ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Find the least positive integer which is a multiple of $13$ and all its digits are the same.
Answer: 111111
Problem: For a positive real number $x$, let $[x]$ denote its integer part. For example, $[2.711] = 2$, $[7] = 7$, $[6.9] = 6$. 

Find the maximum real number $z$ such that $\left[\frac{5}{z}\right] + \left[\frac{6}{z}\right] = 7$. 

Calculate the value of $20z$.
Answer: 30
Problem: A triangle $ABC$ is given where $\angle BAC = 40^\circ$ and $\angle ABC = 20^\circ$. Find the length of the angle bisector drawn from vertex $C$, given that the sides $AB$ and $BC$ differ by $4$ centimeters.
Answer: 4
Problem: Using the digits $1, 2, 3, 4, 5, 6$ without repetition, form three two-digit numbers. Calculate the number of different sums that can be obtained when these numbers are added together.
Answer: 10
Problem: Let $f(n)$ be the number of times you have to hit the $\sqrt{}$ key on a calculator to get a number less than $2$ starting from $n$. For instance, $f(2) = 1$, $f(5) = 2$. For how many $1 < m < 2008$ is $f(m)$ odd?
Answer: 242
Problem: Find the number of distinct values obtained from the expression:

$$ \frac{n^2 - 2}{n^2 - n + 2} $$

where \( n \in \{1, 2, 3, \ldots, 100\} \).
Answer: 98
Problem: In the following alpha-numeric puzzle, each letter represents a different non-zero digit. What are all possible values for $b+e+h$?

$$ \begin{array}{cccc}
 & a & b & c \\
 & d & e & f \\
+ & g & h & i \\
\hline
1 & 6 & 6 & 5 \\
\end{array} $$
Answer: 15
Problem: The figure below was created by joining together 5 non-overlapping congruent squares. The total area of the figure is 45. Determine the perimeter of the figure.
Answer: 36
Problem: Compute the sum of all positive integers $a \leq 26$ for which there exist integers $b$ and $c$ such that $a + 23b + 15c - 2$ and $2a + 5b + 14c - 8$ are both multiples of $26$.
Answer: 31
Problem: Let $P(x) = x^3 + 8x^2 - x + 3$ and let the roots of $P$ be $a, b,$ and $c.$ The roots of a monic polynomial $Q(x)$ are $ab - c^2, ac - b^2, bc - a^2.$ Find $Q(-1).$
Answer: 1536
Problem: If 
$$\sum_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = a + \frac {b}{c}$$
where $a, b, c \in \mathbb{N}, b < c, \gcd(b,c) = 1,$ then what is the value of $a + b?$
Answer: 80
Problem: Let $x_0, x_1, \ldots$ be a sequence of real numbers such that \(x_n = \frac{1 + x_{n -1}}{x_{n - 2}}\) for \(n \geq 2\). Find the number of ordered pairs of positive integers \((x_0, x_1)\) such that the sequence gives \(x_{2018} = \frac{1}{1000}\).
Answer: 128
Problem: Today is Barbara's birthday, and Alberto wants to give her a gift by playing the following game. The numbers $0, 1, 2, \ldots, 1024$ are written on a blackboard. 

1. First, Barbara erases $2^{9}$ numbers.
2. Then, Alberto erases $2^{8}$ numbers.
3. Next, Barbara erases $2^{7}$ numbers.
4. This process continues until there are only two numbers $a$ and $b$ left.

Barbara earns $|a-b|$ euros. 

Find the maximum number of euros that Barbara can always win, regardless of Alberto's strategy.
Answer: 32
Problem: Let $P(x)$ be the unique polynomial of degree four for which $P(165) = 20$, and \[ P(42) = P(69) = P(96) = P(123) = 13. \] Compute $P(1) - P(2) + P(3) - P(4) + \dots + P(165)$.
Answer: 20
Problem: Natural numbers $a \leq b \leq c \leq d$ satisfy $a + b + c + d = 30$. Find the maximum value of the product $P = abcd$.
Answer: 3136
Problem: For positive real numbers $x$ and $y$, define their special mean to be the average of their arithmetic and geometric means. Find the total number of pairs of integers $(x, y)$, with $x \le y$, from the set of numbers $\{1,2,\ldots,2016\}$, such that the special mean of $x$ and $y$ is a perfect square.
Answer: 506
Problem: Determine the maximal size of a set of positive integers with the following properties:

1. The integers consist of digits from the set $\{ 1,2,3,4,5,6\}$.

2. No digit occurs more than once in the same integer.

3. The digits in each integer are in increasing order.

4. Any two integers have at least one digit in common (possibly at different positions).

5. There is no digit which appears in all the integers.
Answer: 32
Problem: The polynomial $f(x) = x^3 + rx^2 + sx + t$ has $r$, $s$, and $t$ as its roots (with multiplicity), where $f(1)$ is rational and $t \neq 0$. Compute $|f(0)|$.
Answer: 1
Problem: Positive integers $a$ and $b$ satisfy the equations $a^3 + 32b + 2c = 2018$ and $b^3 + 32a + 2c = 1115$. Find $a^2 + b^2 + c^2$.
Answer: 226
Problem: On a blackboard, a stranger writes the values of $s_7(n)^2$ for $n=0,1,\ldots,7^{20}-1$, where $s_7(n)$ denotes the sum of digits of $n$ in base $7$. Compute the average value of all the numbers on the board.
Answer: 3680
Problem: Let $a, b, c, x, y,$ and $z$ be complex numbers such that:
\[ a = \frac{b+c}{x-2}, \quad b = \frac{c+a}{y-2}, \quad c = \frac{a+b}{z-2}. \]
Given that:
\[ xy + yz + zx = 1000 \quad \text{and} \quad x + y + z = 2016, \]
find the value of $xyz$.
Answer: -4044
Problem: On a board, there are integers written from $1$ to $119$. Two players, $A$ and $B$, take turns making a move. A move consists of erasing $9$ numbers from the board. The player after whose move two numbers remain on the board wins, and their score is equal to the positive difference between the two remaining numbers. Player $A$ makes the first move. Find the highest integer $k$ such that player $A$ can ensure that their score is not smaller than $k$.
Answer: 64
Problem: Five rays $\overrightarrow{OA}$, $\overrightarrow{OB}$, $\overrightarrow{OC}$, $\overrightarrow{OD}$, and $\overrightarrow{OE}$ radiate in a clockwise order from $O$ forming four non-overlapping angles such that:
- $\angle EOD = 2\angle COB$,
- $\angle COB = 2\angle BOA$,
- $\angle DOC = 3\angle BOA$.

If $E$, $O$, $A$ are collinear with $O$ between $A$ and $E$, what is the degree measure of $\angle DOB$?
Answer: 90
Problem: Andrew rolls two fair six-sided dice, each numbered from $1$ to $6$. Brian rolls one fair twelve-sided die, numbered from $1$ to $12$. The probability that the sum of the numbers obtained from Andrew's two rolls is less than the number obtained from Brian's roll can be expressed as a common fraction in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 17
Problem: In triangle $ABC$, where $AB = 52$, $BC = 56$, and $CA = 60$, 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 the length of $DE$.
Answer: 6
Problem: PUMaCDonalds, a newly-opened fast food restaurant, has 5 menu items. If the first 4 customers each choose one menu item at random, what is the probability that the 4th customer orders a previously unordered item? The probability is given by \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m+n \).
Answer: 189
Problem: We call a path Valid if:

1. It only comprises the following types of steps:
   - $(x, y) \rightarrow (x + 1, y + 1)$
   - $(x, y) \rightarrow (x + 1, y - 1)$

2. It never goes below the x-axis.

Let $M(n)$ be the set of all valid paths from $(0,0)$ to $(2n,0)$, where $n$ is a natural number. 

Consider a Valid path $T \in M(n)$. Denote:

$$\phi(T) = \prod_{i=1}^{2n} \mu_i$$

where $\mu_i$ is defined as:

- $1$, if the $i^{th}$ step is $(x, y) \rightarrow (x + 1, y + 1)$
- $y$, if the $i^{th}$ step is $(x, y) \rightarrow (x + 1, y - 1)$

Now, let $f(n) = \sum_{T \in M(n)} \phi(T)$. Evaluate the number of zeroes at the end of the decimal expansion of $f(2021)$.
Answer: 0
Problem: David, Kevin, and Michael each choose an integer from the set $\{1, 2, \ldots, 100\}$ randomly, uniformly, and independently of each other. The probability that the positive difference between David's and Kevin's numbers is \emph{strictly} less than that of Kevin's and Michael's numbers is $\frac{m}{n}$, for coprime positive integers $m$ and $n$. Find $100m + n$.
Answer: 1005100
Problem: A substring of a number $n$ is a number formed by removing some digits from the beginning and end of $n$ (possibly a different number of digits is removed from each side). Find the sum of all prime numbers $p$ such that any substring of $p$ is also prime.
Answer: 576
Problem: Three roots of the quartic polynomial $f(x)=x^4 + ax^3 + bx + c$ are $-1$, $3$, and $5$. What is $a + b - c$?
Answer: 1
Problem: Mr. Ambulando is at the intersection of $5^{\text{th}}$ and $\text{A St}$, and needs to walk to the intersection of $1^{\text{st}}$ and $\text{F St}$. There's an accident at the intersection of $4^{\text{th}}$ and $\text{B St}$, which he'd like to avoid. 

