ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: For how many (not necessarily positive) integer values of $n$ is the value of $4000 \cdot \left(\frac{2}{5}\right)^n$ an integer? Provide your answer as an integer.
Answer: 9
Problem: A large rectangle is partitioned into four smaller rectangles by two segments parallel to its sides. The areas of three of these rectangles are given. Determine the area of the fourth rectangle.

```
\begin{asy}
draw((0,0)--(10,0)--(10,7)--(0,7)--cycle);
draw((0,5)--(10,5));
draw((3,0)--(3,7));
label("6", (1.5,6));
label("?", (1.5,2.5));
label("14", (6.5,6));
label("35", (6.5,2.5));
\end{asy}
```

Find the area of the fourth rectangle.
Answer: 15
Problem: What is the difference between the maximum value and the minimum value of the sum $a_1 + 2a_2 + 3a_3 + 4a_4 + 5a_5$ where $\{a_1,a_2,a_3,a_4,a_5\} = \{1,2,3,4,5\}$? Provide your answer as an integer.
Answer: 20
Problem: Find the largest number among the following expressions: \(30^{30}\), \(50^{10}\), \(40^{20}\), \(45^{15}\), and \(5^{60}\). The original answer is in the form a^b, where a is the base and b is the exponent. Please provide the sum of a and b.
Answer: 60
Problem: 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 how many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position.
Answer: 12
Problem: Find the sum of distinct real roots of the polynomial $x^5 + x^4 - 4x^3 - 7x^2 - 7x - 2$.
Answer: 0
Problem: A semi-palindrome is a four-digit number whose first two digits and last two digits are identical. For instance, $2323$ and $5757$ are semi-palindromes, but $1001$ and $2324$ are not. Find the difference between the largest semi-palindrome and the smallest semi-palindrome.
Answer: 8989
Problem: What is the least integer $n \geq 100$ such that $77$ divides $1+2+2^2+2^3+\dots + 2^n$? Find the value of $n$.
Answer: 119
Problem: Triangle $BAD$ is right-angled at $B$. On $AD$, there is a point $C$ such that $AC = CD$ and $AB = BC$. Determine the magnitude of angle $DAB$ in degrees. If the answer is $x^\circ$, what is the value of $x$?
Answer: 60
Problem: When $\left(1 - \frac{1}{a}\right)^6$ is expanded, find the sum of the last three coefficients.
Answer: 10
Problem: Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What angle $\theta=\angle PAB=\angle QPC=\angle RQB=\cdots$ will produce a path length of $120$ meters? (Do not assume the zigzag path has exactly four segments as shown; there could be more or fewer.) If the answer is in the form of \(\arccos\frac{k}{m}\), provide the value of k + m.
Answer: 11
Problem: Find the number of distinct points common to the graphs of $x^2+y^2=9$ and $y^2=9$.
Answer: 2
Problem: Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{\left(\frac{n}{2}\right)}$ when $n$ is even and by $t_n=\frac{1}{t_{(n-1)}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$, find the sum of the digits of $n$.
Answer: 15
Problem: A circle centered at $O$ has radius $1$ and contains the point $A$. Segment $AB$ is tangent to the circle at $A$ and $\angle AOB = \theta$. If point $C$ lies on $\overline{OA}$ and $\overline{BC}$ bisects $\angle ABO$, find the value of $OC$. The original answer is in the form \frac{k}{1+m\sin(\theta)}, please give the value of k + m.
Answer: 2
Problem: How many odd positive 3-digit integers are divisible by 3 but do not contain the digit 3? Fill in the number of such integers.
Answer: 96
Problem: A house and a store were each sold for $\$12000$. The house was sold at a loss of $20\%$ of its cost, and the store at a gain of $20\%$ of its cost. Find the entire transaction's net loss or gain in dollars.
Answer: 1000
Problem: Point $F$ is taken on side $AD$ of square $ABCD$. At $C$, a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is $256$ square inches, and the area of triangle $CEF$ is $200$ square inches. Determine the number of inches in $BE$.
Answer: 12
Problem: Let $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1) = 0$, $50 < f(7) < 60$, $70 < f(8) < 80$, and $5000k < f(100) < 5000(k+1)$ for some integer $k$. Find the integer $k$.
Answer: 3
Problem: Points $A$ and $B$ lie on a circle centered at $O$, and $\angle AOB = 60^\circ$. A second circle is internally tangent to the first and tangent to both $\overline{OA}$ and $\overline{OB}$. The ratio of the area of the smaller circle to that of the larger circle is in the form \frac{k}{m}. Please find the value of k + m.
Answer: 10
Problem: Delete all perfect squares from the sequence $1, 2, 3, \ldots$. Determine the 2003rd number in the remaining sequence.
Answer: 2048
Problem: In \(\triangle ABC\), we have \(AB = 1\) and \(AC = 2\). Side \(BC\) and the median from \(A\) to \(BC\) have the same length. The original answer is in the form \(\sqrt{k}\). Please determine the value of \(k\).
Answer: 2
Problem: Given a square $ABCD$ with side length $8$ feet, a circle is drawn through vertices $A$ and $D$ and is tangent to side $BC$. Determine the radius of the circle, in feet. What is the value of the radius?
Answer: 5
Problem: Tom, Dick, and Harry started out on a $100$-mile journey. Tom and Harry traveled by automobile at a speed of $25$ mph, while Dick walked at a speed of $5$ mph. After a certain distance, Harry got off and continued walking at $5$ mph, while Tom went back for Dick and brought him to the destination at the same time that Harry arrived. Find the number of hours required for the trip.
Answer: 8
Problem: If \(b > 1\), \(\sin x > 0\), \(\cos x > 0\), and \(\log_b \sin x = a\), find the value of \(\log_b \cos x\). The original answer is in the form \(\frac{m}{n} \log_b(1-b^{p})\). Please provide the value of \(m + n + p\).
Answer: 5
Problem: In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$?
Answer: 14
Problem: Given that $a_{i} \in \{0,1,2,3,4\}$ for every $0 \le i \le 9$, if $6\sum_{i=0}^{9}a_{i} 5^{i} \equiv 1 \pmod{5^{10}}$, find the value of $a_{9}$.
Answer: 4
Problem: On an algebra quiz, $10\%$ of the students scored $70$ points, $35\%$ scored $80$ points, $30\%$ scored $90$ points, and the rest scored $100$ points. What is the difference between the mean and median score of the students' scores on this quiz?
Answer: 3
Problem: How many ordered pairs $(p,n)$ are there such that $(1+p)^n = 1+pn + n^p$ where $p$ is a prime and $n$ is a positive integer?
Answer: 1
Problem: On Halloween, 31 children walked into the principal's office asking for candy. They can be classified into three types: some always lie, some always tell the truth, and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order:

