Let $a, b$ be positive integers. Prove that if $a^b + b^a \equiv 3 \pmod{4}$, then either $a+1$ or $b+1$ can't be written as the sum of two integer squares. [i](Proposed by Orestis Lignos, Greece)[/i]

Determine all possible digits $z$ for which $\underbrace{9...9}_{100}z\underbrace{0...0}_{100}9$ is a square number.

Tanya chose a natural number $X\le100$, and Sasha is trying to guess this number. He can select two natural numbers $M$ and $N$ less than $100$ and ask about $\gcd(X+M,N)$. Show that Sasha can determine Tanya's number with at most seven questions.

Sone of vertices of the infinite grid $\mathbb{Z}^{2}$ are missing. Let's take the remainder as a graph. Connect two edges of the graph if they are the same in one component and their other components have a difference equal to one. Call every connected component of this graph a [b]branch[/b]. Suppose that for every natural $n$ the number of missing vertices in the $(2n+1)\times(2n+1)$ square centered by the origin is less than $\frac{n}{2}$. Prove that among the branches of the graph, exactly one has an infinite number of vertices. Time allowed for this problem was 90 minutes.

Find the largest real constant $k$ for which the inequality \[(a^2 + 3)(b^2 + 3)(c^2 + 3)(d^2 + 3) + k(a - 1)(b - 1)(c - 1)(d - 1) \ge 0\] holds for all real numbers $a$, $b$, $c$, and $d$.

There are $128$ numbered seats arranged around a circle in a palaestra. The first person to enter the place would sit on seat number $1$. Since a contagious disease is infecting the people of the city, each person who enters the palaestra would sit on a seat whose distance is the longest to the nearest occupied seat. If there are several such seats, the newly entered person would sit on the seat with the smallest number. What is the number of the seat on which the $39$th person sits?

Let $a, d$ be integers such that $a,a + d, a+ 2d$ are all prime numbers larger than $3$. Prove that $d$ is a multiple of $6$.

Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.

Find all real numbers $ x$ which satisfy the following equation: $ [2x]\plus{}[3x]\plus{}[7x]\equal{}2008$. Note: $ [x]$ means the greatest integer less or equal than $ x$.

In a committee there are $n$ members. Each pair of members are either friends or enemies. Each committee member has exactly three enemies. It is also known that for each committee member, an enemy of his friend is automatically his own enemy. Find all possible value(s) of $n$

Determine all non-negative integral solutions $ (n_{1},n_{2},\dots , n_{14}) $ if any, apart from permutations, of the Diophantine Equation \[n_{1}^{4}+n_{2}^{4}+\cdots+n_{14}^{4}=1,599.\]

The smallest positive integer greater than 1 that leaves a remainder of 1 when divided by 4, 5, and 6 lies between which of the following pairs of numbers? $\textbf{(A) }2\text{ and }19\qquad\textbf{(B) }20\text{ and }39\qquad\textbf{(C) }40\text{ and }59\qquad\textbf{(D) }60\text{ and }79\qquad\textbf{(E) }80\text{ and }124$

Point $E$ is on side $AB$ of square $ABCD$. If $EB$ has length one and $EC$ has length two, then the area of the square is $\text{(A)}\ \sqrt{3} \qquad \text{(B)}\ \sqrt{5} \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 2\sqrt{3} \qquad \text{(E)}\ 5$

A circle with area $40$ is tangent to a circle with area $10$. Let R be the smallest rectangle containing both circles. The area of $R$ is $\frac{n}{\pi}$. Find $n$. [asy] defaultpen(linewidth(0.7)); size(120); real R = sqrt(40/pi), r = sqrt(10/pi); draw(circle((0,0), R)); draw(circle((R+r,0), r)); draw((-R,-R)--(-R,R)--(R+2*r,R)--(R+2*r,-R)--cycle);[/asy]

The Hawking Space Agency operates $n-1$ space flights between the $n$ habitable planets of the Local Galaxy Cluster. Each flight has a fixed price which is the same in both directions, and we know that using these flights, we can travel from any habitable planet to any habitable planet. In the headquarters of the Agency, there is a clearly visible board on a wall, with a portrait, containing all the pairs of different habitable planets with the total price of the cheapest possible sequence of flights connecting them. Suppose that these prices are precisely $1,2, ... , \binom{n}{2}$ monetary units in some order. prove that $n$ or $n-2$ is a square number.