Given that Mr. Ambulando wants to walk the shortest distance possible, how many different routes through downtown can he take?
Answer: 56
Problem: Find the largest positive integer $k$ such that $\phi(\sigma(2^k)) = 2^k$. Here, $\phi(n)$ denotes the number of positive integers that are smaller than $n$ and relatively prime to $n$, and $\sigma(n)$ denotes the sum of divisors of $n$. You are given the hint that $641 \mid 2^{32} + 1$.
Answer: 31
Problem: There exists a unique pair of polynomials $P(x)$ and $Q(x)$ such that:
\[ \begin{align*} P(Q(x)) &= P(x)(x^2-6x+7) \\ Q(P(x)) &= Q(x)(x^2-3x-2) \end{align*} \]
Compute $P(10) + Q(-10)$.
Answer: -90
Problem: Let $S$ be the set of natural numbers whose digits are different and belong to the set $\{1, 3, 5, 7\}$. Calculate the sum of the elements of $S$.
Answer: 117856
Problem: How many ways are there to choose three digits $A, B, C$ with $1 \leq A \leq 9$ and $0 \leq B, C \leq 9$ such that $\overline{ABC}_b$ is even for all choices of base $b$ with $b \geq 10$?
Answer: 225
Problem: Let $X_0$ be the interior of a triangle with side lengths $3$, $4$, and $5$. For all positive integers $n$, define $X_n$ to be the set of points within $1$ unit of some point in $X_{n-1}$. The area of the region outside $X_{20}$ but inside $X_{21}$ can be written as $a\pi + b$, for integers $a$ and $b$. Compute $100a + b$.
Answer: 4112
Problem: Let $N$ be the number of ways of distributing $52$ identical balls into $4$ distinguishable boxes such that no box is empty and the difference between the number of balls in any two of the boxes is not a multiple of $6$. If $N = 100a + b$, where $a, b$ are positive integers less than $100$, find $a + b$.
Answer: 81
Problem: Let $ABCD$ be a cyclic quadrilateral, with $AB = 7$, $BC = 11$, $CD = 13$, and $DA = 17$. Let the incircle of $\triangle ABD$ touch $BD$ at $R$ and the incircle of $\triangle CBD$ touch $BD$ at $S$. What is $RS$?
Answer: 4
Problem: Let $a_1, a_2, \ldots$ be an infinite sequence of integers such that $a_i$ divides $a_{i+1}$ for all $i \geq 1$. Let $b_i$ be the remainder when $a_i$ is divided by $210$. What is the maximal number of distinct terms in the sequence $b_1, b_2, \ldots$?
Answer: 16
Problem: Let $a$, $b$, $c$, and $d$ be real numbers such that \[ a^2 + 3b^2 + \frac{c^2 + 3d^2}{2} = a + b + c + d - 1. \] Find $1000a + 100b + 10c + d$.
Answer: 527
Problem: An up-right path from lattice points $P$ and $Q$ on the $xy$-plane is a path in which every move is either one unit right or one unit up. The probability that a randomly chosen up-right path from $(0,0)$ to $(10,3)$ does not intersect the graph of $y=x^2+0.5$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 248
Problem: A triangle $\triangle ABC$ is situated on the plane and a point $E$ is given on segment $AC$. Let $D$ be a point in the plane such that lines $AD$ and $BE$ are parallel. Suppose that $\angle EBC = 25^\circ$, $\angle BCA = 32^\circ$, and $\angle CAB = 60^\circ$. Find the smallest possible value of $\angle DAB$ in degrees.
Answer: 63
Problem: Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to $2$ and the sum of its terms is $200$. How many progressions does Omar's list have?
Answer: 6
Problem: Find the smallest prime number $p$ that cannot be represented in the form $|3^{a} - 2^{b}|$, where $a$ and $b$ are non-negative integers.
Answer: 41
Problem: The license plate of each automobile in Iran consists of a two-digit and a three-digit number, as well as a letter from the Persian alphabet. The digit $0$ is not used in these numbers. To each license plate, we assign the product of the two numbers on it. For example, if the two numbers are $12$ and $365$ on a license plate, the assigned number would be $12 \times 365 = 4380$. What is the average of all the assigned numbers to all possible license plates?
Answer: 30525
Problem: Colin has $900$ Choco Pies. He realizes that for some integer values of $n \le 900$, if he eats $n$ pies a day, he will be able to eat the same number of pies every day until he runs out. How many possible values of $n$ are there?
Answer: 27
Problem: Find the sum of all positive integers $n$ for which $|2^n + 5^n - 65|$ is a perfect square.
Answer: 6
Problem: Let $A = (2, 0)$, $B = (0, 2)$, $C = (-2, 0)$, and $D = (0, -2)$. Compute the greatest possible value of the product $PA \cdot PB \cdot PC \cdot PD$, where $P$ is a point on the circle $x^2 + y^2 = 9$.
Answer: 97
Problem: Among all polynomials $P(x)$ with integer coefficients for which $P(-10) = 145$ and $P(9) = 164$, compute the smallest possible value of $|P(0)|.$
Answer: 25
Problem: Julius has a set of five positive integers whose mean is 100. If Julius removes the median of the set, the mean of the remaining four numbers increases by 5, and the median decreases by 5. Find the maximum possible value of the largest of the five numbers.
Answer: 269
Problem: Consider all the positive integers $N$ with the property that all of the divisors of $N$ can be written as $p-2$ for some prime number $p$. Then, there exists an integer $m$ such that $m$ is the maximum possible number of divisors of all numbers $N$ with such property. Find the sum of all possible values of $N$ such that $N$ has $m$ divisors.
Answer: 135
Problem: Circle $\omega_1$ and $\omega_2$ have centers $(0,6)$ and $(20,0)$, respectively. Both circles have radius $30$, and intersect at two points $X$ and $Y$. The line through $X$ and $Y$ can be written in the form $y = mx+b$. Compute $100m+b$.
Answer: 303
Problem: We are given a series of similar triangles with areas $1^2, 3^2, 5^2, \ldots, 49^2$. If the smallest triangle has a perimeter of 4, what is the sum of the perimeters of all the triangles?
Answer: 2500
Problem: Let $\triangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. The altitude from $A$ intersects $BC$ at $D$.

