Let $\mathcal F$ be the set of all $2013 \times 2013$ arrays whose entries are $0$ and $1$. A transformation $K : \mathcal F \to \mathcal F$ is defined as follows: for each entry $a_{ij}$ in an array $A \in \mathcal F$, let $S_{ij}$ denote the sum of all the entries of $A$ sharing either a row or column (or both) with $a_{ij}$. Then $a_{ij}$ is replaced by the remainder when $S_{ij}$ is divided by two. Prove that for any $A \in \mathcal F$, $K(A) = K(K(A))$. [i]Proposed by Aaron Lin[/i]

Let an acute angled triangle $ABC$ be given. Prove that the circles whose diameters are $AB$ and $AC$ have a point of intersection on $BC$.

Determine the values of $n$ for which an $n\times n$ square can be tiled with pieces of the type [img]http://oi53.tinypic.com/v3pqoh.jpg[/img].

Let $ABC$ be a non-isosceles acute triangle with $(I)$ be it's incircle. $D, E, F$ are the touchpoints of $(I)$ and $BC, CA, AB,$ respectively. $P$ is the perpendicular projection of $D$ on $EF.$ $DP$ intersects $(I)$ at the second point $K,L$ is the perpendicular projection of $A$ on $IK.$ $(LEC), (LFB) $ intersects $(I)$ the second time at $M, N,$ respectively. Prove that $M, N, P$ are collinear.

Find all real numbers $x,y,z$ such that satisfied the following equalities at same time: $\sqrt{x^3-y}=z-1 \wedge \sqrt{y^3-z}=x-1\wedge \sqrt{z^3-x}=y-1$

An integer $n$ such that $\Bigl\lfloor \frac{n}{9} \Bigr\rfloor$ is a three digit number with equal digits, and $\Bigl\lfloor \frac{n-172}{4} \Bigr\rfloor$ is a $4$ digit number with the digits $2,0,2,4$ in some order. What is the remainder when $n$ is divided by $100$?

Two circles $O_1$ and $O_2$ intersect at $M,N$. The common tangent line nearer to $M$ of the two circles touches $O_1,O_2$ at $A,B$ respectively. Let $C,D$ be the symmetric points of $A,B$ with respect to $M$ respectively. The circumcircle of triangle $DCM$ intersects circles $O_1$ and $O_2$ at points $E,F$ respectively which are distinct from $M$. Prove that the circumradii of the triangles $MEF$ and $NEF$ are equal.

Let $G$ be centroid and $R$ the circunradius of a triangle $ABC$. The extensions of $GA,GB,GC$ meet the circuncircle again at $D,E,F$. Prove that: $\frac{3}{R} \leq \frac{1}{GD} + \frac{1}{GE} + \frac{1}{GF} \leq \sqrt{3} \leq \frac{1}{AB} + \frac{1}{BC} + \frac{1}{CA}$

The unfolding of the lateral surface of a cone is a sector of angle $120^o$. The angles at the base of a pyramid constitute an arithmetic progression with a difference of $15^o$. The pyramid is inscribed in the cone. Consider a lateral face of the pyramid with the smallest area. Find the angle $\alpha$ between the plane of this face and the base.

Find all polynomials $p(x)$ satisfying $p(x^3+1)=p(x+1)^3$ for all $x$.

In the square $ABCD$ of side length $2$ the point $M$ is the midpoint of $BC$ and $P$ a point on $DC$. Determine the smallest value of $AP+PM$. [img]https://1.bp.blogspot.com/-WD8WXIE6DK4/XzcC9GYsa6I/AAAAAAAAMXg/vl2OrbAdChEYrRpemYmj6DiOrdOSqj_IgCLcBGAsYHQ/s178/2001%2BMohr%2Bp3.png[/img]

All positive integers whose binary representations (excluding leading zeroes) have at least as many $1$’s as $0$’s are put in increasing order. Compute the number of digits in the binary representation of the $200$th number.

Given several integers, the allowed operation is to replace two of them by their non-negative difference. The operation is repeated until only one number remains. If the initial numbers are $1, 2, … , 2010$, what can be the last remaining number?

Let $r$ be the inradius of an acute triangle. Prove that the sum of the distances from the orthocenter to the sides of the triangle does not exceed $3r$

For all $a$ and $b$, let $a\clubsuit b = 3a + 2b + 1$. Compute $c$ such that $(2c)\clubsuit (5\clubsuit (c + 3)) = 60$.

Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad \textbf{(E) }10$

Five runners, $P$, $Q$, $R$, $S$, $T$, have a race, and $P$ beats $Q$, $P$ beats $R$, $Q$ beats $S$, and $T$ finishes after $P$ and before $Q$. Who could NOT have finished third in the race? $\text{(A)}\ P\text{ and }Q \qquad \text{(B)}\ P\text{ and }R \qquad \text{(C)}\ P\text{ and }S \qquad \text{(D)}\ P\text{ and }T \qquad \text{(E)}\ P,S\text{ and }T$

Find all positive integers $n$ such that $n^4 - n^3 + 3n^2 + 5$ is a perfect square.

Five mathematicians find a bag of $100$ gold coins in a room. They agree to split up the coins according to the following plan: • The oldest person in the room proposes a division of the coins among those present. (No coin may be split.) Then all present, including the proposer, vote on the proposal. • If at least $50\%$ of those present vote in favor of the proposal, the coins are distributed accordingly and everyone goes home. (In particular, a proposal wins on a tie vote.) • If fewer than $50\%$ of those present vote in favor of the proposal, the proposer must leave the room, receiving no coins. Then the process is repeated: the oldest person remaining proposes a division, and so on. • There is no communication or discussion of any kind allowed, other than what is needed for the proposer to state his or her proposal, and the voters to cast their vote. Assume that each person is equally intelligent and each behaves optimally to maximize his or her share. How much will each person get?

Find all functions $ f:\mathbb{N} \rightarrow \mathbb{ N }_0 $ satisfy the following conditions: i) $ f(ab)=f(a)+f(b)-f(\gcd(a,b)), \forall a,b \in \mathbb{N} $ ii) For all primes $ p $ and natural numbers $ a $, $ f(a)\ge f(ap) \Rightarrow f(a)+f(p) \ge f(a)f(p)+1 $

An (unordered) partition $P$ of a positive integer $n$ is an $n$-tuple of nonnegative integers $P=(x_1,x_2,\cdots,x_n)$ such that $\sum_{k=1}^n kx_k=n$. For positive integer $m\leq n$, and a partition $Q=(y_1,y_2,\cdots,y_m)$ of $m$, $Q$ is called compatible to $P$ if $y_i\leq x_i$ for $i=1,2,\cdots,m$. Let $S(n)$ be the number of partitions $P$ of $n$ such that for each odd $m<n$, $m$ has exactly one partition compatible to $P$ and for each even $m<n$, $m$ has exactly two partitions compatible to $P$. Find $S(2010)$.

Find all positive integers $ k$ with the following property: There exists an integer $ a$ so that $ (a\plus{}k)^{3}\minus{}a^{3}$ is a multiple of $ 2007$.

The vertices of $\triangle ABC$ are $A = (0,0)$, $B = (0,420)$, and $C = (560,0)$. The six faces of a die are labeled with two $A$'s, two $B$'s, and two $C$'s. Point $P_1 = (k,m)$ is chosen in the interior of $\triangle ABC$, and points $P_2$, $P_3$, $P_4, \dots$ are generated by rolling the die repeatedly and applying the rule: If the die shows label $L$, where $L \in \{A, B, C\}$, and $P_n$ is the most recently obtained point, then $P_{n + 1}$ is the midpoint of $\overline{P_n L}$. Given that $P_7 = (14,92)$, what is $k + m$?

Let $m,n$ and $a_1,a_2,\dots,a_m$ be arbitrary positive integers. Ali and Mohammad Play the following game. At each step, Ali chooses $b_1,b_2,\dots,b_m \in \mathbb{N}$ and then Mohammad chosses a positive integers $s$ and obtains a new sequence $\{c_i=a_i+b_{i+s}\}_{i=1}^m$, where $$b_{m+1}=b_1,\ b_{m+2}=b_2, \dots,\ b_{m+s}=b_s$$ The goal of Ali is to make all the numbers divisible by $n$ in a finite number of steps. FInd all positive integers $m$ and $n$ such that Ali has a winning strategy, no matter how the initial values $a_1, a_2,\dots,a_m$ are. [hide=clarification] after we create the $c_i$ s, this sequence becomes the sequence that we continue playing on, as in it is our 'new' $a_i$[/hide] Proposed by Shayan Gholami

A regular pentagon $A_1A_2A_3A_4A_5$ with side length $s$ is given. At each point $A_i$, a sphere $K_i$ of radius $\frac{s}{2}$ is constructed. There are two spheres $K_1$ and $K_2$ each of radius $\frac{s}{2}$ touching all the five spheres $K_i.$ Decide whether $K_1$ and $K_2$ intersect each other, touch each other, or have no common points.