1. "Are you a truth-teller?" The principal gave a piece of candy to each of the 22 children who answered yes.
2. "Are you an alternater?" The principal gave a piece of candy to each of the 15 children who answered yes.
3. "Are you a liar?" The principal gave a piece of candy to each of the 9 children who answered yes.

How many pieces of candy in all did the principal give to the children who always tell the truth?
Answer: 7
Problem: 在\(\triangle ABC\)中，已知边长\(|AB|=3\)，且从点\(C\)作出的高的长度为\(2\)。求另外两条高的长度乘积的最大值。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
Answer: 169
Problem: If the Highest Common Divisor of $6432$ and $132$ is diminished by $8$, what is the resulting value?
Answer: 4
Problem: How many positive integers not exceeding 2001 are multiples of 3 or 4 but not 5? Provide your answer as an integer.
Answer: 801
Problem: At the end of a professional bowling tournament, the top 5 bowlers have a playoff. First, bowler #5 competes against bowler #4. The loser receives the 5th prize, and the winner competes against bowler #3 in another game. The loser of this game receives the 4th prize, and the winner competes against bowler #2. The loser of this game receives the 3rd prize, and the winner competes against bowler #1. The winner of this game gets the 1st prize, and the loser gets the 2nd prize. In how many orders can bowlers #1 through #5 receive the prizes? Provide your answer as an integer.
Answer: 16
Problem: The student locker numbers at Olympic High are numbered consecutively beginning with locker number $1$. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $9$ and four cents to label locker number $10$. If it costs $137.94$ to label all the lockers, how many lockers are there at the school? Provide your answer as an integer.
Answer: 2001
Problem: 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 answer is $x$, what is the value of $10x$?
Answer: 19765
Problem: In $\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D, E,$ and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. Find the perimeter of parallelogram $ADEF$.
Answer: 56
Problem: Consider two solid spherical balls:

1. **First Ball**:
   - Centered at $(0, 0, \frac{21}{2})$
   - Radius $6$

2. **Second Ball**:
   - Centered at $(0, 0, 1)$
   - Radius $\frac{9}{2}$

Determine the number of lattice points (points with integer coordinates $(x, y, z)$) that lie in the intersection of these two spheres.
Answer: 13
Problem: We define a positive integer as "special" if it is divisible by all of its non-zero digits. Determine the maximum number of consecutive special numbers that can exist.
Answer: 12
Problem: Find the largest integer not exceeding $\lfloor (n+1)a \rfloor - \lfloor na \rfloor$, where $n$ is a natural number and $a=\frac{\sqrt{2013}}{\sqrt{2014}}$.
Answer: 1
Problem: The yearly changes in the population census of a town for four consecutive years are, respectively, $25\%$ increase, $25\%$ increase, $25\%$ decrease, $25\%$ decrease. Find the net change over the four years, to the nearest percent. If the net change is $x\%$, what is the value of $x$?
Answer: -12
Problem: A man makes a trip by automobile at an average speed of $50$ mph. He returns over the same route at an average speed of $45$ mph. What is his average speed for the entire trip in miles per hour? Express your answer as a fraction in the form $\frac{k}{m}$ where $k$ and $m$ are integers. Please provide the value of k + m.
Answer: 919
Problem: If the remainder is $2013$ when a polynomial with coefficients from the set $\{0,1,2,3,4,5\}$ is divided by $x-6$, find the least possible value of the coefficient of $x$ in this polynomial.
Answer: 5
Problem: Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral? The original answer is in the form k\sqrt{m}, please provide the value of k + m.
Answer: 8
Problem: What is the sum of the digits of the first $2010$ positive integers?
Answer: 28068
Problem: A circular disk is divided by $2n$ equally spaced radii ($n > 0$) and one secant line. What is the maximum number of non-overlapping areas into which the disk can be divided? The original answer is in the form $kn + m$. Please find the value of $k + m$.
Answer: 4
Problem: A traffic light runs repeatedly through the following cycle: green for \(30\) seconds, then yellow for \(3\) seconds, and then red for \(30\) seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching? Provide your answer in the form \(\frac{k}{m}\) where the fraction is in simplest form, and find the value of \(k + m\).
Answer: 8
Problem: A data set consists of $6$ (not distinct) positive integers: $1$, $7$, $5$, $2$, $5$, and $X$. The average (arithmetic mean) of the $6$ numbers equals a value in the data set. What is the sum of all positive values of $X$?
Answer: 36
Problem: The numbers $\log(a^3b^7)$, $\log(a^5b^{12})$, and $\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $12^{\text{th}}$ term of the sequence is $\log{b^n}$. What is the value of $n$?
Answer: 112
Problem: The lines \(x = \frac{1}{4}y + a\) and \(y = \frac{1}{4}x + b\) intersect at the point \((1,2)\). The original answer is in \(\frac{k}{m}\) format, please find the value of \(k + m\).
Answer: 13
Problem: The number of distinct pairs \((x,y)\) of real numbers satisfying both of the following equations: 
\[ \begin{align*} x &= x^2 + y^2, \\ y &= 2xy \end{align*} \]
is \(\underline{\phantom{4}}\).
Answer: 4
Problem: How many numbers \( (x_1 x_2 \ldots x_{1998}) \) are there such that:

\[
(x_1 x_2 \ldots x_{1998}) = 7 \cdot 10^{1996} (x_1 + x_2 + \ldots + x_{1998})
\]

