Find the number of ways to put a number in every unit square of a $3 \times 3$ square such that any number is divisible by the number directly to the top and the number directly to the left of it, and the top-left number is $1$ and the bottom right number is $2013$.

An octomino is made by joining $8$ congruent squares edge to edge. Three examples are shown below. How many octominoes have at least $2$ lines of symmetry? [asy] size(8cm,0); filldraw((0,1)--(0,2)--(1,2)--(1,1)--cycle,grey); filldraw((0,2)--(0,3)--(1,3)--(1,2)--cycle,grey); filldraw((0,3)--(0,4)--(1,4)--(1,3)--cycle,grey); filldraw((1,0)--(1,1)--(2,1)--(2,0)--cycle,grey); filldraw((1,1)--(1,2)--(2,2)--(2,1)--cycle,grey); filldraw((1,2)--(1,3)--(2,3)--(2,2)--cycle,grey); filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle,grey); filldraw((2,2)--(2,3)--(3,3)--(3,2)--cycle,grey); filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle,grey); filldraw((4,1)--(4,2)--(5,2)--(5,1)--cycle,grey); filldraw((4,2)--(4,3)--(5,3)--(5,2)--cycle,grey); filldraw((4,3)--(4,4)--(5,4)--(5,3)--cycle,grey); filldraw((5,1)--(5,2)--(6,2)--(6,1)--cycle,grey); filldraw((5,3)--(5,4)--(6,4)--(6,3)--cycle,grey); filldraw((6,2)--(6,3)--(7,3)--(7,2)--cycle,grey); filldraw((6,3)--(6,4)--(7,4)--(7,3)--cycle,grey); filldraw((8,3)--(8,4)--(9,4)--(9,3)--cycle,grey); filldraw((9,3)--(9,4)--(10,4)--(10,3)--cycle,grey); filldraw((10,3)--(10,4)--(11,4)--(11,3)--cycle,grey); filldraw((11,2)--(11,3)--(12,3)--(12,2)--cycle,grey); filldraw((11,3)--(11,4)--(12,4)--(12,3)--cycle,grey); filldraw((12,2)--(12,3)--(13,3)--(13,2)--cycle,grey); filldraw((12,3)--(12,4)--(13,4)--(13,3)--cycle,grey); filldraw((13,3)--(13,4)--(14,4)--(14,3)--cycle,grey); [/asy]

Let $p = (a_{1}, a_{2}, \ldots , a_{12})$ be a permutation of $1, 2, \ldots, 12$. We will denote \[S_{p} = |a_{1}-a_{2}|+|a_{2}-a_{3}|+\ldots+|a_{11}-a_{12}|\]We'll call $p$ $\textit{optimistic}$ if $a_{i} > \min(a_{i-1}, a_{i+1})$ $\forall i = 2, \ldots, 11$. $a)$ What is the maximum possible value of $S_{p}$. How many permutations $p$ achieve this maximum?$\newline$ $b)$ What is the number of $\textit{optimistic}$ permtations $p$? $c)$ What is the maximum possible value of $S_{p}$ for an $\textit{optimistic}$ $p$? How many $\textit{optimistic}$ permutations $p$ achieve this maximum?

Let $p>2$ be a prime number, $m>1$ and $n$ be positive integers such that $\frac {m^{pn}-1}{m^n-1}$ is a prime number. Show that: $$pn\mid (p-1)^n+1$$

$2010$ ships deliver bananas, lemons and pineapples from South America to Russia. The total number of bananas on each ship equals the number of lemons on all other ships combined, while the total number of lemons on each ship equals the total number of pineapples on all other ships combined. Prove that the total number of fruits is a multiple of $31$.

Find all functions $f$ from reals to reals which satisfy $f (f(x) + f(y)) = f(x^2) + 2x^2 f(y) + (f(y))^2$ for all real numbers $x$ and $y$.

Given isosceles $\triangle ABC$ with $AB = AC$ and $\angle BAC = 22^{\circ}$. On the side $BC$ point $D$ is chosen such that $BD = 2CD$. The foots of perpendiculars from $B$ to lines $AD$ and $AC$ are points $E$, $F$ respectively. Find with the proof value of the angle $\angle CEF$.

Four perpendiculars are drawn from four vertices of a convex pentagon to the opposite sides. If these four lines pass through the same point, prove that the perpendicular from the fifth vertex to the opposite side also passes through this point.

Given a triangle $PQR$, the inscribed circle $\omega$ which touches the sides $QR, RP$ and $PQ$ at points $A, B$ and $C$, respectively, and $AB^2 + AC^2 = 2BC^2$. Prove that the point of intersection of the segments $PA, QB$ and $RC$, the center of the circle $\omega$, the point of intersection of the medians of the triangle $ABC$, the point $A$ and the midpoints of the segments $AC$ and $AB$ lie on one circle.

a) Denote by $S(n)$ the sum of digits of a positive integer $n$. After the decimal point, we write one after the other the numbers $S(1),S(2),...$. Show that the number obtained is irrational. b) Denote by $P(n)$ the product of digits of a positive integer $n$. After the decimal point, we write one after the other the numbers $P(1),P(2),...$. Show that the number obtained is irrational.

