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?

Find all functions $ f:[0,1]\longrightarrow \mathbb{R} $ that are continuous and have the property that, for any continuous function $ g:[0,1]\longrightarrow [0,1] , $ the following equality holds. $$ \int_0^1 f\left( g(x) \right) dx =\int_0^1 g(x) dx $$

There are $12$ candies on the table, four of which are rare candies. Chad has a friend who can tell rare candies apart from regular candies, but Chad can’t. Chad’s friend is allowed to take four candies from the table, but may not take any rare candies. Can his friend always take four candies in such a way that Chad will then be able to identify the four rare candies? If so, describe a strategy. If not, prove that it cannot be done. Note that Chad does not know anything about how the candies were selected (e.g. the order in which they were selected). However, Chad and his friend may communicate beforehand.

Initially the numbers 1 through 2005 are marked. A finite set of marked consecutive integers is called a block if it is not contained in any larger set of marked consecutive integers. In each step we select a set of marked integers which does not contain the first or last element of any block, unmark the selected integers, and mark the same number of consecutive integers starting with the integer two greater than the largest marked integer. What is the minimum number of steps necessary to obtain 2005 single integer blocks?

The altitude from one vertex of a triangle, the bisector from the another one and the median from the remaining vertex were drawn, the common points of these three lines were marked, and after this everything was erased except three marked points. Restore the triangle. (For every two erased segments, it is known which of the three points was their intersection point.) (A. Zaslavsky)

Let $ABC$ be a triangle such that $AB < AC$. Let the excircle opposite to A be tangent to the lines $AB, AC$, and $BC$ at points $D, E$, and $F$, respectively, and let $J$ be its centre. Let $P$ be a point on the side $BC$. The circumcircles of the triangles $BDP$ and $CEP$ intersect for the second time at $Q$. Let $R$ be the foot of the perpendicular from $A$ to the line $FJ$. Prove that the points $P, Q$, and $R$ are collinear. (The [i]excircle[/i] of a triangle $ABC$ opposite to $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.) [i]Proposed by Bozhidar Dimitrov, Bulgaria[/i]

Denote by $\mathbb{V}$ the real vector space of all real polynomials in one variable, and let $\gamma :\mathbb{V}\to \mathbb{R}$ be a linear map. Suppose that for all $f,g\in \mathbb{V}$ with $\gamma(fg)=0$ we have $\gamma(f)=0$ or $\gamma(g)=0$. Prove that there exist $c,x_0\in \mathbb{R}$ such that \[ \gamma(f)=cf(x_0)\quad \forall f\in \mathbb{V}\]

Let $ABC$ be a triangle ($\angle A\neq 90^\circ$). $BE,CF$ are the altitudes of the triangle. The bisector of $\angle A$ intersects $EF,BC$ at $M,N$. Let $P$ be a point such that $MP\perp EF$ and $NP\perp BC$. Prove that $AP$ passes through the midpoint of $BC$. [i]Proposed by Iman Maghsoudi, Hooman Fattahi[/i]

Determine the possible interior angles of isosceles triangles that can be subdivided in two isosceles triangles with disjoint interior.