Markers in the colors violet, cyan, octarine and gamma were placed on all fields of a $41\times 5$ chessboard. Show that there are four squares of the same color that form the vertices of a rectangle whose edges are parallel to those of the board.

There are $36$ cards in a deck arranged in the sequence spades, clubs, hearts, diamonds, spades, clubs, hearts, diamonds, etc. Somebody took part of this deck off the top, turned it upside down, and cut this part into the remaining part of the deck (i.e. inserted it between two consecutive cards). Then four cards were taken off the top, then another four, etc. Prove that in any of these sets of four cards, all the cards are of different suits. (A Merkov, Moscow)

Let $k$ be a positive integer with the following property: For every subset $A$ of $\{1,2,\ldots, 25\}$ with $|A|=k$, we can find distinct elements $x$ and $y$ of $A$ such that $\tfrac23\leq\tfrac xy\leq\tfrac 32$. Find the smallest possible value of $k$.

Calculate the volume $ V $ of tetrahedron $ ABCD $ given the length $ d $ of edge $ AB $ and the area $ S $ of the projection of the tetrahedron on the plane perpendicular to the line $ AB $.

Let $P(x)=a_d x^d+\dots+a_1 x+a_0$ be a non-constant polynomial with non-negative integer coefficients having $d$ rational roots.Prove that $$\text{lcm} \left(P(m),P(m+1),\dots,P(n) \right)\geq m \dbinom{n}{m}$$ for all $n>m$ [i](Navid Safaei, Iran)[/i]

Let $n$ be a positive integer. $n$ letters are written around a circle, each $A$, $B$, or $C$. When the letters are read in clockwise order, the sequence $AB$ appears $100$ times, the sequence $BA$ appears $99$ times, and the sequence $BC$ appears $17$ times. How many times does the sequence $CB$ appear?

Let $\vartriangle ABC$ be a triangle with $\angle ABC > \angle BCA \ge 30^o$ . The angle bisectors of $\angle ABC$ and $\angle BCA$ intersect $CA$ and $AB$ at $D$ and $E$ respectively, and $BD$ and $CE$ intersect at $P$. Suppose that $P D = P E$ and the incircle of $\vartriangle ABC$ has unit radius. What is the maximum possible length of $BC$?

Let $n$ be a positive integer. Ana and Banana are playing the following game: First, Ana arranges $2n$ cups in a row on a table, each facing upside-down. She then places a ball under a cup and makes a hole in the table under some other cup. Banana then gives a finite sequence of commands to Ana, where each command consists of swapping two adjacent cups in the row. Her goal is to achieve that the ball has fallen into the hole during the game. Assuming Banana has no information about the position of the hole and the position of the ball at any point, what is the smallest number of commands she has to give in order to achieve her goal?

Let $ABC$ be an acute-angled triangle with height intersection $H$. Let $G$ be the intersection of parallel of $AB$ through $H$ with the parallel of $AH$ through $B$. Let $I$ be the point on the line $GH$, so that $AC$ bisects segment $HI$. Let $J$ be the second intersection of $AC$ and the circumcircle of the triangle $CGI$. Show that $IJ = AH$

The diagram shows a $20$ by $20$ square $ABCD$. The points $E$, $F$, and $G$ are equally spaced on side $BC$. The points $H$, $I$, $J$, and $K$ on side $DA$ are placed so that the triangles $BKE$, $EJF$, $FIG$, and $GHC$ are isosceles. Points $L$ and $M$ are midpoints of the sides $AB$ and $CD$, respectively. Find the total area of the shaded regions. [asy] size(175); defaultpen(linewidth(0.8)); real r=20/8; pair x[]; draw(origin--(0,20)--(20,20)--(20,0)--cycle); string label[]={"B","K","E","J","F","I","G","H","C"}; for(int i=1;i<=7;i=i+1) { if(floor(i/2)==i/2) { x[i]=(i*r,0); label("$"+label[i]+"$",x[i],S); } else { x[i]=(i*r,20); label("$"+label[i]+"$",x[i],N); } } filldraw(origin--x[1]--x[2]--x[3]--x[4]--x[5]--x[6]--x[7]--(20,0)--(20,10)--(0,10)--cycle,gray); label("$B$",origin,SW); label("$C$",(20,0),SE); label("$A$",(0,20),NW); label("$D$",(20,20),NE); label("$M$",(20,10),E); label("$L$",(0,10),W); [/asy]

Let $ABC$ be a triangle and $D$ point on side $BC$ such that $AD$ is angle bisector of angle $\angle BAC$. Let $E$ be the intersection of the side $AB$ with circle $\omega_1$ which has diameter $CD$ and let $F$ be the intersection of the side $AC$ with circle $\omega_2$ which has diameter $BD$. Suppose that there exist points $P\in\omega_1$ and $Q\in\omega_2$ such that $E, P, Q$ and $F$ are collinear and on this order. Prove that $AD, BQ$ and $CP$ are concurrent. [i]Proposed by Dorlir Ahmeti, Kosovo and Noah Walsh, U.S.A.[/i]

In an acute traingle $ABC$ with $AB< BC$ let $BH_b$ be its altitude, and let $O$ be the circumcenter. A line through $H_b$ parallel to $CO$ meets $BO$ at $X$. Prove that $X$ and the midpoints of $AB$ and $AC$ are collinear.

