user: For a certain complex number \(c\), the polynomial
\[P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\]
has exactly 4 distinct roots. The original answer is in \(\sqrt{m}\) format, please give the value of \(m\).
Answer: 10
user: Let $S$ be the set of all positive integer divisors of $100,000.$ Find how many numbers are the product of two distinct elements of $S.$
Answer: 117
user: Right triangles \(T_1\) and \(T_2\), have areas of \(1\) and \(2\), respectively. A side of \(T_1\) is congruent to a side of \(T_2\), and a different side of \(T_1\) is congruent to a different side of \(T_2\). Find the square of the product of the lengths of the other (third) sides of \(T_1\) and \(T_2\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 31
user: Positive real numbers \(a\) and \(b\) have the property that  
\[
\sqrt{\log{a}} + \sqrt{\log{b}} + \log \sqrt{a} + \log \sqrt{b} = 100
\]
and all four terms on the left are positive integers, where \(\log\) denotes the base \(10\) logarithm. The original answer is in the form \(10^k\). Please find the value of \(k\).
Answer: 164
user: In $\triangle ABC$ with a right angle at $C$, point $D$ lies in the interior of $\overline{AB}$ and point $E$ lies in the interior of $\overline{BC}$ so that $AC=CD,$ $DE=EB,$ and the ratio $AC:DE=4:3$. The original answer is in the form x:y for the ratio $AD:DB$. Please give the value of x + y.
Answer: 5
user: The numbers \(1,2,\cdots,9\) are randomly placed into the \(9\) squares of a \(3 \times 3\) grid. Each square gets one number, and each of the numbers is used once. Find the probability that the sum of the numbers in each row and each column is odd. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 15
user: There are lily pads in a row numbered $0$ to $11$, in that order. There are predators on lily pads $3$ and $6$, and a morsel of food on lily pad $10$. Fiona the frog starts on pad $0$, and from any given lily pad, has a $\frac{1}{2}$ chance to hop to the next pad, and an equal chance to jump $2$ pads. What is the probability that Fiona reaches pad $10$ without landing on either pad $3$ or pad $6$? Express your answer as a fraction in simplest form in the format \frac{k}{m}, and find the value of k + m.
Answer: 271
user: A child builds towers using identically shaped cubes of different colors. How many different towers with a height of $8$ cubes can the child build using $2$ red cubes, $3$ blue cubes, and $4$ green cubes? (One cube will be left out.) Please provide your answer as an integer.
Answer: 1260
user: A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin \(k\) is \(2^{-k}\) for \(k = 1,2,3,\ldots\). Find the probability that the red ball is tossed into a higher-numbered bin than the green ball. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 4
user: Let $s_k$ denote the sum of the $\textit{k}$th powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a \, s_k + b \, s_{k-1} + c \, s_{k-2}$ for $k = 2$, $3$, $....$ Find the value of $a+b+c$.
Answer: 10
user: What is the value of the product 

\[\left(\frac{1\cdot3}{2\cdot2}\right)\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{97\cdot99}{98\cdot98}\right)\left(\frac{98\cdot100}{99\cdot99}\right)?\]

Provide your answer as a fraction in simplest form \(\frac{k}{m}\). Please calculate the value of \(k + m\).
Answer: 149
user: A sphere with center \(O\) has radius \(6\). A triangle with sides of length \(15, 15,\) and \(24\) is situated in space so that each of its sides is tangent to the sphere. The distance between \(O\) and the plane determined by the triangle is in the form \(k\sqrt{m}\). Find the value of \(k + m\).
Answer: 7
user: Square pyramid $ABCDE$ has base $ABCD$, which measures $3$ cm on a side, and altitude $AE$ perpendicular to the base, which measures $6$ cm. Point $P$ lies on $BE$, one third of the way from $B$ to $E$; point $Q$ lies on $DE$, one third of the way from $D$ to $E$; and point $R$ lies on $CE$, two thirds of the way from $C$ to $E$. What is the area, in square centimeters, of $\triangle{PQR}$? Express your answer in the form of $a\sqrt{b}$ and find the value of $a + b$.
Answer: 4
user: Raashan, Sylvia, and Ted play the following game. Each starts with $＄$ $1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $＄$ $1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $＄$ $1$? Provide your answer as a fraction in simplest form, \(\frac{k}{m}\), and then give the value of \(k + m\).
