We define a [i]chessboard polygon[/i] to be a polygon whose sides are situated along lines of the form $ x \equal{} a$ or $ y \equal{} b$, where $ a$ and $ b$ are integers. These lines divide the interior into unit squares, which are shaded alternately grey and white so that adjacent squares have different colors. To tile a chessboard polygon by dominoes is to exactly cover the polygon by non-overlapping $ 1 \times 2$ rectangles. Finally, a [i]tasteful tiling[/i] is one which avoids the two configurations of dominoes shown on the left below. Two tilings of a $ 3 \times 4$ rectangle are shown; the first one is tasteful, while the second is not, due to the vertical dominoes in the upper right corner. [asy]size(300); pathpen = linewidth(2.5); void chessboard(int a, int b, pair P){ for(int i = 0; i < a; ++i) for(int j = 0; j < b; ++j) if((i+j) % 2 == 1) fill(shift(P.x+i,P.y+j)*unitsquare,rgb(0.6,0.6,0.6)); D(P--P+(a,0)--P+(a,b)--P+(0,b)--cycle); } chessboard(2,2,(2.5,0));fill(unitsquare,rgb(0.6,0.6,0.6));fill(shift(1,1)*unitsquare,rgb(0.6,0.6,0.6)); chessboard(4,3,(6,0)); chessboard(4,3,(11,0)); MP("\mathrm{Distasteful\ tilings}",(2.25,3),fontsize(12)); /* draw lines */ D((0,0)--(2,0)--(2,2)--(0,2)--cycle); D((1,0)--(1,2)); D((2.5,1)--(4.5,1)); D((7,0)--(7,2)--(6,2)--(10,2)--(9,2)--(9,0)--(9,1)--(7,1)); D((8,2)--(8,3)); D((12,0)--(12,2)--(11,2)--(13,2)); D((13,1)--(15,1)--(14,1)--(14,3)); D((13,0)--(13,3));[/asy] a) Prove that if a chessboard polygon can be tiled by dominoes, then it can be done so tastefully. b) Prove that such a tasteful tiling is unique.

Given an odd $n\in\mathbb N$. In an $n\times n$ chessboard, you may place many $2\times2$ squares. How many grids, at most, are covered by exactly one square?

The area of triangle $ XYZ$ is 8 square inches. Points $ A$ and $ B$ are midpoints of congruent segments $ \overline{XY}$ and $ \overline{XZ}$. Altitude $ \overline{XC}$ bisects $ \overline{YZ}$. What is the area (in square inches) of the shaded region? [asy]/* AMC8 2002 #20 Problem */ draw((0,0)--(10,0)--(5,4)--cycle); draw((2.5,2)--(7.5,2)); draw((5,4)--(5,0)); fill((0,0)--(2.5,2)--(5,2)--(5,0)--cycle, mediumgrey); label(scale(0.8)*"$X$", (5,4), N); label(scale(0.8)*"$Y$", (0,0), W); label(scale(0.8)*"$Z$", (10,0), E); label(scale(0.8)*"$A$", (2.5,2.2), W); label(scale(0.8)*"$B$", (7.5,2.2), E); label(scale(0.8)*"$C$", (5,0), S); fill((0,-.8)--(1,-.8)--(1,-.95)--cycle, white);[/asy] $ \textbf{(A)}\ 1\frac12\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 2\frac12\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 3\frac12$

On the sides $AB,BC,CD$ and $DA$ of a unit square $ABCD$ points $P,Q,R$ and $S$ are chosen respectively. It turned out that the perimeter of $PQRS$ is $2\sqrt{2}$. Find the sum of perpendiculars from $A,B,C,D$ to $SP,PQ,QR,RS$ respectively.

We consider the sequences strictely increasing $(a_0,a_1,...)$ of naturals which have the following property : For every natural $n$, there is exactly one representation of $n$ as $a_i+2a_j+4a_k$, where $i,j,k$ can be equal. Prove that there is exactly a such sequence and find $a_{2002}$

There are $n$ coins in the pile. Two players play a game by alternately performing a move. A move consists of taking $5,7$ or $11$ coins away from the pile. The player unable to perform a move loses the game. Which player - the one playing first or second - has the winning strategy if: (a) $n=2001$; (b) $n=5000$?

There are $5$ yellow pegs, $4$ red pegs, $3$ green pegs, $2$ blue pegs, and $1$ orange peg on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color? [asy] unitsize(20); dot((0,0)); dot((1,0)); dot((2,0)); dot((3,0)); dot((4,0)); dot((0,1)); dot((1,1)); dot((2,1)); dot((3,1)); dot((0,2)); dot((1,2)); dot((2,2)); dot((0,3)); dot((1,3)); dot((0,4));[/asy] $\text{(A)}\ 0\qquad\text{(B)}\ 1\qquad\text{(C)}\ 5!\cdot4!\cdot3!\cdot2!\cdot1!\qquad\text{(D)}\ \frac{15!}{5!\cdot4!\cdot3!\cdot2!\cdot1!}\qquad\text{(E)}\ 15!$

Let $a, b, c \geq 1$. Prove that $$\frac{1}{1+a^2}+\frac{1}{1+b^2}+\frac{1}{1+c^2}\geq\frac{1}{1+ab}+\frac{1}{1+bc}+\frac{1}{1+ca}.$$

