ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$, tens digit $y$, and units digit $z$, and similarly let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$.  How many three-digit numbers $\underline{abc}$, none of whose digits are 0, are there such that $\underline{ab}>\underline{bc}>\underline{ca}$?
Answer: 120
Problem: Let $p(x) = x^2 + x + 1$.  Find the fourth smallest prime $q$ such that $p(n)$ is divisble by $q$ for some integer $n$.
Answer: 19
Problem: In regular hexagon $ABCDEF$, $AC$, $CE$ are two diagonals. Points $M$, $N$ are on $AC$, $CE$ respectively and satisfy $AC: AM = CE: CN = r$. Suppose $B, M, N$ are collinear, find $100r^2$.
[asy]
size(120); defaultpen(linewidth(0.7)+fontsize(10)); 
pair D2(pair P) {
dot(P,linewidth(3)); return P;
}
pair A=dir(0), B=dir(60), C=dir(120), D=dir(180), E=dir(240), F=dir(300), N=(4*E+C)/5,M=intersectionpoints(A--C,B--N)[0];
draw(A--B--C--D--E--F--cycle); draw(A--C--E); draw(B--N);
label("$A$",D2(A),plain.E);
label("$B$",D2(B),NE);
label("$C$",D2(C),NW);
label("$D$",D2(D),W);
label("$E$",D2(E),SW);
label("$F$",D2(F),SE);
label("$M$",D2(M),(0,-1.5));
label("$N$",D2(N),SE);
[/asy]
Answer: 300
Problem: All the diagonals of a regular decagon are drawn.  A regular decagon satisfies the property that if three diagonals concur, then one of the three diagonals is a diameter of the circumcircle of the decagon.  How many distinct intersection points of diagonals are in the interior of the decagon?
Answer: 161
Problem: We say that a rook is "attacking" another rook on a chessboard if the two rooks are in the same row or column of the chessboard and there is no piece directly between them.  Let $n$ be the maximum number of rooks that can be placed on a $6\times 6$ chessboard such that each rook is attacking at most one other.  How many ways can $n$ rooks be placed on a $6\times 6$ chessboard such that each rook is attacking at most one other?
Answer: 8100
Problem: Square $ABCD$ is divided into four rectangles by $EF$ and $GH$. $EF$ is parallel to $AB$ and $GH$ parallel to $BC$.  $\angle BAF = 18^\circ$. $EF$ and $GH$ meet at point $P$. The area of rectangle $PFCH$ is twice that of rectangle $AGPE$. Given that the value of $\angle FAH$ in degrees is $x$, find the nearest integer to $x$.
[asy]
size(100); defaultpen(linewidth(0.7)+fontsize(10)); 
pair D2(pair P) {
dot(P,linewidth(3)); return P;
}
// NOTE: I've tampered with the angles to make the diagram not-to-scale. The correct numbers should be 72 instead of 76, and 45 instead of 55. 
pair A=(0,1), B=(0,0), C=(1,0), D=(1,1), F=intersectionpoints(A--A+2*dir(-76),B--C)[0], H=intersectionpoints(A--A+2*dir(-76+55),D--C)[0], E=F+(0,1), G=H-(1,0), P=intersectionpoints(E--F,G--H)[0];
draw(A--B--C--D--cycle);
draw(F--A--H); draw(E--F); draw(G--H); 
label("$A$",D2(A),NW);
label("$B$",D2(B),SW);
label("$C$",D2(C),SE);
label("$D$",D2(D),NE);
label("$E$",D2(E),plain.N);
label("$F$",D2(F),S);
label("$G$",D2(G),W);
label("$H$",D2(H),plain.E);
label("$P$",D2(P),SE);
[/asy]
Answer: 45
Problem: Assume that $f(a+b) = f(a) + f(b) + ab$, and that $f(75) - f(51) = 1230$.  Find $f(100)$.
