ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: If $x$ is a real number, let $\lfloor x \rfloor$ be the greatest integer that is less than or equal to $x$. If $n$ is a positive integer, let $S(n)$ be defined by
\[
  S(n) 
    = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
        + 10 \left( n - 10^{\lfloor \log n \rfloor} 
               \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
             \right) \, .
\]
(All the logarithms are base 10.) How many integers $n$ from 1 to 2011 (inclusive) satisfy $S(S(n)) = n$?
Answer: 108
Problem: Find the number of subsets of $\{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}$ where the sum of the elements in the subset is $49$.
Answer: 22
Problem: Pairwise distinct points $P_1,P_2,\ldots, P_{16}$ lie on the perimeter of a square with side length $4$ centered at $O$ such that $\lvert P_iP_{i+1} \rvert = 1$ for $i=1,2,\ldots, 16$. (We take $P_{17}$ to be the point $P_1$.) We construct points $Q_1,Q_2,\ldots,Q_{16}$ as follows: for each $i$, a fair coin is flipped. If it lands heads, we define $Q_i$ to be $P_i$; otherwise, we define $Q_i$ to be the reflection of $P_i$ over $O$. (So, it is possible for some of the $Q_i$ to coincide.) Let $D$ be the length of the vector $\overrightarrow{OQ_1} + \overrightarrow{OQ_2} + \cdots + \overrightarrow{OQ_{16}}$. Compute the expected value of $D^2$.
Answer: 88
Problem: Find the smallest positive integer $n$ such that $\underbrace{2^{2^{\cdots^{2}}}}_{n} > 3^{3^{3^3}}$. 

(The notation $\underbrace{2^{2^{\cdots^{2}}}}_{n}$ is used to denote a power tower with $n$ $2$’s. For example, $\underbrace{2^{2^{\cdots^{2}}}}_{4}$ would equal $2^{2^{2^2}}$.)
Answer: 5
Problem: Let $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$, where $n$ is an integer that is not a multiple of $3$. What is the largest integer that is a divisor of every number in $S$?
Answer: 360
Problem: Let $a$, $b$, $c$ be positive integers such that $29a + 30b + 31c = 366$. Find $19a + 20b + 21c$.
Answer: 246
Problem: Find the 1000th number, in ascending order, that is coprime to $105$.
Answer: 2186
Problem: For every three-digit natural number $n$ (where the leading digit of $n$ is nonzero), consider the number $n_0$ obtained by eliminating all zero digits from $n$. For example, if $n = 207$, then $n_0 = 27$. Determine the number of three-digit positive integers $n$ for which $n_0$ is a divisor of $n$ and $n_0 \neq n$.
Answer: 93
Problem: Let $ABCD$ be a rectangle with $AB=10$ and $BC=26$. Let $\omega_1$ be the circle with diameter $\overline{AB}$ and $\omega_2$ be the circle with diameter $\overline{CD}$. Suppose $\ell$ is a common internal tangent to $\omega_1$ and $\omega_2$ and that $\ell$ intersects $AD$ and $BC$ at $E$ and $F$ respectively. What is $EF$?
Answer: 26
Problem: Kris is asked to compute $\log_{10} (x^y)$, where $y$ is a positive integer and $x$ is a positive real number. However, they misread this as $(\log_{10} x)^y$ and compute this value. Despite the reading error, Kris still got the right answer. Given that $x > 10^{1.5}$, determine the largest possible value of $y$.
Answer: 4
Problem: Compute the number of ordered triples \((x, y, z)\) of integers satisfying the equation:  
\[ x^2 + y^2 + z^2 = 9. \]
Answer: 30
Problem: Suppose the polynomial $f(x) = x^{2014}$ is expressed as $f(x) = \sum^{2014}_{k=0} a_k \binom{x}{k}$ for some real numbers $a_0, a_1, \ldots, a_{2014}$. Find the largest integer $m$ such that $2^m$ divides $a_{2013}$.
Answer: 2004
Problem: Let $s_1, s_2, s_3$ be the three roots of the polynomial $x^3 + x^2 + \frac{9}{2}x + 9$.

Find the value of $a + b + c$ if the expression $\prod_{i=1}^{3}(4s^4_i + 81)$ can be written as $2^a3^b5^c$.
Answer: 16
Problem: For a positive integer $n$, an $n$-branch $B$ is an ordered tuple $(S_1, S_2, \dots, S_m)$ of nonempty sets (where $m$ is any positive integer) satisfying $S_1 \subset S_2 \subset \dots \subset S_m \subseteq \{1,2,\dots,n\}$. An integer $x$ is said to appear in $B$ if it is an element of the last set $S_m$. Define an $n$-plant to be an (unordered) set of $n$-branches $\{ B_1, B_2, \dots, B_k\}$, and call it perfect if each of $1$, $2$, \dots, $n$ appears in exactly one of its branches.

Let $T_n$ be the number of distinct perfect $n$-plants (where $T_0=1$), and suppose that for some positive real number $x$ we have the convergence \[ \ln \left( \sum_{n \ge 0} T_n \cdot \frac{\left( \ln x \right)^n}{n!} \right) = \frac{6}{29}. \] If $x = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
Answer: 76
Problem: Given that $20^{22} + 1$ has exactly four prime divisors $p_1 < p_2 < p_3 < p_4$, determine $p_1 + p_2$.
Answer: 490
Problem: Two ants crawl along the sides of the $49$ squares of a $7 \times 7$ board. Each ant passes through all $64$ vertices exactly once and returns to its starting point. What is the smallest possible number of sides covered by both ants?
Answer: 16
Problem: Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers $a, b, c$, respectively. They then compute the polynomial $f(x) = x^{3} + px^{2} + qx + r$ with roots $a, b, c$. If the expected value of the sum of the squares of the coefficients of $f(x)$ is $\frac{m}{n}$ for relatively prime positive integers $m, n$, find the remainder when $m+n$ is divided by 1000.
Answer: 551
Problem: Let $S \subset [0, 1]$ be a set of 5 points with $\{0, 1\} \subset S$. The graph of a real function $f : [0, 1] \to [0, 1]$ is continuous and increasing, and it is linear on every subinterval $I$ in $[0, 1]$ such that the endpoints but no interior points of $I$ are in $S$.

We want to compute, using a computer, the extreme values of 
\[ g(x, t) = \frac{f(x+t)-f(x)}{f(x)-f(x-t)} \]
for $x - t, x + t \in [0, 1]$. At how many points $(x, t)$ is it necessary to compute $g(x, t)$ with the computer?
Answer: 75
Problem: The positive integers $a$ and $b$ are such that the numbers $15a + 16b$ and $16a - 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
Answer: 231361
Problem: Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?

\[ \textbf{(A)}\ 37 \quad \textbf{(B)}\ 63 \quad \textbf{(C)}\ 117 \quad \textbf{(D)}\ 143 \quad \textbf{(E)}\ 163 \]
Answer: 143
Problem: Find the largest natural number $n$ for which there exist different sets $S_1, S_2, \ldots, S_n$ such that:

1. $|S_i \cup S_j| \leq 2004$ for each pair $1 \leq i, j \leq n$.
2. $S_i \cup S_j \cup S_k = \{1, 2, \ldots, 2008\}$ for each three integers $1 \leq i < j < k \leq n$.
Answer: 32
Problem: A marble is placed on each $33$ unit square of a $10 \times 10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?
Answer: 438
Problem: Let the roots of \(x^{2022} - 7x^{2021} + 8x^2 + 4x + 2\) be \(r_1, r_2, \ldots, r_{2022}\), the roots of \(x^{2022} - 8x^{2021} + 27x^2 + 9x + 3\) be \(s_1, s_2, \ldots, s_{2022}\), and the roots of \(x^{2022} - 9x^{2021} + 64x^2 + 16x + 4\) be \(t_1, t_2, \ldots, t_{2022}\). Compute the value of:
\[
\sum_{1 \le i,j \le 2022} r_i s_j + \sum_{1 \le i,j \le 2022} s_i t_j + \sum_{1 \le i,j \le 2022} t_i r_j.
\]
Answer: 191
Problem: For all positive integers $n$, let
\[ f(n) = \sum_{k=1}^n \varphi(k) \left\lfloor \frac{n}{k} \right\rfloor^2. \]
Compute $f(2019) - f(2018)$. Here, $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$.
Answer: 11431
Problem: For positive real numbers $x$ and $y$, let $f(x, y) = x^{\log_2 y}$. The sum of the solutions to the equation \[ 4096 f(f(x, x), x) = x^{13} \] can be written in simplest form as $\frac{m}{n}$. Compute $m + n$.
Answer: 177
Problem: A subset $X$ of $\{1, 2, 3, \ldots, 10000\}$ has the following property: If $a, b$ are distinct elements of $X$, then $ab \not\in X$. What is the maximal number of elements in $X$?
Answer: 9901
Problem: Neo has an infinite supply of red pills and blue pills. When he takes a red pill, his weight will double, and when he takes a blue pill, he will lose one pound. If Neo originally weighs one pound, what is the minimum number of pills he must take to make his weight \(2015\) pounds?
Answer: 13
Problem: Find the largest prime number less than $2008$ that is a divisor of some integer in the infinite sequence \[ \left\lfloor\frac{2008^1}{1}\right\rfloor, \left\lfloor\frac{2008^2}{2}\right\rfloor, \left\lfloor\frac{2008^3}{3}\right\rfloor, \left\lfloor\frac{2008^4}{4}\right\rfloor, \ldots. \]
Answer: 2003
Problem: Find a multiple of $2018$ whose decimal expansion's first four digits are $2017$.
Answer: 20171928
Problem: Compute the remainder when $2^{30}$ is divided by $1000$.
Answer: 824
Problem: Let $\alpha$ and $\beta$ be positive integers such that $\frac{43}{197} < \frac{\alpha}{\beta} < \frac{17}{77}$. Find the minimum possible value of $\beta$.
Answer: 32
Problem: A two-digit positive integer is $\textit{primeable}$ if one of its digits can be deleted to produce a prime number. A two-digit positive integer that is prime, yet not primeable, is $\textit{unripe}$. Compute the total number of unripe integers.
Answer: 5
Problem: If $57a + 88b + 125c \geq 1148$, where $a, b, c > 0$, what is the minimum value of
\[
a^3 + b^3 + c^3 + 5a^2 + 5b^2 + 5c^2?
\]
Answer: 466
Problem: Let $f(n)$ denote the largest odd factor of $n$, including possibly $n$ itself. Determine the value of \[ \frac{f(1)}{1} + \frac{f(2)}{2} + \frac{f(3)}{3} + \cdots + \frac{f(2048)}{2048}, \] rounded to the nearest integer.
Answer: 1365
Problem: Consider the following three lines in the Cartesian plane:

\[
\begin{cases}
\ell_1: & 2x - y = 7\\
\ell_2: & 5x + y = 42\\
\ell_3: & x + y = 14
\end{cases}
\]