where \( (x_1 x_2 \ldots x_{1998}) \) shows a number with 1998 digits in the decimal system? Provide your answer as an integer.
Answer: 4
Problem: Juan and Mary play a two-person game where the winner gains 2 points and the loser loses 1 point. If Juan won exactly 3 games and Mary had a final score of 5 points, find the total number of games they played.
Answer: 7
Problem: Regular octagon \(ABCDEFGH\) has area \(n\). Let \(m\) be the area of quadrilateral \(ACEG\). The answer is in the form \(\frac{\sqrt{k}}{l}\). Find the value of \(k + l\).
Answer: 4
Problem: Find the number of scalene triangles having all sides of integral lengths, and a perimeter less than $13$.
Answer: 3
Problem: If \(10^{2y} = 25\), then find the value of \(10^{-y}\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 6
Problem: $ABCDE$ is a convex pentagon. Given the following measurements: \( m(\widehat{B}) = m(\widehat{D}) = 90^\circ \), \( m(\widehat{C}) = 120^\circ \), \( |AB| = 2 \), \( |BC| = |CD| = \sqrt{3} \), and \( |ED| = 1 \). Find the length of \( |AE| \). The original answer is in the form \(\sqrt{k}\), please give the value of \(k\).
Answer: 3
Problem: Let $D$ be the foot of the internal angle bisector of the angle $A$ of a triangle $ABC$. Let $E$ be a point on side $[AC]$ such that $|CE|= |CD|$ and $|AE|=6\sqrt{5}$; let $F$ be a point on the ray $[AB$ such that $|DB|=|BF|$ and $|AB|<|AF| = 8\sqrt{5}$. The original answer for the length of $|AD|$ is in the form $k\sqrt{m}$. Please find the value of k + m.
Answer: 19
Problem: $A$ and $B$ travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after $B$ has travelled $100$ yards, and meet a second time $60$ yards before $A$ completes one lap, find the circumference of the track in yards.
Answer: 480
Problem: Let $r$ be the speed in miles per hour at which a wheel, 11 feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\frac{1}{4}$ of a second, the speed $r$ is increased by 5 miles per hour. What is the value of $r$?
Answer: 10
Problem: Let $ABC$ be a triangle such that $|BC| = 7$ and $|AB| = 9$. If $m(\widehat{ABC}) = 2m(\widehat{BCA})$, then find the area of the triangle. The original answer is in the form of $m\sqrt{n}$, please give the value of m + n.
Answer: 19
Problem: For how many different values of integer $n$, can one find $n$ different lines in the plane such that each line intersects with exactly $2004$ of the other lines? Provide your answer as an integer.
Answer: 12
Problem: In Cartesian coordinates, two areas \(M\) and \(N\) are defined as follows:  
- \(M: y \geq 0, y \leq x, y \leq 2-x\);  
- \(N: t \leq x \leq t+1\).  
Here, \(t\) is a real number such that \(t \in [0,1]\).  