Answer: 3825
Problem: Let $f$ be a function such that $f(x)+f(x+1)=2^x$ and $f(0)=2010$. Find the last two digits of $f(2010)$.
Answer: 51
Problem: The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$, where $p$ is an integer. Find $p$.
Answer: 192
Problem: Let $n$ be the number of polynomial functions from the integers modulo $2010$ to the integers modulo $2010$.  $n$ can be written as $n = p_1 p_2 \cdots p_k$, where the $p_i$s are (not necessarily distinct) primes.  Find $p_1 + p_2 + \cdots + p_n$.
Answer: 4527
Problem: Triangle $ABC$ has $AB = 4$, $AC = 5$, and $BC = 6$.  An angle bisector is drawn from angle $A$, and meets $BC$ at $M$.  What is the nearest integer to $100 \frac{AM}{CM}$?
Answer: 100
Problem: On rectangular coordinates, point $A = (1,2)$, $B = (3,4)$. $P = (a, 0)$ is on $x$-axis. Given that $P$ is chosen such that $AP + PB$ is minimized, compute $60a$.
Answer: 100
Problem: Find the largest positive integer $n$ such that $n\varphi(n)$ is a perfect square.  ($\varphi(n)$ is the number of integers $k$, $1 \leq k \leq n$ that are relatively prime to $n$)
Answer: 1
Problem: The Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play.  If the Band can't play any two tiring songs in a row, how many ways can the band play its 8 songs?
Answer: 14400
Problem: Let $f(x)=3x^3-5x^2+2x-6$. If the roots of $f$ are given by $\alpha$, $\beta$, and $\gamma$, find
\[
\left(\frac{1}{\alpha-2}\right)^2+\left(\frac{1}{\beta-2}\right)^2+\left(\frac{1}{\gamma-2}\right)^2.
\]
Answer: 68
Problem: A sequence of real numbers $\{a_n\}_{n = 1}^\infty (n=1,2,...)$ has the following property:
\begin{align*}
6a_n+5a_{n-2}=20+11a_{n-1}\  (\text{for }n\geq3).
\end{align*}
The first two elements are $a_1=0, a_2=1$. Find the integer closest to $a_{2011}$.
Answer: 40086
Problem: Let
\[f_1(x) = \frac{1}{x}\quad\text{and}\quad f_2(x) = 1 - x\]
Let $H$ be the set of all compositions of the form $h_1 \circ h_2 \circ \ldots \circ h_k$, where each $h_i$ is either $f_1$ or $f_2$.  For all $h$ in $H$, let $h^{(n)}$ denote $h$ composed with itself $n$ times.  Find the greatest integer $N$ such that $\pi, h(\pi), \ldots, h^{(N)}(\pi)$ are all distinct for some $h$ in $H$.
Answer: 2
Problem: Let $\sigma$ be a random permutation of $\{0, 1, \ldots, 6\}$. Let $L(\sigma)$ be the length of the longest initial monotonic consecutive subsequence of $\sigma$ not containing $0$; for example, \[L(\underline{2,3,4},6,5,1,0) = 3,\ L(\underline{3,2},4,5,6,1,0) = 2,\ L(0,1,2,3,4,5,6) = 0.\] If the expected value of $L(\sigma)$ can be written as $\frac mn$, where $m$ and $n$ are relatively prime positive integers, then find $m + n$.
Answer: 5399
Problem: Suppose the polynomial $x^3 - x^2 + bx + c$ has real roots $a, b, c$. What is the square of the minimum value of $abc$?
Answer: 1
Problem: Let $ABC$ be a triangle with $AB = 2, BC = 5, AC = 4$. Let $M$ be the projection of $C$ onto the external angle bisector at vertex $B$. Similarly, let $N$ be the projection of $B$ onto the external angle bisector at vertex $C$. If the ratio of the area of quadrilateral $BCNM$ to the area of triangle $ABC$ is $a/b$, where $a$ and $b$ are positive integers and $\gcd(a, b) = 1$, find $a + b$.