Let $\omega_1$ and $\omega_2$ be the incircles of $\triangle ABD$ and $\triangle ACD$, respectively. The common external tangent of $\omega_1$ and $\omega_2$ (other than $BC$) intersects $AD$ at $E$. Compute the length of $AE$.
Answer: 7
Problem: Determine the number of rational numbers whose decimal expansion is of the form $0.\overline{abcdef}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are digits. Each of the digits $2$, $0$, $1$, and $5$ must appear at least once in the sequence $abcdef$.
Answer: 23160
Problem: Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly 1987 zeros.
Answer: 7960
Problem: If $n$ is a positive even number, find the last two digits of $(2^{6n} + 26) - (6^{2n} - 62)$.
Answer: 88
Problem: The positive real numbers $a, b, c$ satisfy the following equations:

$$\frac{a}{2b+1} + \frac{2b}{3c+1} + \frac{3c}{a+1} = 1$$

$$\frac{1}{a+1} + \frac{1}{2b+1} + \frac{1}{3c+1} = 2$$

What is the value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$?
Answer: 12
Problem: 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
Problem: The diagram below shows $\triangle ABC$ with area $64$, where $D, E$, and $F$ are the midpoints of $BC, CA$, and $AB$, respectively. Point $G$ is the intersection of $DF$ and $BE$. Find the area of quadrilateral $AFGE$.
Answer: 24
Problem: Find the smallest value of the expression $|253^m - 40^n|$ over all pairs of positive integers $(m, n)$.
Answer: 9
Problem: Triangle $ABC$ has side lengths $AB=231$, $BC=160$, and $AC=281$. Point $D$ is constructed on the opposite side of line $AC$ as point $B$ such that $AD=178$ and $CD=153$. Compute the distance from $B$ to the midpoint of segment $AD$.
Answer: 208
Problem: Let $f(x) = \sin(\sin(x))$. Evaluate:
\[
\lim_{h \to 0} \frac{f(x+h) - f(h)}{x}
\] at $x = \pi$.
Answer: 0
Problem: Find the sum of the $2007$ roots of 
\[(x-1)^{2007} + 2(x-2)^{2006} + 3(x-3)^{2005} + \cdots + 2006(x-2006)^2 + 2007(x-2007).\]
Answer: 2005
Problem: Let $n$ be an integer greater than 1. If all digits of $97n$ are odd, find the smallest possible value of $n$.
Answer: 35
Problem: A prime number $q$ is called a 'Kowai' number if $q = p^2 + 10$, where $q$, $p$, $p^2-2$, $p^2-8$, and $p^3+6$ are all prime numbers. We know that at least one 'Kowai' number can be found. Find the summation of all 'Kowai' numbers.
Answer: 59
Problem: 1296 CMU students sit in a circle. Every pair of adjacent students rolls a standard six-sided die, and the 'score' of any individual student is the sum of their two dice rolls. A 'matched pair' of students is an (unordered) pair of distinct students with the same score. What is the expected value of the number of matched pairs of students?
Answer: 94605
Problem: What is the remainder of the number $1 \binom{2008}{0} + 2 \binom{2008}{1} + \ldots + 2009 \binom{2008}{2008}$ when divided by $2008$?