Let $x,y,z$ be nonzero numbers, not necessarily real, such that \[(x-y)^2+(y-z)^2+(z-x)^2=24yz\] and \[\tfrac{x^2}{yz}+\tfrac{y^2}{zx}+\tfrac{z^2}{xy}=3.\] Compute $\tfrac{x^2}{yz}$.

An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions $f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$). A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.

Anna and Ben are playing with a permutation $p$ of length $2020$, initially $p_i = 2021 - i$ for $1\le i \le 2020$. Anna has power $A$, and Ben has power $B$. Players are moving in turns, with Anna moving first. In his turn player with power $P$ can choose any $P$ elements of the permutation and rearrange them in the way he/she wants. Ben wants to sort the permutation, and Anna wants to not let this happen. Determine if Ben can make sure that the permutation will be sorted (of form $p_i = i$ for $1\le i \le 2020$) in finitely many turns, if a) $A = 1000, B = 1000$ b) $A = 1000, B = 1001$ c) $A = 1000, B = 1002$ [i]Anton Trygub[/i]

Let $\omega$ be a circle with center $O$, and let $l$ be a line not intersecting $\omega$. $E$ is a point on $l$ such that $OE$ is perpendicular with $l$. Let $M$ be an arbitrary point on $M$ different from $E$. Let $A$ and $B$ be distinct points on the circle $\omega$ such that $MA$ and $MB$ are tangents to $\omega$. Let $C$ and $D$ be the foot of perpendiculars from $E$ to $MA$ and $MB$ respectively. Let $F$ be the intersection of $CD$ and $OE$. As $M$ moves, determine the locus of $F$.

Find the smallest number of squares on an $8\times8$ board that should be colored so that every $L$-tromino on the board contains at least one colored square.

$\vartriangle ABC$ is an isosceles triangle with $AB = AC$. Let $M$ be the midpoint of $BC$ and $E$ be the point on AC such that $AE :CE = 5 : 3$. Let $X$ be the intersection of $BE$ and $AM$. Given that the area of $\vartriangle CM X$ is $15$, find the area of $\vartriangle ABC$.

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Rays $AO$, $CO$ intersect sides $BC, BA$ in points $A_1, C_1$ respectively, $K$ is the projection of $O$ onto the segment $A_1C_1$, $M$ is the midpoint of $AC$. Prove that $\angle HMA = \angle BKC_1$. [i]Proposed by Anton Trygub[/i]

Given an $n$-tuple of numbers $(x_1,x_2,\dots ,x_n)$ where each $x_i=+1$ or $-1$, form a new $n$-tuple $$(x_1x_2,x_2x_3,x_3x_4,\dots ,x_nx_1),$$ and continue to repeat this operation. Show that if $n=2^k$ for some integer $k\ge 1$, then after a certain number of repetitions of the operation, we obtain the $n$-tuple $$(1,1,1,\dots ,1).$$

Let $n$ be a positive integer. Ilya and Sasha both choose a pair of different polynomials of degree $n$ with real coefficients. Lenya knows $n$, his goal is to find out whether Ilya and Sasha have the same pair of polynomials. Lenya selects a set of $k$ real numbers $x_1<x_2<\dots<x_k$ and reports these numbers. Then Ilya fills out a $2 \times k$ table: For each $i=1,2,\dots,k$ he writes a pair of numbers $P(x_i),Q(x_i)$ (in any of the two possible orders) intwo the two cells of the $i$-th column, where $P$ and $Q$ are his polynomials. Sasha fills out a similar table. What is the minimal $k$ such that Lenya can surely achieve the goal by looking at the tables? [i]Proposed by L. Shatunov[/i]

Let $\triangle ABC$ be a triangle and points $P, Q, R$ on the sides $AB, BC,$ and $CA$ respectively, such that: \[ \frac{AP}{AB} = \frac{BQ}{BC} = \frac{CR}{CA} = \frac{1}{n} \] for $n \in \mathbb{N}$. The segments $AQ$ and $CP$ intersect at $D$, the segments $BR$ and $AQ$ intersect at $E$, and the segments $BR$ and $CP$ intersect at $F$. Compute the ratio: \[ \frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle DEF)}. \]