Answer: 87
Problem: Two logs of length 10 are laying on the ground touching each other. Their radii are 3 and 1, and the smaller log is fastened to the ground. The bigger log rolls over the smaller log without slipping, and stops as soon as it touches the ground again. The volume of the set of points swept out by the larger log as it rolls over the smaller one can be expressed as $n \pi$, where $n$ is an integer. Find $n$.
Answer: 250
Problem: Let $d(n)$ denote the number of divisors of $n$ (including itself). You are given that \[\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.\]  Find $p(6)$, where $p(x)$ is the unique polynomial with rational coefficients satisfying \[p(\pi) = \sum_{n=1}^{\infty} \frac{d(n)}{n^2}.\]
Answer: 36
Problem: Let $ABCD$ be a trapezoid with $AD$ parallel to $BC$, $AD = 2$,  and $BC = 1$. Let $M$ be the midpoint of $AD$, and let $P$ be the intersection of $BD$ with $CM$. Extend $AP$ to meet segment $CD$ at point $Q$. If the ratio $CQ/QD = a/b$, where $a$ and $b$ are positive integers and $\text{gcd}(a, b) = 1$, find $a + b$.
Answer: 3
Problem: Let $\ell_1$ and $\ell_2$ be two parallel lines, a distance of 15 apart. Points $A$ and $B$ lie on $\ell_1$ while points $C$  and $D$ lie on $\ell_2$ such that $\angle BAC = 30^\circ$ and $\angle ABD = 60^\circ$. The minimum value of $AD + BC$ is $a\sqrt b$, where $a$ and $b$ are integers and $b$ is squarefree. Find $a + b$.
Answer: 23
Problem: A function $S(m, n)$ satisfies the initial conditions $S(1, n) = n$, $S(m, 1) = 1$, and the recurrence $S(m, n) = S(m - 1, n)S(m, n - 1)$ for $m\geq 2, n\geq 2$. Find the largest integer $k$ such that $2^k$ divides $S(7, 7)$.
Answer: 370
Problem: Let $1,\alpha_1,\alpha_2,...,\alpha_{10}$ be the roots of the polynomial $x^{11}-1$. It is a fact that there exists a unique polynomial of the form $f(x) = x^{10}+c_9x^9+ \dots + c_1x$ such that each $c_i$ is an integer, $f(0) = f(1) = 0$, and for any $1 \leq i \leq 10$ we have $(f(\alpha_i))^2 = -11$. Find $\left|c_1+2c_2c_9+3c_3c_8+4c_4c_7+5c_5c_6\right|$.
Answer: 15
Problem: Shirley has a magical machine.  If she inputs a positive even integer $n$, the machine will output $n/2$, but if she inputs a positive odd integer $m$, the machine will output $m+3$. The machine keeps going by automatically using its output as a new input, stopping immediately before it obtains a number already processed. Shirley wants to create the longest possible output sequence possible with initial input at most $100$. What number should she input?
Answer: 67
Problem: Let $N$ be the number of ways to place $4$ bishops on a $5 \times 5$ chessboard such that no $3$ are on the same diagonal. Find the remainder when $N$ is divided by $100$. (Note: the length of a diagonal on a $5 \times 5$ chessboard can be 2, 3, 4, or 5.)
Answer: 12
Problem: Let $f(x) = x^3-7x^2+16x-10$. As $x$ ranges over all integers, find the sum of distinct prime values taken on by $f(x)$.
Answer: 2
Problem: Let $ABC$ be a triangle with $AB = 15, BC = 17$, $CA = 21$, and incenter $I$. If the circumcircle of triangle $IBC$ intersects side $AC$ again at $P$, find $CP$.
Answer: 6
Problem: Consider the sum $\overline{a b} + \overline{ c d e}$, where each of the letters is a distinct digit between $1$ and $5$. How many values are possible for this sum?