Determine the area of \(M \cap N\). The original answer is in the form \(-at^2 + bt + \frac{c}{d}\), where \(\frac{c}{d}\) is a simplified fraction. Please give the value of \(a + b + c + d\).
Answer: 5
Problem: The graph of \( y = x^2 + 2x - 15 \) intersects the \( x \)-axis at points \( A \) and \( C \) and the \( y \)-axis at point \( B \). Find the value of \( \tan(\angle ABC) \). The original answer is in \(\frac{m}{n}\) format, please give the value of \(m + n\).
Answer: 11
Problem: How many ordered integer pairs \((a, b)\) satisfy the equation \(a^2b + ab^2 = 2009201020092010\)?
Answer: 0
Problem: Let $P$ be a polynomial such that $(x-4)P(2x) = 4(x-1)P(x)$, for every real $x$. If $P(0) \neq 0$, what is the degree of $P$?
Answer: 2
Problem: Three friends are to divide five different jobs between each other so that nobody is left without a job. In how many different ways can this be done?
Answer: 150
Problem: If the integers $m,n,k$ satisfy the equation $221m + 247n + 323k = 2001$, find the smallest possible value of $k$ greater than $100$.
Answer: 111
Problem: A square's area is equal to the perimeter of a $15 \times 17$ rectangle. What is the value of the square's perimeter?
Answer: 32
Problem: Calculate the difference between the number of hours in $10$ years and the number of seconds in $1$ day.
Answer: 1200
Problem: Let \(S\) be the set of ordered triples \((x,y,z)\) of real numbers for which
\[ \log_{10} (x + y) = z \text{ and } \log_{10} (x^2 + y^2) = z + 1. \]
There are real numbers \(a\) and \(b\) such that for all ordered triples \((x,y,z)\) in \(S\) we have 
\[ x^3 + y^3 = a \cdot 10^{3z} + b \cdot 10^{2z}. \]
The original answer is in \(\frac{m}{n}\) format. Please find the value of \(m + n\).
Answer: 31
Problem: How many positive integers divide $5n^{11} - 2n^5 - 3n$ for all positive integers $n$?
Answer: 12
Problem: It is now between $10:00$ and $11:00$ o'clock, and six minutes from now, the minute hand of the watch will be exactly opposite the place where the hour hand was three minutes ago. What is the exact time now? Express your answer in the format of "HH:MM", where HH is the hour and MM is the minute, both as two-digit numbers. Calculate the sum of HH and MM.
Answer: 25
Problem: The area of a trapezoidal field is $1400$ square yards. Its altitude is $50$ yards. Find the two bases, if the number of yards in each base is an integer divisible by $8$. How many solutions are there to this problem?
Answer: 3
Problem: In the figure, polygons $A$, $E$, and $F$ are isosceles right triangles; $B$, $C$, and $D$ are squares with sides of length $1$; and $G$ is an equilateral triangle. The figure can be folded along its edges to form a polyhedron having the polygons as faces. The volume of this polyhedron is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Answer: 11
Problem: What is the largest integer $n$ that satisfies $(100^2-99^2)(99^2-98^2)\cdots(3^2-2^2)(2^2-1^2)$ is divisible by $3^n$?
Answer: 49
Problem: A square and a circle have equal perimeters. Find the ratio of the area of the circle to the area of the square. The original answer is in \(\frac{k}{m}\) format, where k and m are integers, and the fraction is in its simplest form. Please provide the value of k + m.
Answer: 7
Problem: A small block with an initial speed $v_0$ smoothly moves onto a sloped big block of mass $M$. After the small block reaches a height $h$ on the slope, it slides back down. Find the height $h$ in terms of $v_0$, $g$, $m$, and $M$. The original answer is in the form \frac{k}{m} \cdot \frac{n v_0^2}{p + M}. Please give the value of k + m + n + p.
Answer: 5
Problem: How many $3$-digit positive integers have digits whose product equals $24$?
Answer: 21
Problem: Rectangle \(ABCD\) has \(AB=4\) and \(BC=3\). Segment \(EF\) is constructed through \(B\) so that \(EF\) is perpendicular to \(DB\), and \(A\) and \(C\) lie on \(DE\) and \(DF\), respectively. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 137
Problem: Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from \((c,0)\) to \((3,3)\), divides the entire region into two regions of equal area. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 5
Problem: The positive integers $x$ and $y$ are the two smallest positive integers for which the product of $360$ and $x$ is a square and the product of $360$ and $y$ is a cube. What is the sum of $x$ and $y$?
Answer: 85
Problem: Rectangle \(ABCD\) has \(AB=8\) and \(BC=6\). Point \(M\) is the midpoint of diagonal \(\overline{AC}\), and \(E\) is on \(AB\) with \(\overline{ME}\perp\overline{AC}\). Find the area of \(\triangle AME\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 83
Problem: For $k > 0$, let $I_k = 10\ldots 064$, where there are $k$ zeros between the $1$ and the $6$. Let $N(k)$ be the number of factors of $2$ in the prime factorization of $I_k$. What is the maximum value of $N(k)$?
Answer: 7
Problem: Circle $A$ has radius $100$. Circle $B$ has an integer radius $r<100$ and remains internally tangent to circle $A$ as it rolls once around the circumference of circle $A$. The two circles have the same points of tangency at the beginning and end of circle $B$'s trip. How many possible values can $r$ have?
Answer: 8
Problem: A particular $12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $1$, it mistakenly displays a $9$. For example, when it is $1:16$ PM the clock incorrectly shows $9:96$ PM. Find the fraction of the day during which the clock shows the correct time. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 3
Problem: Two angles of an isosceles triangle measure $70^\circ$ and $x^\circ$. What is the sum of the three possible values of $x$?
Answer: 165
Problem: Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is the value of $AE$?
Answer: 6
Problem: A convex polyhedron $Q$ has vertices $V_1,V_2,\cdots,V_n$, and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\cdots,P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$. In addition, no two planes intersect inside or on $Q$. The cuts produce $n$ pyramids and a new polyhedron $R$. How many edges does $R$ have?
Answer: 300
Problem: What is the remainder when $3^0 + 3^1 + 3^2 + \cdots + 3^{2009}$ is divided by $8$? Fill in the blank with your answer.
Answer: 4
Problem: Let $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are complex numbers. Suppose that  

$$p(2009 + 9002\pi i) = p(2009) = p(9002) = 0$$
Find the number of nonreal zeros of $x^{12} + ax^8 + bx^4 + c$.
Answer: 8
Problem: Andy and Bethany have a rectangular array of numbers with $40$ rows and $75$ columns. Andy adds the numbers in each row. The average of his $40$ sums is $A$. Bethany adds the numbers in each column. The average of her $75$ sums is $B$. Find the value of \(\frac{A}{B}\). The answer should be in the form \(\frac{k}{m}\), where k and m are integers, and please give the value of k + m.
Answer: 23
Problem: Two cubical dice each have removable numbers $1$ through $6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is $7$? Express your answer in the form \(\frac{k}{m}\), where k and m are coprime, and find the value of k + m.
Answer: 13