Let \( f_i(P) \) correspond to the reflection of the point \( P \) across \( \ell_i \). Suppose \( X \) and \( Y \) are points on the \( x \) and \( y \) axes, respectively, such that \( f_1(f_2(f_3(X)))= Y \). Let \( t \) be the length of segment \( XY \); what is the sum of all possible values of \( t^2 \)?
Answer: 260
Problem: Find the maximal positive integer $p$ such that $5^7$ is the sum of $p$ consecutive positive integers.
Answer: 250
Problem: Let $d_1, d_2, \ldots , d_{k}$ be the distinct positive integer divisors of $6^8$. Find the number of ordered pairs $(i, j)$ such that $d_i - d_j$ is divisible by $11$.
Answer: 665
Problem: Let $\alpha$ be a root of $x^6-x-1$, and call two polynomials $p$ and $q$ with integer coefficients \textit{equivalent} if $p(\alpha)\equiv q(\alpha)\pmod{3}$. It is known that every such polynomial is equivalent to exactly one of $0, 1, x, x^2, \ldots, x^{727}$. Find the largest integer $n < 728$ for which there exists a polynomial $p$ such that $p^3 - p - x^n$ is equivalent to $0$.
Answer: 727
Problem: How many ways are there to fill in a $2 \times 2$ square grid with the numbers $1, 2, 3,$ and $4$ such that the numbers in any two grid squares that share an edge have an absolute difference of at most $2$?
Answer: 8
Problem: Find the $2016$th smallest positive integer that satisfies the congruence $x^x \equiv x \pmod{5}$.
Answer: 3360
Problem: How many positive integers $n$ are there such that $n+2$ divides $(n+18)^2$?
Answer: 7
Problem: A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get \(-1\) mark. He will get 0 mark for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.)
Answer: 495
Problem: Triangle $ABC$ satisfies $AB=104$, $BC=112$, and $CA=120$. Let $\omega$ and $\omega_A$ denote the incircle and $A$-excircle of $\triangle ABC$, respectively. There exists a unique circle $\Omega$ passing through $A$ which is internally tangent to $\omega$ and externally tangent to $\omega_A$. Compute the radius of $\Omega$.
Answer: 49
Problem: Let $\triangle ABC$ be an isosceles triangle with $AB=AC$ and $BC=14$. The triangle is inscribed in a circle $\omega$. Let $D$ be a point on the ray $BC$ such that $CD=6$. The intersection of $AD$ and $\omega$ is point $E$. Given that $AE=7$, find $AC^2$.
Answer: 105
Problem: Find the positive integer $k$ such that the roots of the polynomial $x^3 - 15x^2 + kx - 1105$ are three distinct collinear points in the complex plane.
Answer: 271
Problem: A right regular hexagonal prism has bases $ABCDEF$ and $A'B'C'D'E'F'$ with edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, and $FF'$, each perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. A plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contained within the prism can be expressed as $m\sqrt{n}$, where $n$ is not divisible by the square of any prime. Find $m+n$.
Answer: 323
Problem: How many triples of integers $(a, b, c)$ with $-10 \leq a, b, c \leq 10$ satisfy $a^2 + b^2 + c^2 = (a + b + c)^2$?
Answer: 103
Problem: For positive integers $n$, let $S_n$ be the set of integers $x$ such that $n$ distinct lines, with no three concurrent, can divide a plane into $x$ regions. For example, $S_2 = \{3, 4\}$ because the plane is divided into 3 regions if the two lines are parallel, and 4 regions otherwise. What is the minimum $i$ such that $S_i$ contains at least 4 elements?
Answer: 4
Problem: Evaluate \( \frac{1}{\int _0^{\frac{\pi}{2}} \cos ^{2006}x \cdot \sin 2008 x\, dx} \).
Answer: 2007