Answer: 27
Problem: Let $\omega_1$ be a circle of radius 6, and let $\omega_2$ be a circle of radius 5 that passes through the center $O$ of $\omega_1$. Let $A$ and $B$ be the points of intersection of the two circles, and let $P$ be a point on major arc $AB$ of $\omega_2$. Let $M$ and $N$ be the second intersections of $PA$ and $PB$ with $\omega_1$, respectively. Let $S$ be the midpoint of $MN$. As $P$ ranges over major arc $AB$ of $\omega_2$, the minimum length of segment $SA$ is $a/b$, where $a$ and $b$ are positive integers and $\gcd(a, b) = 1$. Find $a+b$.
Answer: 17
Problem: Calculate the sum of the coordinates of all pairs of positive integers $(n, k)$ such that $k\equiv 0, 3\pmod 4$, $n > k$, and $\displaystyle\sum^n_{i = k + 1} i^3 = (96^2\cdot3 - 1)\displaystyle\left(\sum^k_{i = 1} i\right)^2 + 48^2$
Answer: 103
Problem: A set of $n$ dominoes, each colored with one white square and one black square, is used to cover a $2 \times n$ board of squares. For $n = 6$, how many different patterns of colors can the board have? (For $n = 2$, this number is $6$.)
Answer: 328
Problem: A function $f:\{1,2, \ldots, n\} \to \{1, \ldots, m\}$ is [i]multiplication-preserving[/i] if $f(i)f(j) = f(ij)$ for all $1 \le i \le j \le ij \le n$, and [i]injective[/i] if $f(i) = f(j)$ only when $i = j$. For $n = 9, m = 88$, the number of injective, multiplication-preserving functions is $N$. Find the sum of the prime factors of $N$, including multiplicity. (For example, if $N = 12$, the answer would be $2 + 2 + 3 = 7$.)
Answer: 33
Problem: Let $\omega$ be a circle of radius $6$ with center $O$. Let $AB$ be a chord of $\omega$ having length $5$. For any real constant $c$, consider the locus $\mathcal{L}(c) $ of all points $P$ such that $PA^2 - PB^2 = c$. Find the largest value of $c$ for which the intersection of $\mathcal{L}(c)$ and $\omega$ consists of just one point.
Answer: 60
Problem: Let $a$ and $b$ be positive integers such that $a + bz = x^3 + y^4$ has no solutions for any integers $x, y, z$, with $b$ as small as possible, and $a$ as small as possible for the minimum $b$. Find $ab$.
Answer: 91
Problem: A polynomial $p$ can be written as 
\begin{align*}
p(x) = x^6+3x^5-3x^4+ax^3+bx^2+cx+d.
\end{align*}
Given that all roots of $p(x)$ are equal to either $m$ or $n$ where $m$ and $n$ are integers, compute $p(2)$.
Answer: 64
Problem: At the start of the PUMaC opening ceremony in McCosh auditorium, the speaker counts $90$ people in the audience. Every minute afterwards, either one person enters the auditorium (due to waking up late) or leaves (in order to take a dreadful math contest). The speaker observes that in this time, exactly $100$ people enter the auditorium, $100$ leave, and $100$ was the largest audience size he saw. Find the largest integer $m$ such that $2^m$ divides the number of different possible sequences of entries and exits given the above information.
Answer: 6
Problem: Let $f$ be an invertible function defined on the complex numbers such that \[z^2 = f(z + f(iz + f(-z + f(-iz + f(z + \ldots)))))\]
for all complex numbers $z$. Suppose $z_0 \neq 0$ satisfies $f(z_0) = z_0$. Find $1/z_0$.
(Note: an invertible function is one that has an inverse).
Answer: 4
Problem: A road company is trying to build a system of highways in a country with $21$ cities. Each highway runs between two cities. A trip is a sequence of distinct cities $C_1,\dots, C_n$, for which there is a highway between $C_i$ and $C_{i+1}$. The company wants to fulfill the following two constraints: 
(1) for any ordered pair of distinct cities $(C_i, C_j)$, there is exactly one trip starting at $C_i$ and ending at $C_j$. 
