Let ${a_n}$ be the sequence of rational numbers defined by $a_0 = 2021,$ and $a_{n+1}=a_{n}+\frac{2}{a_{n}},$ for all $n \geq 0.$ Can any $a_{n}, n \geq 1,$ be a square of a rational number?

Let $a, b, c, d$ be the roots of the equation $x^4+x+1=0.$ Let $a^5+2a+1, b^5+2b+1, c^5+2c+1, d^5+2d+1$ be roots of the equation $x^4+px^3+qx^2+rx+s=0.$ Find the value of $p+2q+4r+8s.$

Four non-overlapping families dine at a restaurant together. Each family consists of one father, one mother and three children. There are five tables, which are red, blue, yellow, green and brown respectively. Every table can accomodate four people, and every person must sit at some table. How many ways are there to assign the 20 people to tables so that each child sits at the same table as at least one of his/her parents? (We do not distinguish between the four seats at the same table.)

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O.$ Diagonals $AC$ and $BD$ meet at $E.$ $F$ and $G$ are points on segments $AB$ and $CD$ respectively. Suppose $AF=15,$ $FB=13,$ $BE=30,$ $ED=13,$ $DG=7.5,$ and $GC=6.5.$ Let $P$ be a point such that $PF\perp AB$ and $PG\perp CD.$ Find $\frac{PE}{PO}.$

Let $f(n)=\left|\binom{n}{0}+\binom{n}{3}+\dots +\binom{n}{3[\frac{n}{3}]}-\frac{2^n}{3}\right|,$ where $[x]$ is the greatest integer not exceeding $x.$ Find $$f(1)+f(2)+\dots+f(2021).$$

There is a set of $n$ $01$-sequences of length $200.$ Every pair of $01$-sequences differ at least at $101$ positions. (For example, the two $01$-sequences of length $6,$ $111100$ and $010001$ differ at four positions, $1$st, $3$rd, $4$th and $6$th positions, counting from the left.) Is it possible that $n \geq 101?

Let $ABC$ be a triangle. Let $M$ be the midpoint of $BC,$ and let $G$ be the centroid of $\triangle ABC.$ Let $D$ be a point on the segment $GM.$ A straight line passing through $D$ meets the sides $AB$ and $AC$ at $E$ and $F$ respectively (with $E, F \neq A$). