ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Toothpicks of equal length are used to build a rectangular grid. If the grid is 20 toothpicks high and 10 toothpicks wide, find the total number of toothpicks used.
Answer: 430
Problem: Let $c = \frac{2\pi}{11}.$ What is the value of \[ \frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}? \]
Answer: 1
Problem: Let $D$ be a point on the side $[BC]$ of $\triangle ABC$ such that $|AB| + |BD| = |AC|$ and $m(\widehat{BAD}) = m(\widehat{DAC}) = 30^\circ$. If the answer is $x^\circ$, what is the value of $x$ for $m(\widehat{ACB})$? Please provide the value of $x$ as an integer.
Answer: 40
Problem: My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only $4$ years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha's age and the mean of my grandparents’ ages?
Answer: 2
Problem: The sides of a triangle have lengths $11$, $15$, and $k$, where $k$ is an integer. Find the number of integer values of $k$ for which the triangle is obtuse.
Answer: 13
Problem: A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex \(W\)? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 13
Problem: Given $a_1 = 1$, $a_{n+1} = \frac{a_n}{\sqrt{1 + 4a_n^2}}$ for $n \ge 1$, find the least $k$ such that $a_k < 10^{-2}$.
Answer: 2501
Problem: How many solutions does the equation $\sin \left( \frac{\pi}{2} \cos x\right)=\cos \left( \frac{\pi}{2} \sin x\right)$ have in the closed interval $[0,\pi]$? Provide your answer as an integer.
Answer: 2
Problem: How many different solutions does the congruence $x^3 + 3x^2 + x + 3 \equiv 0 \pmod{25}$ have? Provide your answer as an integer.
Answer: 6
Problem: Four positive integers are given. Select any three of these integers, find their arithmetic average, and add this result to the fourth integer. Thus, the numbers $29$, $23$, $21$, and $17$ are obtained. Find one of the original integers.
Answer: 21
Problem: The minimum value of the quotient of a (base ten) number of three different nonzero digits divided by the sum of its digits is x. If the answer is x, what is the value of 10x?
Answer: 105
Problem: The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.

\[\text{[asy]}\]
\text{size(300);}
\text{defaultpen(linewidth(0.8));}
\text{real r = 0.35;}
\text{path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r);}
\text{path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r);}
\text{for(int i=0;i <= 4;i=i+1)}
\text{{}}
\text{draw(shift((4*i,0)) * P);}
\text{draw(shift((4*i,0)) * Q);}
\text{{}}
\text{for(int i=1;i <= 4;i=i+1)}
\text{{}}
\text{draw(shift((4*i-2,0)) * Pp);}
\text{draw(shift((4*i-1,0)) * Qp);}
\text{{}}
\text{draw((-1,0)--(18.5,0),Arrows(TeXHead));}
\[\text{[/asy]}\]

How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?

- some rotation around a point of line $\ell$
- some translation in the direction parallel to line $\ell$
- the reflection across line $\ell$
- some reflection across a line perpendicular to line $\ell$

Fill in the blank with the number of transformations: \_\_
Answer: 2
Problem: Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $8:00 \text{ AM}$, and all three always take the same amount of time to eat lunch. On Monday, the three of them painted $50\%$ of a house, quitting at $4:00 \text{ PM}$. On Tuesday, when Paula wasn't there, the two helpers painted only $24\%$ of the house and quit at $2:12 \text{ PM}$. On Wednesday, Paula worked by herself and finished the house by working until $7:12 \text{ PM}$. How long, in minutes, was each day's lunch break?
Answer: 48
Problem: The largest prime factor of $16384$ is $2$, because $16384 = 2^{14}$. What is the sum of the digits of the largest prime factor of $16383$?
Answer: 10
Problem: Given the linear fractional transformation of \(x\) into \(f_1(x) = \frac{2x-1}{x+1}\). Define \(f_{n+1}(x) = f_1(f_n(x))\) for \(n=1,2,3,\ldots\). Assuming that \(f_{35}(x) = f_5(x)\), find the expression for \(f_{28}(x)\). The original answer is in the form \(\frac{k}{m-nx}\). Please find the value of k + m + n.
Answer: 3
Problem: The expression \((x + y + z)^{2006} + (x - y - z)^{2006}\) is simplified by expanding it and combining like terms. How many terms are in the simplified expression? Please provide your answer as an integer.
Answer: 1008016
Problem: For a geometric series $(a_n)$ with all terms real, if $S_{10} = 10$ and $S_{30} = 70$, find the value of $S_{40}$.
Answer: 150
Problem: The base-nine representation of the number $N$ is $27,006,000,052_{\text{nine}}$. Find the remainder when $N$ is divided by $5$.
Answer: 3
Problem: A simplified model of a bicycle of mass \(M\) has two tires that each comes into contact with the ground at a point. The wheelbase of this bicycle (the distance between the points of contact with the ground) is \(w\), and the center of mass of the bicycle is located midway between the tires and a height \(h\) above the ground. The bicycle is moving to the right, but slowing down at a constant rate. The acceleration has a magnitude \(a\). Air resistance may be ignored.

Case 1: Assume that the coefficient of sliding friction between each tire and the ground is \(\mu\), and that both tires are skidding: sliding without rotating. Express your answers in terms of \(w\), \(h\), \(M\), and \(g\).

What is the maximum value of \(a\) so that both tires remain in contact with the ground? The original answer is in the form \(\frac{kwg}{mh}\), please find the value of \(k + m\).
Answer: 2
Problem: A massless elastic cord (that obeys Hooke's Law) will break if the tension in the cord exceeds \(T_{max}\). One end of the cord is attached to a fixed point, the other is attached to an object of mass \(3m\). If a second, smaller object of mass \(m\) moving at an initial speed \(v_0\) strikes the larger mass and the two stick together, the cord will stretch and break, but the final kinetic energy of the two masses will be zero. If instead the two collide with a perfectly elastic one-dimensional collision, the cord will still break, and the larger mass will move off with a final speed of \(v_f\). All motion occurs on a horizontal, frictionless surface.

Find \(\frac{v_f}{v_0}\). The answer should be in the form \(\frac{k}{\sqrt{m}}\). Please calculate the value of k + m.
Answer: 7
Problem: Let the bisectors of the exterior angles at \(B\) and \(C\) of triangle \(ABC\) meet at \(D\). Then, if all measurements are in degrees, find the value of angle \(BDC\). The original answer is in the form \(\frac{k}{m}(n - A)\). Please provide the value of k + m + n.
Answer: 183
Problem: The parabola $P$ has its focus at $(0,0)$ and passes through the points $(4,3)$ and $(-4,-3)$. Determine the number of points $(x,y) \in P$ with integer coefficients such that $|4x + 3y| \leq 1000$.
Answer: 40
Problem: At most, how many elements can a set have such that all elements are less than $102$ and the set does not contain the sum of any two of its elements? Provide your answer as an integer.
Answer: 51
Problem: In a bag of marbles, \(\frac{3}{5}\) of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? The original answer is in the form \(\frac{k}{m}\), where the fraction is in simplest form. Please provide the value of \(k + m\).
Answer: 11
Problem: Find the number of noncongruent integer-sided triangles with positive area and perimeter less than $15$ that are neither equilateral, isosceles, nor right triangles.
Answer: 5
Problem: For what value of $k$ does the pair of equations $y = x^2$ and $y = 3x + k$ have two identical solutions? Provide your answer as a fraction in the form $-\frac{m}{n}$, where m and n are integers. Please find the value of m + n.
Answer: 13
Problem: Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool, one at the rate of $3$ feet per second, the other at $2$ feet per second. They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other.
Answer: 20
Problem: The number $2013$ has the property that it includes four consecutive digits ($0$, $1$, $2$, and $3$). Find how many $4$-digit numbers include $4$ consecutive digits.
Answer: 162
Problem: If the sum of digits of only $m$ and $m+n$ from the numbers $m$, $m+1$, \ldots, $m+n$ are divisible by $8$ where $m$ and $n$ are positive integers, what is the largest possible value of $n$?
Answer: 15
Problem: In triangle $ABC$, $D$ is the midpoint of $AB$; $E$ is the midpoint of $DB$; and $F$ is the midpoint of $BC$. If the area of $\triangle ABC$ is $96$, then what is the area of $\triangle AEF$?
Answer: 36
Problem: Let $\overline{AB}$ be a diameter in a circle of radius $5\sqrt{2}$. Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at a point $E$ such that $BE=2\sqrt{5}$ and $\angle AEC = 45^{\circ}$. Find the value of $CE^2+DE^2$.
Answer: 100
Problem: Find the product of uncommon real roots of the two polynomials $x^4 + 2x^3 - 8x^2 - 6x + 15$ and $x^3 + 4x^2 - x - 10$.
Answer: 6
Problem: Find the sum of the $2$ smallest prime factors of $2^{1024} - 1$.
Answer: 8
Problem: What is the least positive integer $k$ such that $n+k \in S$ for every $n \in S$, where $S = \{ n : n3^n + (2n+1)5^n \equiv 0 \pmod{7} \}$?
Answer: 21
Problem: If $x_1=5, x_2=401$, and \[ x_n=x_{n-2}-\frac{1}{x_{n-1}} \] for every $3\leq n \leq m$, find the largest value of $m$.
Answer: 2007
Problem: A sequence of complex numbers $z_0, z_1, z_2, \ldots$ is defined by the rule \[ z_{n+1} = \frac{i z_n}{\overline{z_n}} \] where $\overline{z_n}$ is the complex conjugate of $z_n$ and $i^2 = -1$. Suppose that $|z_0| = 1$ and $z_{2005} = 1$. Find the number of possible values for $z_0$. The original answer is in the form of $x * 2^{y}$, please give the value of $x + y$.
Answer: 2006
Problem: Let $N$ be the second smallest positive integer that is divisible by every positive integer less than 7. What is the sum of the digits of $N$?
Answer: 3
Problem: A painting $18'' \times 24''$ is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, find the ratio of the smaller to the larger dimension of the framed painting in the form $a:b$. Then, calculate the value of $a + b$.
Answer: 5
Problem: A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are \((3,17)\) and \((48,281)\)? Include both endpoints of the segment in your count.
Answer: 4
Problem: Two men at points $R$ and $S$, 76 miles apart, set out at the same time to walk towards each other. The man at $R$ walks uniformly at the rate of $\frac{9}{2}$ miles per hour; the man at $S$ walks at the constant rate of $\frac{13}{4}$ miles per hour for the first hour, $\frac{15}{4}$ miles per hour for the second hour, and so on, in arithmetic progression. If the men meet $x$ miles nearer $R$ than $S$ in an integral number of hours, find the value of $x$.
Answer: 4
Problem: Calculate the product \(\frac{1}{3} \times \frac{2}{4} \times \frac{3}{5} \times \cdots \times \frac{18}{20} \times \frac{19}{21}\). What is the value of this product in its simplest form? The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 211
Problem: The parallelogram bounded by the lines $y = ax + c$, $y = ax + d$, $y = bx + c$, and $y = bx + d$ has an area of 18. The parallelogram bounded by the lines $y = ax + c$, $y = ax - d$, $y = bx + c$, and $y = bx - d$ has an area of 72. Given that $a$, $b$, $c$, and $d$ are positive integers, find the smallest possible value of $a + b + c + d$.
Answer: 16
Problem: Let $D$ be a point on side $[AB]$ of triangle $ABC$ with $|AB| = |AC|$, such that $[CD]$ is an angle bisector and $m(\widehat{ABC}) = 40^\circ$. Let $F$ be a point on the extension of $[AB]$ beyond $B$ such that $|BC| = |AF|$. Let $E$ be the midpoint of $[CF]$. If $G$ is the intersection of lines $ED$ and $AC$, what is the value of $x$ if $m(\widehat{FBG}) = x^\circ$?
Answer: 150
Problem: Find the number of prime numbers $p$ such that $2p^4 - 7p^2 + 1$ is equal to the square of an integer.
Answer: 1
Problem: If the perimeter of a rectangle is \(p\) and its diagonal is \(d\), find the difference between the length and width of the rectangle. The original answer is in the form \(\frac{\sqrt{n \cdot d^2 - p^2}}{m}\). Please calculate the value of \(k + m + n\).
Answer: 11
Problem: A list of integers has mode $32$ and mean $22$. The smallest number in the list is $10$. The median $m$ of the list is a member of the list. If the list member $m$ were replaced by $m + 10$, the mean and median of the new list would be $24$ and $m + 10$, respectively. If $m$ were instead replaced by $m - 8$, the median of the new list would be $m - 4$. What is the value of $m$?
Answer: 20
Problem: Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ such that $\overline{DE} \perp \overline{BC}$. Find the ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 4
Problem: How many functions $f : A \rightarrow A$ are there satisfying $f(f(a)) = a$ for every $a \in A=\{1,2,3,4,5,6,7\}$?
Answer: 232
Problem: The two-digit integers from $19$ to $92$ are written consecutively to form the larger integer $N = 19202122\ldots909192$. If $3^{k}$ is the highest power of $3$ that is a factor of $N$, what is the value of $k$?
Answer: 1
Problem: In order to pass $B$ going $40$ mph on a two-lane highway, $A$, going $50$ mph, must gain $30$ feet. Meanwhile, $C$, who is $210$ feet from $A$, is headed toward him at $50$ mph. If $B$ and $C$ maintain their speeds, then, in order to pass safely, $A$ must increase his speed by how many mph?
Answer: 5
Problem: The probability that event $A$ occurs is \(\frac{3}{4}\); the probability that event $B$ occurs is \(\frac{2}{3}\). Let \(p\) be the probability that both \(A\) and \(B\) occur. Find the smallest interval necessarily containing \(p\) and express it in the form \(\left[\frac{x}{y}, \frac{c}{d}\right]\), where \(x, y, c,\) and \(d\) are integers. Please give the value of \(x + y + c + d\).
Answer: 22
Problem: A truck is initially moving at velocity \(v\). The driver presses the brake to slow the truck to a stop. The brake applies a constant force \(F\) to the truck. The truck rolls a distance \(x\) before coming to a stop, and the time it takes to stop is \(t\).

Find the expression that represents the initial momentum of the truck (i.e., the momentum before the driver starts braking) in terms of \(F\), \(x\), and \(v\). The original answer is in the form \(\frac{kF \cdot x}{mv}\). Please give the value of \(k + m\).
Answer: 3
Problem: How many ways are there to partition $7$ students into the groups of $2$ or $3$? Provide your answer as an integer.
Answer: 105
Problem: Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. Find the probability that the four chords are the sides of a convex quadrilateral. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 92
Problem: Asli will distribute $100$ candies among her brother and $18$ friends of his. Asli splits the friends of her brother into several groups and distributes all the candies into these groups. In each group, the candies are shared in a fair way such that each child in a group takes the same number of candies, and this number is the largest possible. Then, Asli's brother takes the remaining candies of each group. At most, how many candies can Asli's brother have?
Answer: 16
Problem: What is the largest real number $C$ that satisfies the inequality $x^2 \geq C \lfloor x \rfloor (x-\lfloor x \rfloor)$ for every real $x$?
Answer: 4
Problem: The sum of the largest number and the smallest number of a triple of positive integers $(x,y,z)$ is called the power of the triple. Find the sum of powers of all triples $(x,y,z)$ where $x,y,z \leq 9$.
Answer: 7290
Problem: Three mutually tangent spheres of radius \(1\) rest on a horizontal plane. A sphere of radius \(2\) rests on them. What is the distance from the plane to the top of the larger sphere? Express your answer in the form \(3 + \frac{\sqrt{n}}{m}\) where \(n\) and \(m\) are integers. Please give the value of \(n + m\).
Answer: 72
Problem: Find the integer that divides $3^{3n+1} + 5^{3n+2} + 7^{3n+3}$ for every positive integer $n$.
Answer: 7
Problem: Find the radius of the inscribed circle of a triangle with sides of length \(13\), \(30\), and \(37\). Provide your answer in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest form. Calculate the value of \(k + m\).
Answer: 11
Problem: In a geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. What is the common ratio? Provide your answer in the form of \(\frac{\sqrt{k} - m}{n}\) and find the value of \(k + m + n\).
Answer: 8
Problem: Call a number "prime-looking" if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime-looking numbers are there less than 1000? Provide your answer as an integer.
Answer: 100
Problem: Calculate the value of \(\cot 10 + \tan 5\) and express your answer in terms of a trigonometric function. If the answer is \(\csc x\), what is the value of \(x\)?
Answer: 10
Problem: A positive number is mistakenly divided by $6$ instead of being multiplied by $6$. Calculate the error committed, and provide the answer to the nearest percent.
Answer: 97
Problem: Find the sum of all prime numbers between $1$ and $100$ that are simultaneously $1$ greater than a multiple of $4$ and $1$ less than a multiple of $5$. What is the sum?
Answer: 118
Problem: Let $P$ equal the product of $3,659,893,456,789,325,678$ and $342,973,489,379,256$. Find the number of digits in $P$.
Answer: 34
Problem: Find the number of real values of $x$ satisfying the equation $2^{2x^2 - 7x + 5} = 1$.
Answer: 2
Problem: The arithmetic mean of the nine numbers in the set $\{9, 99, 999, 9999, \ldots, 999999999\}$ is a 9-digit number $M$, all of whose digits are distinct. Find the digit that is not present in the number $M$.
Answer: 0
Problem: Let $u$ be a positive number. Consider the set $S$ of all points whose rectangular coordinates $(x, y)$ satisfy all of the following conditions:

1. $\frac{a}{2} \le x \le 2a$
2. $\frac{a}{2} \le y \le 2a$
3. $x + y \ge a$
4. $x + a \ge y$
5. $y + a \ge x$

The boundary of set $S$ is a polygon with how many sides?
Answer: 6
Problem: A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13, 19, 20, 25$, and $31$, although this is not necessarily their order around the pentagon. What is the area of the pentagon?
Answer: 745
Problem: A charity sells 140 benefit tickets for a total of \$2001. Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full-price tickets? If the answer is \$x, what is the value of x?
Answer: 782
Problem: Let \(ABC\) be a triangle such that \(|AB| = 7\), \(|BC| = 8\), \(|AC| = 6\). Let \(D\) be the midpoint of side \([BC]\). If the circle through \(A\), \(B\), and \(D\) cuts \(AC\) at \(A\) and \(E\), find the length of \(|AE|\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 5
Problem: The numbers \(1, 2, \ldots, 9\) are arranged so that the \(1\)st term is not \(1\) and the \(9\)th term is not \(9\). Find the probability that the third term is \(3\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 442
Problem: Find the odd integer that contains the digit 5, is divisible by 3, and lies between $12^2$ and $13^2$.
Answer: 165
Problem: The function $f$ satisfies the functional equation \[ f(x) + f(y) = f(x + y) - xy - 1 \] for every pair $x, y$ of real numbers. If $f(1) = 1$, find the number of integers $n \neq 1$ for which $f(n) = n$.
Answer: 1
Problem: How many integers $n$ satisfy $|n| < 2020$ and the equation $11^3 \mid n^3 + 3n^2 - 107n + 1$? Provide your answer as an integer.
Answer: 0
Problem: Let $m = (abab)$ and $n = (cdcd)$ be four-digit numbers in the decimal system. If $m + n$ is a perfect square, find the largest value of $a \cdot b \cdot c \cdot d$.
Answer: 600
Problem: The number $10!$ (where $10$ is written in base $10$), when converted to the base $12$ system, ends in exactly $k$ zeroes. What is the value of $k$?
Answer: 4
Problem: Suppose that $m$ and $n$ are positive integers such that $m < n$, the geometric mean of $m$ and $n$ is greater than $2007$, and the arithmetic mean of $m$ and $n$ is less than $2007$. Find the number of pairs $(m, n)$ that satisfy these conditions.
Answer: 0
Problem: Eight points are equally spaced around a circle of radius $r$. If we draw a circle of radius $1$ centered at each of the eight points, then each of these circles will be tangent to two of the other eight circles that are next to it. If $r^2 = a + b\sqrt{2}$, where $a$ and $b$ are integers, find the value of $a + b$.
Answer: 6
Problem: How many different prime numbers are factors of $N$ if 
\[ \log_2 (\log_3 (\log_5 (\log_7 N))) = 11? \]
Answer: 1
Problem: Real numbers \(x\) and \(y\) are chosen independently and uniformly at random from the interval \((0,1)\). Find the probability that \(\lfloor \log_2{x} \rfloor = \lfloor \log_2{y} \rfloor\), where \(\lfloor r \rfloor\) denotes the greatest integer less than or equal to the real number \(r\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 4
Problem: Charles has $5q + 1$ quarters and Richard has $q + 5$ quarters. Find the difference in their money in dimes. The answer should be in the format $k(q - m)$. Please calculate the value of $k + m$.
Answer: 11
Problem: On a map with a scale of \( 400 \) miles to an inch and a half, a certain estate is represented by a rhombus with a \( 60^{\circ} \) angle. The diagonal opposite the \( 60^{\circ} \) angle is \( \frac{3}{16} \) inches. Calculate the area of the estate in square miles. The original answer is in the form of \( k\sqrt{m} \). Please provide the value of \( k + m \).
Answer: 1253
Problem: What is the maximum area of a triangle that can be inscribed in an ellipse with semi-axes $a$ and $b$? Express your answer in terms of $a$ and $b$. The original answer is in the form $ab\frac{k\sqrt{3}}{m}$, where $k$ and $m$ are integers. Please give the value of $k + m$.
Answer: 7
Problem: Lori makes a list of all the numbers between $1$ and $999$ inclusive. She first colors all the multiples of $5$ red. Then she colors blue every number which is adjacent to a red number. How many numbers in her list are left uncolored?
Answer: 402
Problem: How many integer triples $(x,y,z)$ are there such that:
\[
\begin{array}{rcl}
 x - yz^2 & \equiv & 1 \pmod{13} \\
 xz + y & \equiv & 4 \pmod{13} \\
\end{array}
\]
where $0 \leq x < 13$, $0 \leq y < 13$, and $0 \leq z < 13$? Provide your answer as an integer.
Answer: 23
Problem: In a right triangle $ABC$, let $D$ be the midpoint of the hypotenuse $AB$. The circumradius of the triangle is $\frac{5}{2}$ and $|BC| = 3$. Find the distance between the circumcenter of $\triangle ACD$ and the incenter of $\triangle BCD$. The answer is in the form \(\frac{k\sqrt{m}}{n}\), please provide the value of \(k + m + n\).
Answer: 51