(2) if $N$ is the number of trips including exactly 5 cities, then $N$ is maximized. 
What is this maximum value of $N$?
Answer: 192
Problem: For every integer $n$ from $0$ to $6$, we have $3$ identical weights with weight $2^n$. How many ways are there to form a total weight of 263 grams using only these given weights?
Answer: 60
Problem: Four circles are situated in the plane so that each is tangent to the other three. If three of the radii are $5$, $5$, and $8$, the largest possible radius of the fourth circle is $a/b$, where $a$ and $b$ are positive integers and gcd$(a, b) = 1$. Find $a + b$.
Answer: 43
Problem: For how many ordered triplets of three positive integers is it true that their product is four more than twice their sum?
Answer: 15
Problem: Let $\{g_i\}_{i=0}^{\infty}$ be a sequence of positive integers such that $g_0=g_1=1$ and the following recursions hold for every positive integer $n$: \begin{align*}
g_{2n+1} &= g_{2n-1}^2+g_{2n-2}^2 \\
g_{2n} &= 2g_{2n-1}g_{2n-2}-g_{2n-2}^2
\end{align*}
Compute the remainder when $g_{2011}$ is divided by $216$.
Answer: 34
Problem: A rectangular piece of paper has corners labeled $A, B, C$, and $D$, with $BC = 80$ and $CD = 120$. Let $M$ be the midpoint of side $AB$. The corner labeled $A$ is folded along line $MD$ and the corner labeled $B$ is folded along line $MC$ until the segments $AM$ and $MB$ coincide. Let $S$ denote the point in space where $A$ and $B$ meet. If $H$ is the foot of the perpendicular from $S$ to the original plane of the paper, find $HM$.
Answer: 45
Problem: Let $ABC$ be a triangle with $\angle BAC = 60^\circ, BA = 2$, and $CA = 3$. A point $M$ is located inside $ABC$ such that $MB = 1$ and $MC = 2$. A semicircle tangent to $MB$ and $MC$ has its center $O$ on $BC$. Let $P$ be the intersection of the angle bisector of $\angle BAC$ and the perpendicular bisector of $AC$. If the ratio $OP/MO$ is $a/b$, where $a$ and $b$ are positive integers and $\gcd(a, b) = 1$, find $a + b$.
Answer: 3
Problem: What is the largest positive integer $n < 1000$ for which there is a positive integer $m$ satisfying \[\text{lcm}(m,n) = 3m \times \gcd(m,n)?\]
Answer: 972
Problem: $5$ people stand in a line facing one direction. In every round, the person at the front moves randomly to any position in the line, including the front or the end. Suppose that $\frac{m}{n}$ is the expected number of rounds needed for the last person of the initial line to appear at the front of the line, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
Answer: 137
Problem: If $n$ is an integer such that $n \ge 2^k$ and $n < 2^{k+1}$, where $k = 1000$, compute the following:
$$n - \left(  \lfloor \frac{n -2^0}{2^1} \rfloor + \lfloor \frac{n -2^1}{2^2} \rfloor + ...+ \lfloor \frac{n -2^{k-1}}{2^k} \rfloor \right)$$
Answer: 2037
Problem: Let $a, b$, and $c$ be positive integers satisfying
$a^4 + a^2b^2 + b^4 = 9633$
$2a^2 + a^2b^2 + 2b^2 + c^5 = 3605$.
What is the sum of all distinct values of $a + b + c$?
Answer: 18
Problem: Three circles, with radii of $1, 1$, and $2$, are externally tangent to each other. The minimum possible area of a quadrilateral that contains and is tangent to all three circles can be written as $a + b\sqrt{c}$ where $c$ is not divisible by any perfect square larger than $1$. Find $a + b + c$
Answer: 22
Problem: Find the largest possible sum $ m + n$ for positive integers $m, n \le 100$ such that $m + 1 \equiv 3$ (mod $4$) and there exists a prime number $p$ and nonnegative integer $a$ such $\frac{m^{2n-1}-1}{m-1} = m^n+p^a$
.
Answer: 84
Problem: Considering all numbers of the form $n = \lfloor \frac{k^3}{2012} \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $k$ ranges from $1$ to $2012$, how many of these $n$’s are distinct?
Answer: 1995
Problem: Find the sum of all possible sums $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is a perfect square.
Answer: 9
Problem: Two circles centered at $O$ and $P$ have radii of length $5$ and $6$ respectively. Circle $O$ passes through point $P$. Let the intersection points of circles $O$ and $P$ be $M$ and $N$. The area of triangle $\vartriangle MNP$ can be written in simplest form as $a/b$. Find $a + b$.
Answer: 457
Problem: When some number $a^2$ is written in base $b$, the result is $144_b$. $a$ and $b$ also happen to be integer side lengths of a right triangle. If $a$ and $b$ are both less than $20$, find the sum of all possible values of $a$.
Answer: 169
Problem: Cyclic quadrilateral $ABCD$ has side lengths $AB = 2, BC = 3, CD = 5, AD = 4$. 
Find $\sin A \sin B(\cot A/2 + \cot B/2 + \cot C/2 + \cot D/2)^2$. 
Your answer can be written in simplest form as $a/b$. Find $a + b$.
Answer: 5101
Problem: Two white pentagonal pyramids, with side lengths all the same, are glued to each other at their regular pentagon bases. Some of the resulting $10$ faces are colored black. How many rotationally distinguishable colorings may result?
Answer: 120
Problem: Compute the smallest positive integer $a$ for which $$\sqrt{a +\sqrt{a +...}} - \frac{1}{a +\frac{1}{a+...}}> 7$$
Answer: 43
Problem: A PUMaC grader is grading the submissions of forty students $s_1, s_2, ..., s_{40}$ for the individual finals round, which has three problems. After grading a problem of student $s_i$, the grader either:
$\bullet$  grades another problem of the same student, or
$\bullet$ grades the same problem of the student $s_{i-1}$ or $s_{i+1}$ (if $i > 1$ and $i < 40$, respectively).
He grades each problem exactly once, starting with the first problem of $s_1$ and ending with the third problem of $s_{40}$. Let $N$ be the number of different orders the grader may grade the students’ problems in this way. Find the remainder when $N$ is divided by $100$.
Answer: 78
Problem: Assume the earth is a perfect sphere with a circumference of $60$ units. A great circle is a circle on a sphere whose center is also the center of the sphere. There are three train tracks on three great circles of the earth. One is along the equator and the other two pass through the poles, intersecting at a $90$ degree angle. If each track has a train of length $L$ traveling at the same speed, what is the maximum value of $L$ such that the trains can travel without crashing?
Answer: 25
Problem: Let f be a polynomial of degree $3$ with integer coefficients such that $f(0) = 3$ and $f(1) = 11$.
If f has exactly $2$ integer roots, how many such polynomials $f$ exist?
Answer: 0
Problem: How many factors of $(20^{12})^2$ less than $20^{12}$ are not factors of $20^{12}$ ?
Answer: 288
Problem: Call a positive integer $x$ a leader if there exists a positive integer $n$ such that the decimal representation of $x^n$ starts ([u]not ends[/u])  with $2012$. For example, $586$ is a leader since $586^3 =201230056$. How many leaders are there in the set $\{1, 2, 3, ..., 2012\}$?
Answer: 2008
Problem: An octahedron (a solid with 8 triangular faces) has a volume of $1040$. Two of the spatial diagonals intersect, and their plane of intersection contains four edges that form a cyclic quadrilateral. The third spatial diagonal is perpendicularly bisected by this plane and intersects the plane at the circumcenter of the cyclic quadrilateral. Given that the side lengths of the cyclic quadrilateral are $7, 15, 24, 20$, in counterclockwise order, the sum of the edge lengths of the entire octahedron can be written in simplest form as $a/b$. Find $a + b$.