We are given the finite sets $ X$, $ A_1$, $ A_2$, $ \dots$, $ A_{n \minus{} 1}$ and the functions $ f_i: \ X\rightarrow A_i$. A vector $ (x_1,x_2,\dots,x_n)\in X^n$ is called [i]nice[/i], if $ f_i(x_i) \equal{} f_i(x_{i \plus{} 1})$, for each $ i \equal{} 1,2,\dots,n \minus{} 1$. Prove that the number of nice vectors is at least \[ \frac {|X|^n}{\prod\limits_{i \equal{} 1}^{n \minus{} 1} |A_i|}. \]

Find the rational roots (if any) of the equation $$abx^2 + (a^2 + b^2 )x +1 = 0 , \,\,\,\, (a, b \in Z).$$

Solve the system of equations $$\begin{cases} \sqrt{\dfrac{y^2+x}{4x}}+\dfrac{y}{\sqrt{y^2+x}}=\dfrac{y^2}{4}\sqrt{\dfrac{4x}{y^2+x}} \\ \sqrt{x}+ \sqrt{x-y-1}=(y+1)(\sqrt{x}- \sqrt{x-y-1}) \end{cases}$$

Prove that if $ n $ is a natural number and the angle $ \alpha $ is not a multiple of $ \frac{180^{\circ}}{2^n} $, then $$\frac{1}{\sin 2\alpha} + \frac{1}{\sin 4\alpha} + \frac{1}{\sin 8\alpha} + ... + = ctg \alpha - ctg 2^n \alpha.$$

Let $\omega_1$ and $\omega_2$ be two circles such that the center $O$ of $\omega_2$ lies in $\omega_1$. Let $C$ and $D$ be the two intersection points of the circles. Let $A$ be a point on $\omega_1$ and let $B$ be a point on $\omega_2$ such that $AC$ is tangent to $\omega_2$ in C and BC is tangent to $\omega_1$ in $C$. The line segment $AB$ meets $\omega_2$ again in $E$ and also meets $\omega_1$ again in F. The line $CE$ meets $\omega_1$ again in $G$ and the line $CF$ meets the line $GD$ in $H$. Prove that the intersection point of $GO$ and $EH$ is the center of the circumcircle of the triangle $DEF$.

For each non-constant integer polynomial $P(x)$, let’s define $$M_{P(x)} = \underset{x\in [0,2021]}{\max} |P(x)|.$$ 1. Find the minimum value of $M_{P(x)}$ when deg $P(x) = 1$. 2. Suppose that $P(x) \in Z[x]$ when deg $P(x) = n$ and $2 \le n \le 2022$. Prove that $M_{P(x)} \ge 1011$.

Triangle $ ABC$ has $ AC \equal{} 450$ and $ BC \equal{} 300$. Points $ K$ and $ L$ are located on $ \overline{AC}$ and $ \overline{AB}$ respectively so that $ AK \equal{} CK$, and $ \overline{CL}$ is the angle bisector of angle $ C$. Let $ P$ be the point of intersection of $ \overline{BK}$ and $ \overline{CL}$, and let $ M$ be the point on line $ BK$ for which $ K$ is the midpoint of $ \overline{PM}$. If $ AM \equal{} 180$, find $ LP$.

Squares $ABKL$, $BCMN$, $CAOP$ are drawn externally on the sides of a triangle $ABC$. The line segments $KL$, $MN$, $OP$, when extended, form a triangle $A'B'C'$. Find the area of $A'B'C'$ if $ABC$ is an equilateral triangle of side length $2$.

The set $ \{3,6,9,10\}$ is augmented by a fifth element $ n$, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of $ n$? $ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 19 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 26$

Solve the following system (over the real numbers): \[\begin{cases}5x+5y+5xy-2xy^2-2x^2y=20 &\\ 3x+3y+3xy+xy^2+x^2y=23&\end{cases}\]

Convex polygons $P_1$ and $P_2$ are drawn in the same plane with $n_1$ and $n_2$ sides, respectively, $n_1 \le n_2$. If $P_1$ and $P_2$ do not have any line segment in common, then the maximum number of intersections of $P_1$ and $P_2$ is: $\textbf{(A)}\ 2n_1 \qquad\textbf{(B)}\ 2n_2 \qquad\textbf{(C)}\ n_1n_2 \qquad\textbf{(D)}\ n_1+n_2 \qquad\textbf{(E)}\ \text{none of these} $

For a positive integer $ n $, define $ \phi(n) $ as the number of positive integers less than or equal to $ n $ that are relatively prime to $ n $. Find the sum of all positive integers $ n $ such that $ \phi(n) = 20 $.

Does there exist a function $f: \mathbb{R}\rightarrow \mathbb{R}$ such that: $f(f(x))=-x$, for all $x\in \mathbb{R}$?

The bisectors $AD{}$ and $BE{}$ were drawn in the triangle $ABC{}$ and they intersected at point $I{}.$ Then everything was erased, leaving only the points $D{}$ and $E{}.$ Find the set of possible positions of the point $I{}.$ [i]Proposed by M. Didin[/i]

Find the number of real solutions of the equation $a e^x = x^3$, where $a$ is a real parameter.

Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$ [i]Michael Ren and Ankan Bhattacharya, USA[/i]

A hexagon $ABCDEF$ is inscribed in a circle. Let $P, Q, R, S$ be intersections of $AB$ and $DC$, $BC$ and $ED$, $CD$ and $FE$, $DE$ and $AF$, then $\angle BPC=50^{\circ}$, $\angle CQD=45^{\circ}$, $\angle DRE=40^{\circ}$, $\angle ESF=35^{\circ}$. Let $T$ be an intersection of $BE$ and $CF$. Find $\angle BTC$.