Suppose $P(x,y)$ is a homogenous non-constant polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for all real $t$. Prove that $P(x,y) = (x^2+y^2)^k$ for some positive integer $k$. (A polynomial $A(x,y)$ with real coefficients and having a degree of $n$ is homogenous if it is the sum of $a_ix^iy^{n-i}$ for some real number $a_i$, for all integer $0 \le i \le n$.)

In the acute triangle $ABC$ point $I$ is the incenter, $O$ is the circumcenter, while $I_a$ is the excenter opposite the vertex $A$. Point $A'$ is the reflection of $A$ across the line $BC$. Prove that angles $\angle IOI_a$ and $\angle IA'I_a$ are equal.

Prove that if every subspace of a Hausdorff space $X$ is $\sigma$-compact, then $X$ is countable.

In a $1024$ person randomly seeded single elimination tournament bracket, each player has a unique skill rating. In any given match, the player with the higher rating has a $\frac34$ chance of winning the match. What is the probability the second lowest rated player wins the tournament?

Find prime numbers $p$, $q$, $r$ and $s$, pairwise distinct, such that their sum is prime number and numbers $p^2+qr$ and $p^2+qs$ are perfect squares

Find all pairs of prime numbers $(p,q)$, such that $\frac{(2p^2-1)^q+1}{p+q}$ and $\frac{(2q^2-1)^p+1}{p+q}$ are both integers.

Let $P(x) = x^3 + (t - 1)x^2 - (t + 3)x + 1$. For what values of real $t$ the sum of the squares and the reciprocals of the roots of $ P(x)$ is minimum?

Points $X$ and $Y$ are chosen inside an acute triangle $ABC$ so that: $$\angle AXB = \angle CYB = 180^\circ - \angle ABC, \text{ } \angle ABX = \angle CBY$$ Show that the points $X$ and $Y$ are equidistant from the center of the circumscribed circle of $\triangle ABC$. [i]Proposed by Anton Trygub[/i]

Let $ABC$ be an acute triangle. Let $P$ be a point on the circle $(ABC)$, and $Q$ be a point on the segment $AC$ such that $AP\perp BC$ and $BQ\perp AC$. Lot $O$ be the circumcenter of triangle $APQ$. Find the angle $OBC$.

Find all pairs of functions $f ,g : Z \rightarrow Z $ that satisfy $f (g(x)+y) = g( f (y)+x) $ for all integers $ x,y$ and such that $g(x) = g(y)$ only if $x = y$.

We say that a square-free positive integer $n$ is [i]almost prime[/i] if \[n \mid x^{d_1}+x^{d_2}+\dots+x^{d_k}-kx\] for all integers $x$, where $1=d_1<d_2<\dots<d_k=n$ are all the positive divisors of $n$. Suppose that $r$ is a Fermat prime (i.e. it is a prime of the form $2^{2^m}+1$ for an integer $m \ge 0$), $p$ is a prime divisor of an almost prime integer $n$, and $p \equiv 1 \pmod{r}$. Show that, with the above notation, $d_i \equiv 1 \pmod{r}$ for all $1 \le i \le k$. (An integer $n$ is called [i]square-free[/i] if it is not divisible by $d^2$ for any integer $d>1$.)

Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.

Let $n$ be a positive integer and let $k_0,k_1, \dots,k_{2n}$ be nonzero integers such that $k_0+k_1 +\dots+k_{2n}\neq 0$. Is it always possible to a permutation $(a_0,a_1,\dots,a_{2n})$ of $(k_0,k_1,\dots,k_{2n})$ so that the equation \begin{align*} a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_0=0 \end{align*} has not integer roots?

Let $\alpha$ be the radian measure of the smallest angle in a $3{-}4{-}5$ right triangle. Let $\beta$ be the radian measure of the smallest angle in a $7{-}24{-}25$ right triangle. In terms of $\alpha$, what is $\beta$? $ \textbf{(A) }\frac{\alpha}{3}\qquad \textbf{(B) }\alpha - \frac{\pi}{8}\qquad \textbf{(C) }\frac{\pi}{2} - 2\alpha \qquad \textbf{(D) }\frac{\alpha}{2}\qquad \textbf{(E) }\pi - 4\alpha\qquad $

A $2N\times2N$ board is covered by non-overlapping dominos of $1\times2$ size. A lame rook (which can only move one cell at a time, horizontally or vertically) has visited each cell once on its route across the board. Call a move by the rook longitudinal if it is a move from one cell of a domino to another cell of the same domino. What is: [list=a] [*]the maximum possible number of longitudinal moves? [*]the minimum possible number of longitudinal moves? [/list]