Answer: 541
Problem: If the probability that the sum of three distinct integers between $16$ and $30$ (inclusive) is even can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m + n$.
Answer: 97
Problem: How many (possibly empty) sets of lattice points $\{P_1, P_2, ... , P_M\}$, where each point $P_i =(x_i, y_i)$ for $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$, satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$ ? An infinite slope, e.g. $P_i$ is vertically above $P_j$ , does not count as positive.
Answer: 3432
Problem: Six ants are placed on the vertices of a regular hexagon with an area of $12$. At each point in time, each ant looks at the next ant in the hexagon (in counterclockwise order), and measures the distance, $s$, to the next ant. Each ant then proceeds towards the next ant at a speed of $\frac{s}{100}$ units per year. After T years, the ants’ new positions are the vertices of a new hexagon with an area of $4$. T is of the form $a \ln b$, where $b$ is square-free. Find $a + b$.
Answer: 103
Problem: Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors. 
Suppose $M$ can be written as  $\Pi_{k=1}^{n}p_k^{a_k}$ where the $p_k$’s are distinct primes and the $a_k$’s are positive integers. 
Find $\Sigma_{k=1}^{n}(p_k + a_k)$
Answer: 1012
Problem: Let $a_n$ be a sequence such that $a_1 = 1$ and $a_{n+1} = \lfloor a_n +\sqrt{a_n} +\frac12 \rfloor $, where $\lfloor x \rfloor$  denotes the greatest integer less than or equal to $x$. What are the last four digits of $a_{2012}$?
Answer: 2037
Problem: Let $\vartriangle ABC$ be a triangle with $\angle BAC = 45^o, \angle BCA = 30^o$, and $AB = 1$. Point $D$ lies on segment $AC$ such that $AB = BD$. Find the square of the length of the common external tangent to the circumcircles of triangles $\vartriangle BDC$ and $\vartriangle ABC$.
Answer: 2
Problem: Let $p_1 = 2012$ and $p_n = 2012^{p_{n-1}}$ for $n > 1$. Find the largest integer $k$ such that $p_{2012}- p_{2011}$ is divisible by $2011^k$.
Answer: 2012
Problem: Compute $\Sigma_{n=1}^{\infty}\frac{n + 1}{n^2(n + 2)^2}$ .
Your answer in simplest form can be written as $a/b$, where $a, b$ are relatively-prime positive integers. Find $a + b$.
Answer: 21
Problem: Let A be a regular $12$-sided polygon. A new $12$-gon B is constructed by connecting the midpoints of the sides of A. The ratio of the area of B to the area of A can be written in simplest form as $(a +\sqrt{b})/c$, where $a, b, c$ are integers. Find $a + b + c$.
Answer: 9
Problem: Let the sequence $\{x_n\}$ be defined by $x_1 \in \{5, 7\}$ and, for $k \ge 1, x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$. For example, the possible values of $x_3$ are $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$, and $7^{7^7}$. Determine the sum of all possible values for the last two digits of $x_{2012}$.
Answer: 75
Problem: Let $f_n(x) = n + x^2$. Evaluate the product $gcd\{f_{2001}(2002), f_{2001}(2003)\} \times gcd\{f_{2011}(2012), f_{2011}(2013)\}  \times gcd\{f_{2021}(2022), f_{2021}(2023)\}$, where $gcd\{x, y\}$ is the greatest common divisor of $x$ and $y$
Answer: 125
Problem: What is the smallest natural number $n$ greater than $2012$ such that the polynomial $f(x) =(x^6 + x^4)^n - x^{4n} - x^6$ is divisible by $g(x) = x^4 + x^2 + 1$?
Answer: 2015