A gradian is a unit of measurement of angles much like degrees, except that there are $100$ gradians in a right angle. Suppose that the number of gradians in an interior angle of a regular polygon with $m$ sides equals the number of degrees in an interior angle of a regular polygon with $n$ sides. Compute the number of possible distinct ordered pairs $(m, n)$.

Find the smallest odd integer $ k$ such that: for every $ 3\minus{}$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$.

Let $ABC$ be a right-angled triangle with $\angle A = 90^{\circ}$. Let $K$ be the midpoint of $BC$, and let $AKLM$ be a parallelogram with centre $C$. Let $T$ be the intersection of the line $AC$ and the perpendicular bisector of $BM$. Let $\omega_1$ be the circle with centre $C$ and radius $CA$ and let $\omega_2$ be the circle with centre $T$ and radius $TB$. Prove that one of the points of intersection of $\omega_1$ and $\omega_2$ is on the line $LM$. [i]Proposed by Greece[/i]

Find all $a\in\mathbb{N}$ such that exists a bijective function $g :\mathbb{N} \to \mathbb{N}$ and a function $f:\mathbb{N}\to\mathbb{N}$, such that for all $x\in\mathbb{N}$, $$f(f(f(...f(x)))...)=g(x)+a$$ where $f$ appears $2009$ times. [i](tatari/nightmare)[/i]

It was noted that during one day in a town, each person made at most one phone call. Prove that the people in the town can be divided into three groups such that no two persons in the same group talked by phone that day.

A natural number of five digits is called [i]Ecuadorian [/i]if it satisfies the following conditions: $\bullet$ All its digits are different. $\bullet$ The digit on the far left is equal to the sum of the other four digits. Example: $91350$ is an Ecuadorian number since $9 = 1 + 3 + 5 + 0$, but $54210$ is not since $5 \ne 4 + 2 + 1 + 0$. Find how many Ecuadorian numbers exist.

Let $ABC$ be a triangle with orthocenter $H$. Let $\ell_b, \ell_c$ be the reflection of lines $AB$ and $AC$ about $AH$ respectively. Suppose $\ell_b$ intersect $CH$ at $P$, and $\ell_c$ intersect $BH$ at $Q$. Prove that $AH, PQ, BC$ are concurrent. [i]Proposed by Ivan Chan Kai Chin[/i]

Let $a$ and $b$ be positive integers. Prove that the numbers $an^2+b$ and $a(n+1)^2+b$ are both perfect squares only for finitely many integers $n$.

A point $(x,y)$ is a [i]lattice point[/i] if $x,y\in\Bbb Z$. Let $E=\{(x,y):x,y\in\Bbb Z\}$. In the coordinate plane, $P$ and $Q$ are both sets of points in and on the boundary of a convex polygon with vertices on lattice points. Let $T=P\cap Q$. Prove that if $T\ne\emptyset$ and $T\cap E=\emptyset$, then $T$ is a non-degenerate convex quadrilateral region.

Let be given an integer $n\ge 2$ and a positive real number $p$. Find the maximum of \[\displaystyle\sum_{i=1}^{n-1} x_ix_{i+1},\] where $x_i$ are non-negative real numbers with sum $p$.

The inscribed circle $\Omega$ of triangle $ABC$ touches the sides $AB$ and $AC$ at points $K$ and $ L$, respectively. The line $BL$ intersects the circle $\Omega$ for the second time at the point $M$. The circle $\omega$ passes through the point $M$ and is tangent to the lines $AB$ and $BC$ at the points $P$ and $Q$, respectively. Let $N$ be the second intersection point of circles $\omega$ and $\Omega$, which is different from $M$. Prove that if $KM \parallel AC$ then the points $P, N$ and $L$ lie on one line.

Submit a positive integer $n$ less than $10^5$. Let the sum of the valid submissions from all teams to this question be $S$. If you submit an invalid answer, you will receive $0$ points. Otherwise, your score will be $ \max \left(0,\lfloor 25 - \frac{|S'-n|}{10} \rfloor \right)$ , where $S'$ is the sum of the squares of the digits of $S$.

Let $\vartriangle AMO$ be an equilateral triangle. Let $U$ and $G$ lie on side $AM$, and let $S$ and $N$ lie on side $AO$ such that $AU =UG = GM$ and $AS = SN = NO$. Find the value of $\frac{[MONG]}{[U S A]}$