Some $1\times 2$ dominoes, each covering two adjacent unit squares, are placed on a board of size $n\times n$ such that no two of them touch (not even at a corner). Given that the total area covered by the dominoes is $2008$, find the least possible value of $n$.

A possitive integer is called [i]special[/i] if all its decimal digits are equal and it can be represented as the sum of squares of three consecutive odd integers. (a) Find all $4$-digit [i]special[/i] numbers (b) Are there $2000$-digit [i]special[/i] numbers?

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

Does there exist a function $f : Z^+ \to Z^+$ such that $f(f(n)) = 2n$ for all positive integers $n$? Justify your answer, and if the answer is yes, give an explicit construction.

Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(xf(y)+y)+f(-f(x))=f(yf(x)-y)+y\] for all $x,y\in\mathbb{R}$

Let $\Delta ABC$ be a scalene triangle. Points $D,E$ lie on side $\overline{AC}$ in the order, $A,E,D,C$. Let the parallel through $E$ to $BC$ intersect $\odot (ABD)$ at $F$, such that, $E$ and $F$ lie on the same side of $AB$. Let the parallel through $E$ to $AB$ intersect $\odot (BDC)$ at $G$, such that, $E$ and $G$ lie on the same side of $BC$. Prove, Points $D,F,E,G$ are concyclic

Does there exist a sequence of natural numbers $a_1,a_2,\ldots$ such that the number $a_i+a_j$ has an even number of different prime divisors for any two different natural indices $i{}$ and $j{}$? [i]From the folklore[/i]

There are $5$ vertices labelled $1,2,3,4,5$. For any two pairs of vertices $u, v$, the edge $uv$ is drawn with probability $1/2$. If the probability that the resulting graph is a tree is given by $\dfrac{p}{q}$ where $p, q$ are coprime, then find the value of $q^{1/10} + p$.

There´s a ping pong tournament with $n\geq 3$ participants that we´ll call $1, 2, \dots n$. The tournament rules are the following ones: at the start, all the players form a line, ordered from $1$ to $n$. Players $1$ and $2$ play the first match. The winner is at the beginning of the line and the loser is placed behind the last person in the line.In the next play, the two who at that moment are the first two in line face each other, the winner is first in line and the loser goes to the end of the line, just behind the last loser. And so on. After $N$ matches, the tournament ends.Player number $1$ won $a_1$ matches, player number $2$ won $a_2$, and so on till player $n$, that has won $a_n$ matches (it is trivial that $a_1+a_2+\dots+a_n=N)$.Determine how many games each player has lost, based on $a_1, a_2, \dots , a_n$

In acute triangle $ABC$, let $\odot O$ be its circumcircle, $\odot I$ be its incircle. Tangents at $B,C$ to $\odot O$ meet at $L$, $\odot I$ touches $BC$ at $D$. $AY$ is perpendicular to $BC$ at $Y$, $AO$ meets $BC$ at $X$, and $OI$ meets $\odot O$ at $P,Q$. Prove that $P,Q,X,Y$ are concyclic if and only if $A,D,L$ are collinear.

Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S$, the probability that it is divisible by $9$ is $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Determine the sum of angles $A,B,$ where $0^\circ \leq A,B, \leq 180^\circ$ and \[ \sin A + \sin B = \sqrt{\frac{3}{2}}, \cos A + \cos B = \sqrt{\frac{1}{2}} \]

There are $ n $ natural numbers ($ n\geq 2 $) whose sum is equal to their product. Prove that this common value does not exceed $2n$.

There are sixteen buildings all on the same side of a street. How many ways can we choose a nonempty subset of the buildings such that there is an odd number of buildings between each pair of buildings in the subset? [i]Proposed by Yiming Zheng

Per and Kari each have $n$ pieces of paper. They both write down the numbers from $1$ to $2n$ in an arbitrary order, one number on each side. Afterwards, they place the pieces of paper on a table showing one side. Prove that they can always place them so that all the numbers from $1$ to $2n$ are visible at once.

Let $ABCD$ be a trapezoid of bases $AB$ and $CD$ . Let $O$ be the intersection point of the diagonals $AC$ and $BD$. If the area of the triangle $ABC$ is $150$ and the area of the triangle $ACD$ is $120$, calculate the area of the triangle $BCO$.

Find the smallest constant $C > 1$ such that the following statement holds: for every integer $n \geq 2$ and sequence of non-integer positive real numbers $a_1, a_2, \dots, a_n$ satisfying $$\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n} = 1,$$ it's possible to choose positive integers $b_i$ such that (i) for each $i = 1, 2, \dots, n$, either $b_i = \lfloor a_i \rfloor$ or $b_i = \lfloor a_i \rfloor + 1$, and (ii) we have $$1 < \frac{1}{b_1} + \frac{1}{b_2} + \cdots + \frac{1}{b_n} \leq C.$$ (Here $\lfloor \bullet \rfloor$ denotes the floor function, as usual.) [i]Merlijn Staps[/i]

Given $\vartriangle ABC$. Choose two points $P, Q$ on $AB, AC$ such that $BP = CQ$. Let $M, T$ be the midpoints of $BC, PQ$. Show that $MT$ is parallel to the angle bisevtor of $\angle BAC$ [img]http://4.bp.blogspot.com/-MgMtdnPtq1c/XnSHHFl1LDI/AAAAAAAALdY/8g8541DnyGo_Gqd19-7bMBpVRFhbXeYPACK4BGAYYCw/s1600/imoc2017%2Bg1.png[/img]

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.

The points $X, \, \, Y$are selected on the sides $AB$ and $AD$ of the convex quadrilateral $ABCD$, respectively. Find the ratio $AX \, \,: \, \, BX$ if you know that $CX || DA$, $DX || CB$, $BY || CD$ and $CY || BA$.

Three people, John, Macky, and Rik, play a game of passing a basketball from one to another. Find the number of ways of passing the ball starting with Macky and reaching Macky again at the end of the seventh pass.

Suppose $x$ is a real number such that $\sin(1 + \cos^2 x + \sin^4 x) = \tfrac{13}{14}$. Compute $\cos(1 + \sin^2 x + \cos^4 x)$.

The sequence $(a_n)_{n\in\mathbb{N}}$ is defined by $a_1=3$ and $$a_n=a_1a_2\cdots a_{n-1}-1$$ Show that there exist infinitely many prime number that divide at least one number in this sequences

Let $D, E$, and $F$ be the points at which the incircle, $\omega$, of $\vartriangle ABC$ is tangent to $BC$, $CA$, and $AB$, respectively. $AD$ intersects $\omega$ again at $T$. Extend rays $T E$, $T F$ to hit line $BC$ at $E'$, $F'$, respectively. If $BC = 21$, $CA = 16$, and $AB = 15$, then find $\left|\frac{1}{DE'} -\frac{1}{DF'}\right|$.

Consider a pyramid whose base is a $2013$-sided polygon. On each face of the pyramid the number $0$ is written. The following operation is carried out: a vertex is chosen from the pyramid and add or subtract $1$ from all the faces that contain that vertex. It's possible, after repeating a finite number of times the previous procedure, that all the faces of the pyramid have the number $1$ written?