Answer: 5
user: Real numbers between \(0\) and \(1\), inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is \(0\) if the second flip is heads, and \(1\) if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval \([0,1]\). Two random numbers \(x\) and \(y\) are chosen independently in this manner. Find the probability that \(|x-y| > \tfrac{1}{2}\). The original answer is in \(\frac{m}{n}\) format, please give the value of m + n.
Answer: 23
user: Points \(A=(6,13)\) and \(B=(12,11)\) lie on circle \(\omega\) in the plane. Suppose that the tangent lines to \(\omega\) at \(A\) and \(B\) intersect at a point on the \(x\)-axis. The value of the area of \(\omega\) divided by \(\pi\) is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Answer: 93
user: A sphere with center \( O \) has radius \( 6 \). A triangle with sides of length \( 15, 15, \) and \( 24 \) is situated in space so that each of its sides is tangent to the sphere. The original answer is in the form \( k\sqrt{m} \). Find the distance between \( O \) and the plane determined by the triangle, and give the value of \( k + m \).
Answer: 7
user: Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. Find the probability that she gets two heads in a row but she sees a second tail before she sees a second head. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 25
user: Let \[z=\frac{1+i}{\sqrt{2}}.\] Calculate the value of \[\left(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}\right) \cdot \left(\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}}\right).\]
Answer: 36
user: Real numbers between $0$ and $1$, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is $0$ if the second flip is heads, and $1$ if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. Find the probability that $|x-y| > \tfrac{1}{2}$. The original answer is in the form \frac{k}{m}, please provide the value of k + m.
Answer: 23
user: Raashan, Sylvia, and Ted play the following game. Each starts with $＄$ $1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $＄$ $1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $＄$ $1$? The original answer is in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest form. Please provide the value of \(k + m\).
Answer: 5
user: Circles $\omega$ and $\gamma$, both centered at $O$, have radii $20$ and $17$, respectively. Equilateral triangle $ABC$, whose interior lies in the interior of $\omega$ but in the exterior of $\gamma$, has vertex $A$ on $\omega$, and the line containing side $\overline{BC}$ is tangent to $\gamma$. Segments $\overline{AO}$ and $\overline{BC}$ intersect at $P$, and $\dfrac{BP}{CP} = 3$. Then $AB$ can be written in the form $\dfrac{m}{\sqrt{n}} - \dfrac{p}{\sqrt{q}}$ for positive integers $m$, $n$, $p$, $q$ with ${gcd}(m,n) = {gcd}(p,q) = 1$. What is the value of $m+n+p+q$?
Answer: 130
user: Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number $1$, then Todd must say the next two numbers ($2$ and $3$), then Tucker must say the next three numbers ($4$, $5$, $6$), then Tadd must say the next four numbers ($7$, $8$, $9$, $10$), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number $10,000$ is reached. Find the $2019$th number said by Tadd.
Answer: 5979
user: Points \(A=(6,13)\) and \(B=(12,11)\) lie on circle \(\omega\) in the plane. Suppose that the tangent lines to \(\omega\) at \(A\) and \(B\) intersect at a point on the \(x\)-axis. If the area of \(\omega\) is expressed as \(\frac{m\pi}{n}\), please give the value of m+n.
Answer: 93
user: Define binary operations $\diamondsuit$ and $\heartsuit$ by \[a \, \diamondsuit \, b = a^{\log_{7}(b)} \qquad \text{and} \qquad a  \, \heartsuit \, b = a^{\frac{1}{\log_{7}(b)}}\] for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3 = 3\, \heartsuit\, 2$ and \[a_n = (n\, \heartsuit\, (n-1)) \,\diamondsuit\, a_{n-1}\] for all integers $n \geq 4$. To the nearest integer, what is $\log_{7}(a_{2019})$?
Answer: 11
user: How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s? Find the number of such sequences.
Answer: 65
user: Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$. It is given that there exist real numbers $A$, $B$, and $C$ such that \[\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}\] for all $s\not\in\{p,q,r\}$. What is $\tfrac1A+\tfrac1B+\tfrac1C$?
Answer: 244
user: For how many integers $n$ between $1$ and $50$, inclusive, is  
$$\frac{(n^2-1)!}{(n!)^n}$$ 
an integer? (Recall that $0! = 1$.)
Answer: 34
user: Let \(\omega=-\tfrac{1}{2}+\tfrac{1}{2}i\sqrt{3}.\) Let \(S\) denote all points in the complex plane of the form \(a+b\omega+c\omega^2,\) where \(0\leq a \leq 1,0\leq b\leq 1,\) and \(0\leq c\leq 1.\) Find the area of \(S.\) The original answer is in the form \(\frac{k}{m}\sqrt{n}\), please give the value of k + m + n.
