A game consists of a grid of $4\times 4$ and tiles of two colors (Yellow and White). A player chooses a type of token and gives it to the second player who places it where he wants, then the second player chooses a type of token and gives it to the first who places it where he wants, They continue in this way and the one who manages to form a line with three tiles of the same color wins (horizontal, vertical or diagonal and regardless of whether it is the tile you started with or not). Before starting the game, two yellow and two white pieces are already placed as shows the figure below. [img]https://cdn.artofproblemsolving.com/attachments/b/5/ba11377252c278c4154a8c3257faf363430ef7.png[/img] Yolanda and Xinia play a game. If Yolanda starts (choosing the token and giving it to Xinia for this to place) indicate if there is a winning strategy for either of the two players and, if any, describe the strategy.

Given a rectangle $EFGH$ with $EF=3$ and $FG=2010,$ mark a point $P$ on $FG$ such that $FP=4.$ A laser beam is shot from $E$ to $P,$ which then reflects off $FG,$ then $EH,$ then $FG,$ etc. Once it reaches some point of $GH,$ the beam is absorbed; it stops reflecting. How far does the beam travel?

Let $\triangle ABC$ be scalene, with $BC$ as the largest side. Let $D$ be the foot of the perpendicular from $A$ on side $BC$. Let points $K,L$ be chosen on the lines $AB$ and $AC$ respectively, such that $D$ is the midpoint of segment $KL$. Prove that the points $B,K,C,L$ are concyclic if and only if $\angle BAC=90^{\circ}$.

Let $P(x)$ be a polynomial with integer coefficients such that \[P(\sqrt{2}\sin x) = -P(\sqrt{2}\cos x)\] for all real numbers $x$. What is the largest prime that must divide $P(2019)$?

Let $a$, $b$, $c$ be given integers, $a>0$, $ac-b^2=p$ a squarefree positive integer. Let $M(n)$ denote the number of pairs of integers $(x, y)$ for which $ax^2 +bxy+cy^2=n$. Prove that $M(n)$ is finite and $M(n)=M(p^{k} \cdot n)$ for every integer $k \ge 0$.

For positive integers $j\le n$, prove that $$\sum_{k=j}^n\binom{2n}{2k}\binom kj=\frac{n\cdot4^{n-j}}j\binom{2n-j-1}{j-1}.$$ [i]Proposed by Ángel Plaza[/i]

Integers $a,b$ and $c$ satisfy the equality $a+b+c=0$. Denote $S=ab+bc+ac$, $A=a^2+a+1$, $B=b^2+b+1$ and $C=c^2+c+1$. Prove that the number $(S+A)(S+B)(S+C)$ is a perfect square.

Let $ABC$ be a scalene triangle with circumcircle $\Gamma$, and let $D$,$E$,$F$ be the points where its incircle meets $BC$, $AC$, $AB$ respectively. Let the circumcircles of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$ meet $\Gamma$ a second time at $X,Y,Z$ respectively. Prove that the perpendiculars from $A,B,C$ to $AX,BY,CZ$ respectively are concurrent. [i]Proposed by Michael Kural[/i]

The inscribed circle of triangle $ABC$ is tangent to sides $BC$, $AC$ and $AB$ at points $D$, $E$ and $F$, respectively. Let $E'$ be the reflection of point $E$ across line $DF$, and $F'$ be the reflection of point $F$ across line $DE$. Let line $EF$ intersect the circumcircle of triangle $AE'F'$ at points $X$ and $Y$. Prove that $DX=DY$. [i]Proposed by Márton Lovas, Budapest[/i]

(A.Zaslavsky, B.Frenkin, 10--11) In a triangle, one has drawn perpendicular bisectors to its sides and has measured their segments lying inside the triangle. a) All three segments are equal. Is it true that the triangle is equilateral? b) Two segments are equal. Is it true that the triangle is isosceles? c) Can the segments have length 4, 4 and 3?

Find the surface generated by the solutions of \[ \frac {dx}{yz} = \frac {dy}{zx} = \frac{dz}{xy}, \] which intersects the circle $y^2+ z^2 = 1, x = 0.$

In the table are written the positive integers $1, 2,3,...,2018$. John and Mary have the ability to make together the following move: [i]They select two of the written numbers in the table, let $a,b$ and they replace them with the numbers $5a-2b$ and $3a-4b$.[/i] John claims that after a finite number of such moves, it is possible to triple all the numbers in the table, e.g. have the numbers: $3, 6, 9,...,6054$. Mary thinks a while and replies that this is not possible. Who of them is right?