Problem: Find the smallest positive integer $k$ such that there is exactly one prime number of the form $kx + 60$ for the integers $0 \le x \le 10$.
Answer: 17
Problem: How many ways are there to arrange the letters $A, A, A, H, H$ in a row so that the sequence $HA$ appears at least once?
Answer: 9
Problem: In a tournament, each player played exactly one game against each of the other players. In each game, the winner was awarded $1$ point, the loser got $0$ points, and each of the two players earned $\frac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned in games against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of their points against the other nine of the ten). What was the total number of players in the tournament?
Answer: 25
Problem: Let \[ f(x) = \frac{1}{1 - \frac{1}{1-x}} \]. Compute \( f^{2016}(2016) \), where \( f \) is composed upon itself 2016 times.
Answer: 2016
Problem: The integers from 1 to $2008^2$ are written on each square of a $2008 \times 2008$ board. For every row and column, the difference between the maximum and minimum numbers is computed. Let $S$ be the sum of these 4016 numbers. Find the greatest possible value of $S$.
Answer: 16184704896
Problem: Let $ABCDE$ be a regular pentagon, and let $F$ be a point on $\overline{AB}$ with $\angle CDF=55^\circ$. Suppose $\overline{FC}$ and $\overline{BE}$ meet at $G$, and select $H$ on the extension of $\overline{CE}$ past $E$ such that $\angle DHE=\angle FDG$. Find the measure of $\angle GHD$, in degrees.
Answer: 19
Problem: A circle $\omega$ has center $O$ and radius $r$. A chord $BC$ of $\omega$ also has length $r$, and the tangents to $\omega$ at $B$ and $C$ meet at $A$. Ray $AO$ meets $\omega$ at $D$ past $O$, and ray $OA$ meets the circle centered at $A$ with radius $AB$ at $E$ past $A$. Compute the degree measure of $\angle DBE$.
Answer: 135
Problem: The digits $1$, $4$, $9$, and $2$ are each used exactly once to form a $4$-digit number $N$. What is the sum of all possible values of $N$?
Answer: 106656
Problem: Find the sum of all the positive integers that have at most three (not necessarily distinct) prime factors, where the primes come from the set \( \{ 2, 3, 5, 7 \} \).
Answer: 1932
Problem: If $x$, $y$, and $z$ are real numbers such that $x^2 + z^2 = 1$ and $y^2 + 2y(x + z) = 6$, find the maximum value of $y(z - x)$.
Answer: 4
Problem: Let $1000 \leq n = \text{ABCD}_{10} \leq 9999$ be a positive integer whose digits $\text{ABCD}$ satisfy the divisibility condition: $$1111 \mid (\text{ABCD} + \text{AB} \times \text{CD}).$$ Determine the smallest possible value of $n$.
Answer: 1729
Problem: How many ways are there to color the edges of a hexagon orange and black, assuming that two hexagons are indistinguishable if one can be rotated into the other? Note that colorings such as $OOBBOB$ and $BOBBOO$ are considered distinct. Flips are ignored.
Answer: 14
Problem: Let $z$ be a complex number satisfying $(z+\frac{1}{z})(z+\frac{1}{z}+1)=1$. Evaluate $(3z^{100}+\frac{2}{z^{100}}+1)(z^{100}+\frac{2}{z^{100}}+3)$.
Answer: 36
Problem: Five distinct 2-digit numbers are in a geometric progression. Find the middle term.
Answer: 36
Problem: Find the coefficient of $x^7$ in the polynomial expansion of $(1 + 2x - x^2)^4$.
Answer: -8
Problem: You and your friend play a game on a $7 \times 7$ grid of buckets. Your friend chooses $5$ "lucky" buckets by marking an "X" on the bottom that you cannot see. However, he tells you that they either form a vertical or horizontal line of length $5$. To clarify, he will select either of the following sets of buckets:

- Either $\{(a, b), (a, b + 1), (a, b + 2), (a, b + 3), (a, b + 4)\}$,
- Or $\{(b, a), (b + 1, a), (b + 2, a), (b + 3, a), (b + 4, a)\}$,

where $1 \le a \le 7$, and $1 \le b \le 3$. Your friend lets you pick up at most $n$ buckets, and you win if one of the buckets you picked was a "lucky" bucket. What is the minimum possible value of $n$ such that, if you pick your buckets optimally, you can guarantee that at least one is "lucky"?
Answer: 9
Problem: Let $a_n$ be the last digit of the sum of the digits of the number formed by repeating the block $2005$ exactly $n$ times. Calculate $a_1 + a_2 + \dots + a_{2005}$.
Answer: 9025
Problem: Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as 1993?
Answer: 118
Problem: Frist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is located north and east of Frist can be expressed in the form $\frac{a}{b} \pi - c$, where $a, b, c$ are positive integers and $a$ and $b$ are relatively prime. Find $a + b + c$.
Answer: 30
Problem: Call a number "Sam-azing" if it is equal to the sum of its digits times the product of its digits. The only two three-digit Sam-azing numbers are $n$ and $n + 9$. Find $n$.
Answer: 135
Problem: Let $T$ be a trapezoid with two right angles and side lengths $4$, $4$, $5$, and $\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into $4$ regions. If the difference between the areas of the largest and smallest of these regions is $d$, compute $240d$.
Answer: 120
Problem: Find the number of integers $n$ for which $\sqrt{\frac{(2020 - n)^2}{2020 - n^2}}$ is a real number.
Answer: 90
Problem: Find the maximal cardinality $|S|$ of the subset $S \subset A=\{1, 2, 3, \dots, 9\}$ given that no two sums $a+b$ where $a, b \in S$ and $a \neq b$ are equal.
Answer: 5
Problem: In a directed graph with $2013$ vertices, there is exactly one edge between any two vertices, and for every vertex, there exists an edge outwards from this vertex. We know that whatever the arrangement of the edges, from every vertex, we can reach $k$ vertices using at most two edges. Find the maximum value of $k$.
Answer: 2012
Problem: For any positive integer $n$, let $r_n$ denote the greatest odd divisor of $n$. Compute the sum:
\[ T = r_{100} + r_{101} + r_{102} + \cdots + r_{200} \]
Answer: 10025
Problem: Let $x$, $y$, $z$ be positive integers satisfying $x < y < z$ and $x + xy + xyz = 37$. Find the greatest possible value of $x + y + z$.
Answer: 20
Problem: The diagram below shows a parallelogram $ABCD$ with $AB = 36$ and $AD = 60$. Diagonal $BD$ is perpendicular to side $AB$. Points $E$ and $F$ bisect sides $AD$ and $BC$, respectively. Points $G$ and $H$ are the intersections of $BD$ with $AF$ and $CE$, respectively. Find the area of quadrilateral $EGFH$.
Answer: 288
Problem: A crazy physicist has discovered a new particle called an emon. He starts with two emons in the plane, situated a distance $1$ from each other. He also has a crazy machine which can take any two emons and create a third one in the plane such that the three emons lie at the vertices of an equilateral triangle. After he has five total emons, let $P$ be the product of the $\binom{5}{2} = 10$ distances between the $10$ pairs of emons. Find the greatest possible value of $P^2$.
Answer: 108
Problem: Given the expression:
\[ A = 3\sum_{m=1}^{n^2}\left(\frac{1}{2} - \{\sqrt{m}\}\right) \]
where $n$ is a positive integer. Find the largest integer $k$ such that $n^k$ divides $\lfloor A \rfloor$ (the greatest integer less than or equal to $A$).
Answer: 1
Problem: Let $A$ and $B$ be digits between $0$ and $9$, and suppose that the product of the two-digit numbers $\overline{AB}$ and $\overline{BA}$ is equal to $k$. Given that $k+1$ is a multiple of $101$, find $k$.
Answer: 403
Problem: In triangle $ABC$, $AB=13$, $BC=14$, $CA=15$. Let $\Omega$ and $\omega$ be the circumcircle and incircle of $\triangle ABC$ respectively. Among all circles that are tangent to both $\Omega$ and $\omega$, call those that contain $\omega$ inclusive and those that do not contain $\omega$ exclusive. Let $\mathcal{I}$ and $\mathcal{E}$ denote the set of centers of inclusive circles and exclusive circles respectively, and let $I$ and $E$ be the area of the regions enclosed by $\mathcal{I}$ and $\mathcal{E}$ respectively. The ratio $\frac{I}{E}$ can be expressed as $\sqrt{\frac{m}{n}}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
Answer: 1558057
Problem: Let $d$ and $n$ be positive integers such that $d$ divides $n$, $n > 1000$, and $n$ is not a perfect square. The minimum possible value of $|d - \sqrt{n}|$ can be written in the form $a\sqrt{b} + c$, where $b$ is a positive integer not divisible by the square of any prime, and $a$ and $c$ are nonzero integers (not necessarily positive). Compute $a+b+c$.
Answer: 38
Problem: For a real number $\alpha > 0$, consider the infinite real sequence defined by $x_1 = 1$ and 
\[ \alpha x_n = x_1 + x_2 + \cdots + x_{n+1} \text{ for } n \ge 1. \] 
Determine the smallest $\alpha$ for which all terms of this sequence are positive reals.
Answer: 4
Problem: Find the smallest prime $p$ such that the sum of the digits of $p$ (in base 10) is a prime number greater than $10$.
Answer: 29
Problem: A finite sequence of positive integers $m_i$ for $i = 1, 2, \ldots, 2006$ is defined such that $m_1 = 1$ and $m_i = 10m_{i-1} + 1$ for $i > 1$. How many of these integers are divisible by $37$?
Answer: 668
Problem: In the $xy$-coordinate plane, the $x$-axis and the line $y=x$ are mirrors. If you shoot a laser beam from the point $(126, 21)$ toward a point on the positive $x$-axis, there are $3$ places you can aim at where the beam will bounce off the mirrors and eventually return to $(126, 21)$. They are $(126, 0)$, $(105, 0)$, and a third point $(d, 0)$. What is $d$? 

Note: When light bounces off a mirror, the angle of incidence has the same measure as the angle of reflection.
Answer: 111
Problem: Find the remainder when $3^{2007}$ is divided by $2007$.
Answer: 1620
Problem: Consider the polynomial 
\[P(x) = x^3 + 3x^2 + 6x + 10.\]
Let its three roots be \(a\), \(b\), and \(c\). Define \(Q(x)\) to be the monic cubic polynomial with roots \(ab\), \(bc\), and \(ca\). Compute \(|Q(1)|\).
Answer: 75