Answer: 8
user: Let $\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\circ$, $60^\circ$, and $60.001^\circ$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$ to line $A_{n-1}B_{n-1}$. Find the least positive integer $n$ for which $\triangle A_nB_nC_n$ is obtuse.
Answer: 15
user: Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\triangle ABC,\triangle BCD,$ and $\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$? Express your answer in the form $a + b\sqrt{3}$, where $a$ and $b$ are integers. Please provide the value of a + b.
Answer: 22
user: Four friends do yardwork for their neighbors over the weekend, earning $＄$ $15$, $＄$ $20$, $＄$ $25,$ and $＄$ $40,$ respectively. They decide to split their total earnings equally among themselves. If the answer is x dollars, calculate the amount that the friend who earned $＄$ $40$ will give to the others to achieve this equal distribution. Please provide the value of x.
Answer: 15
user: Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?
Answer: 37
user: How many positive even multiples of $3$ less than $2020$ are perfect squares?
Answer: 7
user: Seven cubes, whose volumes are $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?
Answer: 658
user: How many integers between $2020$ and $2400$ have four distinct digits arranged in increasing order? (For example, $2347$ is one integer.)
Answer: 15
user: What is the value of the alternating sum 
$$1+2+3-4+5+6+7-8+\cdots+197+198+199-200?$$
Calculate the final result as an integer.
Answer: 9900
user: What is the median of the following list of $4040$ numbers?  
$1, 2, 3, \ldots, 2020, 1^2, 2^2, 3^2, \ldots, 2020^2$
Calculate the median of this list and provide the value of the median multiplied by 10.
Answer: 19765
user: How many ordered pairs of integers $(x, y)$ satisfy the equation $x^{2020}+y^2=2y?$
Answer: 4
user: A single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N$?
Answer: 18
user: How many ordered pairs of integers $(x, y)$ satisfy the equation  
$$x^{2020}+y^2=2y?$$
Answer: 4
user: How many solutions does the equation $\tan(2x)=\cos(\tfrac{x}{2})$ have on the interval $[0,2\pi]?$
Answer: 5
user: In unit square \(ABCD,\) the inscribed circle \(\omega\) intersects \(\overline{CD}\) at \(M,\) and \(\overline{AM}\) intersects \(\omega\) at a point \(P\) different from \(M.\) The length of \(AP\) is in the form \(\frac{\sqrt{n}}{k}\). Please find the value of \(k + m + n\).
Answer: 15
user: What is the median of the following list of $4040$ numbers ?
$1, 2, 3, \ldots, 2020, 1^2, 2^2, 3^2, \ldots, 2020^2$
If the median is $x$, what is the value of $10x$?
Answer: 19765
user: Ms. Carr asks her students to read any $5$ of the $10$ books on a reading list. Harold randomly selects $5$ books from this list, and Betty does the same. Find the probability that there are exactly $2$ books that they both select. The answer should be expressed in the form \(\frac{k}{m}\), where the fraction is in simplest form. Please give the value of \(k + m\).
Answer: 88
user: A frog sitting at the point \((1, 2)\) begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length \(1\), and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices \((0,0), (0,4), (4,4),\) and \((4,0)\). What is the probability that the sequence of jumps ends on a vertical side of the square? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 13
user: The decimal representation of 
$$\dfrac{1}{20^{20}}$$ 
consists of a string of zeros after the decimal point, followed by a $9$ and then several more digits. How many zeros are in that initial string of zeros after the decimal point?
Answer: 26
user: Line $l$ in the coordinate plane has equation $3x-5y+40=0$. This line is rotated $45^{\circ}$ counterclockwise about the point $(20,20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k$?
Answer: 15
user: Let $\overline{AB}$ be a diameter in a circle of radius $5\sqrt2.$ Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=2\sqrt5$ and $\angle AEC = 45^{\circ}.$ Find the value of $CE^2+DE^2.$
Answer: 100
user: What is the value of $\sqrt{\log_2{6}+\log_3{6}}$? The original answer is in the form $\sqrt{\log_k{3}}+\sqrt{\log_m{2}}$. Please find the value of k + m.