Find all values of positive integers $a$ and $b$ such that it is possible to put $a$ ones and $b$ zeros in every of vertices in polygon with $a+b$ sides so it is possible to rotate numbers in those vertices with respect to primary position and after rotation one neighboring $0$ and $1$ switch places and in every other vertices other than those two numbers remain the same.

If $a+3b = 9$ and $a+11b =21$, what is the missing coefficient in the expression $2a+\underline{?}b = 27$? $\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 9\qquad\textbf{(D) } 12\qquad\textbf{(E) } 14$

Find all positive integers $m$ with the next property: If $d$ is a positive integer less or equal to $m$ and it isn't coprime to $m$ , then there exist positive integers $a_{1}, a_{2}$,. . ., $a_{2019}$ (where all of them are coprimes to $m$) such that $m+a_{1}d+a_{2}d^{2}+\cdot \cdot \cdot+a_{2019}d^{2019}$ is a perfect power.

A convex polygon has the property that its vertices are coloured by three colors, each colour occurring at least once and any two adjacent vertices having different colours. Prove that the polygon can be divided into triangles by diagonals, no two of which intersect in the [b]interior[/b] of the polygon, in such a way that all the resulting triangles have vertices of all three colours.

Let $D_1$, $D_2$, ..., $D_n$ be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most $2003$ discs $D_i$. Prove that there exists a disc $D_k$ which intersects at most $7\cdot 2003 - 1 = 14020$ other discs $D_i$.

Let $ABCD$ be a square. Consider a variable point $P$ inside the square for which $\angle BAP \ge 60^\circ.$ Let $Q$ be the intersection of the line $AD$ and the perpendicular to $BP$ in $P$. Let $R$ be the intersection of the line $BQ$ and the perpendicular to $BP$ from $C$. [list] [*] [b](a)[/b] Prove that $|BP|\ge |BR|$ [*] [b](b)[/b] For which point(s) $P$ does the inequality in [b](a)[/b] become an equality?[/list]

Harry has $3$ sisters and $5$ brothers. His sister Harriet has $X$ sisters and $Y$ brothers. What is the product of $X$ and $Y$? $ \text{(A)}\ 8\qquad\text{(B)}\ 10\qquad\text{(C)}\ 12\qquad\text{(D)}\ 15\qquad\text{(E)}\ 18 $

Let $a, b, c, d$ be non-negative reals such that $\frac{1}{a+3}+\frac{1}{b+3}+\frac{1}{c+3}+\frac{1}{d+3}=1$. Show that there exists a permutation $(x_1, x_2, x_3, x_4)$ of $(a, b, c, d)$, such that $$x_1x_2+x_2x_3+x_3x_4+x_4x_1 \geq 4.$$

For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\{m, m+4, m+10, n+1, n+2, 2n\}$ are equal to $n$. What is $m+n$? $\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$

Let $a, b, x,$ and $y$ be real numbers with $a>4$ and $b>1$ such that $$\frac{x^2}{a^2}+\frac{y^2}{a^2-16}=\frac{(x-20)^2}{b^2-1}+\frac{(y-11)^2}{b^2}=1.$$ Find the least possible value of $a+b.$

Two right triangles are placed next to each other to form a quadrilateral as shown. What is the perimeter of the quadrilateral? [asy] size(4cm); fill((-5,0)--(0,12)--(0,6)--(8,0)--cycle, gray+opacity(0.3)); draw((0,0)--(0,12)--(-5,0)--cycle); draw((0,0)--(8,0)--(0,6)); label("5", (-2.5,0), S); label("13", (-2.5,6), dir(140)); label("6", (0,3), E); label("8", (4,0), S); [/asy] [i]Proposed by Nathan Xiong[/i]

A $20 \times 20 \times 20$ cube is composed of $2000$ bricks of size $2 \times 2 \times 1$ . Prove that it is possible to pierce the cube with a needle so that the needle passes through the cube without passing through a brick . (A . Andjans , Riga)

The function $\mu: \mathbb{N}\to \mathbb{C}$ is defined by \[\mu(n) = \sum^{}_{k \in R_{n}}\left( \cos \frac{2k\pi}{n}+i \sin \frac{2k\pi}{n}\right),\] where $R_{n}=\{ k \in \mathbb{N}\vert 1 \le k \le n, \gcd(k, n)=1 \}$. Show that $\mu(n)$ is an integer for all positive integer $n$.