Answer: 256
Problem: An ordered pair $(n,p)$ is defined as "juicy" if the following conditions are satisfied for a positive integer $n$ and an odd prime $p$:

1. $n^{2} \equiv 1 \pmod{p^{2}}$
2. $n \equiv -1 \pmod{p}$

Determine the number of such "juicy" pairs $(n,p)$ where $n, p \leq 200$.
Answer: 36
Problem: Let $LOVER$ be a convex pentagon such that $LOVE$ is a rectangle. Given that $OV = 20$ and $LO = VE = RE = RL = 23$, compute the radius of the circle passing through $R$, $O$, and $V$.
Answer: 23
Problem: Suppose two circles $\Omega_1$ and $\Omega_2$ with centers $O_1$ and $O_2$ have radii $3$ and $4$, respectively. Suppose that points $A$ and $B$ lie on circles $\Omega_1$ and $\Omega_2$, respectively, such that segments $AB$ and $O_1O_2$ intersect and that $AB$ is tangent to $\Omega_1$ and $\Omega_2$. If $O_1O_2=25$, find the area of quadrilateral $O_1AO_2B$.
Answer: 84
Problem: Let $k = 2^6 \cdot 3^5 \cdot 5^2 \cdot 7^3 \cdot 53$. Let $S$ be the sum of \(\frac{\text{gcd}(m,n)}{\text{lcm}(m,n)}\) over all ordered pairs of positive integers \((m, n)\) where \(mn = k\). If $S$ can be written in simplest form as \(\frac{r}{s}\), compute $r + s$.
Answer: 895
Problem: The natural number $n > 1$ is called "heavy" if it is coprime with the sum of its divisors. What is the maximal number of consecutive "heavy" numbers?
Answer: 4
Problem: Let $ABCD$ be a square, and let $l$ be a line passing through the midpoint of segment $AB$ that intersects segment $BC$. Given that the distances from $A$ and $C$ to $l$ are $4$ and $7$, respectively, compute the area of $ABCD$.
Answer: 185
Problem: We say that a polynomial $p$ is respectful if $\forall x, y \in \mathbb{Z}$, $y - x$ divides $p(y) - p(x)$, and $\forall x \in \mathbb{Z}$, $p(x)  \in \mathbb{Z}$. We say that a respectful polynomial is disguising if it is nonzero, and all of its non-zero coefficients lie between $0$ and $1$, exclusive. Determine $\sum \deg(f)\cdot  f(2)$, where the sum includes all disguising polynomials $f$ of degree at most $5$.
Answer: 0
Problem: Consider the $2 \times 3$ rectangle below. We fill in the small squares with the numbers $1, 2, 3, 4, 5, 6$ (one per square). Define a tasty filling to be one such that each row is not in numerical order from left to right and each column is not in numerical order from top to bottom. If the probability that a randomly selected filling is tasty is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, what is $m+n$?
Answer: 157
Problem: Compute \( \sum_{k=1}^{420} \gcd(k,420) \).
Answer: 4680
Problem: $x, y$ are real numbers such that \[x^2 + y^2 = 1, \quad 20x^3 - 15x = 3\] Find the value of \(|20y^3 - 15y|\).
Answer: 4
Problem: Evaluate \[ \frac{1}{729} \sum_{a=1}^{9} \sum_{b=1}^9 \sum_{c=1}^9 \left( abc+ab+bc+ca+a+b+c \right). \]
Answer: 215
Problem: Evaluate the integral:

$$ \int_0^1 \frac{6x + 5a\sqrt{x}}{4\sqrt{x + a\sqrt{x}}}\, dx $$

where $a = 2007 \times 2009$.
Answer: 2008
Problem: For $0 \leq d \leq 9$, we define the numbers 
\[S_{d}=1+d+d^{2}+\cdots+d^{2006}\]
Find the last digit of the number 
\[S_{0}+S_{1}+\cdots+S_{9}.\]
Answer: 8
Problem: Let $N = 2014! + 2015! + 2016! + \dots + 9999!$. How many zeros are at the end of the decimal representation of $N$?
Answer: 501
Problem: Determine the absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. 

You may use the fact that $\sin{n^\circ}$ is irrational for positive integers $n$ not divisible by $30$.
Answer: 178
Problem: There are three men and eleven women taking a dance class. In how many different ways can each man be paired with a woman partner and then have the eight remaining women be paired into four pairs of two?
Answer: 103950
Problem: Isaac repeatedly flips a fair coin. Whenever a particular face appears for the $2n+1$th time, for any nonnegative integer $n$, he earns a point. The expected number of flips it takes for Isaac to get $10$ points is $\frac{a}{b}$ for coprime positive integers $a$ and $b$. Find $a + b$.
Answer: 39
Problem: Find the constant $k$ such that the sum of all $x \ge 0$ satisfying $\sqrt{x}(x+12)=17x-k$ is $256.$
Answer: 216
Problem: In how many ways can you rearrange the letters of "Alejandro" such that it contains one of the words "ned" or "den"?
Answer: 5040
Problem: Find the distance $\overline{CF}$ in the diagram below where $ABDE$ is a square, and angles and lengths are as given:

\[\text{[asy]}\]
\[\text{markscalefactor=0.15;}\]
\[\text{size(8cm);}\]
\[\text{pair A = (0,0);}\]
\[\text{pair B = (17,0);}\]
\[\text{pair E = (0,17);}\]
\[\text{pair D = (17,17);}\]
\[\text{pair F = (-120/17,225/17);}\]
\[\text{pair C = (17+120/17, 64/17);}\]
\[\text{draw(A--B--D--E--cycle^^E--F--A--cycle^^D--C--B--cycle);}\]
\[\text{label("$A$", A, S);}\]
\[\text{label("$B$", B, S);}\]
\[\text{label("$C$", C, dir(0));}\]
\[\text{label("$D$", D, N);}\]
\[\text{label("$E$", E, N);}\]
\[\text{label("$F$", F, W);}\]
\[\text{label("$8$", (F+E)/2, NW);}\]
\[\text{label("$15$", (F+A)/2, SW);}\]
\[\text{label("$8$", (C+B)/2, SE);}\]
\[\text{label("$15$", (D+C)/2, NE);}\]
\[\text{draw(rightanglemark(E,F,A));}\]
\[\text{draw(rightanglemark(D,C,B));}\]
\[\text{[/asy]}\]

The length $\overline{CF}$ is of the form $a\sqrt{b}$ for integers $a, b$ such that no integer square greater than $1$ divides $b$. What is $a + b$?
Answer: 25
Problem: Consider the following increasing sequence $1, 3, 5, 7, 9, \ldots$ of all positive integers consisting only of odd digits. Find the $2017^{th}$ term of the above sequence.
Answer: 39953
Problem: Find the remainder when \( \left \lfloor \frac{149^{151} + 151^{149}}{22499} \right \rfloor \) is divided by \(10^4\).
Answer: 7800
Problem: Let $T=TNFTPP$. Points $A$ and $B$ lie on a circle centered at $O$ such that $\angle AOB$ is right. Points $C$ and $D$ lie on radii $OA$ and $OB$ respectively such that $AC = T-3$, $CD = 5$, and $BD = 6$. Determine the area of quadrilateral $ACDB$.
Answer: 44
Problem: If $x, y, z \in \mathbb{R}$ are solutions to the system of equations:
\[
\begin{cases}
  x - y + z - 1 = 0 \\
  xy + 2z^2 - 6z + 1 = 0
\end{cases}
\]
What is the greatest value of $(x - 1)^2 + (y + 1)^2$?
Answer: 11
Problem: Two identical blue blocks, two identical red blocks, two identical green blocks, and two identical purple blocks are placed next to each other in a row. Find the number of distinct arrangements of these blocks where no blue block is placed next to a red block, and no green block is placed next to a purple block.
Answer: 36