Answer: 5
user: A frog sitting at the point \((1, 2)\) begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length \(1\), and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices \((0,0), (0,4), (4,4),\) and \((4,0)\). The probability that the sequence of jumps ends on a vertical side of the square is given in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Answer: 13
user: Andy the Ant lives on a coordinate plane and is currently at $(-20, 20)$ facing east (that is, in the positive $x$-direction). Andy moves $1$ unit and then turns $90^{\circ}$ left. From there, Andy moves $2$ units (north) and then turns $90^{\circ}$ left. He then moves $3$ units (west) and again turns $90^{\circ}$ left. Andy continues his progress, increasing his distance each time by $1$ unit and always turning left. What are the coordinates of the point at which Andy makes the $2020$th left turn? The original answer is in the form $(m, n)$, please provide the value of $m + n$.
Answer: -2020
user: There are integers $a, b,$ and $c,$ each greater than $1,$ such that  
 $\sqrt[a]{N\sqrt[b]{N\sqrt[c]{N}}} = \sqrt[36]{N^{25}}$  
for all $N \neq 1$. What is the value of $b$?
Answer: 3
user: Regular octagon \(ABCDEFGH\) has area \(n\). Let \(m\) be the area of quadrilateral \(ACEG\). The answer to the question is in the form \(\frac{\sqrt{k}}{m}\). What is the value of \(k + m\)?
Answer: 4
user: A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$.
Answer: 23
user: Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\ldots$ digits from the left), then erased every fourth digit from the resulting list (that is, the $4$th, $8$th, $12$th, $\ldots$ digits from the left in what remained), and then erased every fifth digit from what remained at that point. Find the sum of the three digits that were then in the positions $2019, 2020, 2021$.
Answer: 11
user: In the complex plane, let $A$ be the set of solutions to $z^{3}-8=0$ and let $B$ be the set of solutions to $z^{3}-8z^{2}-8z+64=0.$ Find the greatest distance between a point of $A$ and a point of $B$. The original answer is in $k\sqrt{m}$ format, please give the value of k + m.
Answer: 23
user: A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) If $d$ to the nearest tenth is $x$, what is $10x$?
Answer: 4
user: A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) If $d$ to the nearest tenth is $x$, what is the value of $10x$?
Answer: 4
user: An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations, the urn contains six balls. What is the probability that the urn contains three balls of each color? Express your answer in the form \(\frac{k}{m}\), where k and m are integers. Then find the value of k + m.
Answer: 6
user: Define \[P(x) =(x-1^2)(x-2^2)\cdots(x-100^2).\] Find the number of integers $n$ such that $P(n)\leq 0$.
Answer: 5100
user: The vertices of a quadrilateral lie on the graph of $y=\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln{\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex?
Answer: 12
user: How many positive integer factors of $2020$ have more than $3$ factors? (As an example, $12$ has $6$ factors, namely $1,2,3,4,6,$ and $12.$) Provide your answer as an integer.
Answer: 7
user: Let $(a,b,c,d)$ be an ordered quadruple of not necessarily distinct integers, each one of them in the set $\{0,1,2,3\}.$ Find the number of such quadruples for which $a\cdot d-b\cdot c$ is odd. (For example, $(0,3,1,1)$ is one such quadruple, because $0\cdot 1-3\cdot 1 = -3$ is odd.)
Answer: 96
user: Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}$, $AC=20$, and $CD=30$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$, and $AE=5$. Find the area of quadrilateral $ABCD$.
Answer: 360
user: 已知 $F_{1}, F_{2}$ 为椭圆 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ 的左、右焦点，该椭圆上存在两点 $A, B$ ，使得 $\overrightarrow{F_{1} A}=3 \overrightarrow{F_{2} B}$ 。求该椭圆的离心率的取值范围。答案的形式为\left(\frac{c}{d}, 1\right)，请给出c+d的值。
Answer: 3
user: 设有三个函数，第一个是 $y=\varphi(x)$ ，它的反函数是第二个函数，而第三个函数的图象与第二个函数的图象关于 $x+y=0$ 对称。第三个函数的表达式为 $y=ax+b\varphi(cx)$，请给出a+b+c的值。
Answer: -2
user: Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ What is the area of $ABCD$?
Answer: 240
user: In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. Find the digit $A$.
Answer: 2
user: Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC=20,$ and $CD=30.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD$?
Answer: 360
user: Let $B$ be a right rectangular prism (box) with edge lengths $1,$ $3,$ and $4$, together with its interior. For real $r\geq0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$, where $a,$ $b,$ $c,$ and $d$ are positive real numbers. Find the value of $\frac{bc}{ad}$.
Answer: 19
user: Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. Find the number of the $125$ sequences of three of these transformations (not necessarily distinct) that will return $T$ to its original position.
Answer: 12
user: Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. Find the probability that after they are painted, the cubes can be rotated to be identical in appearance. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 1171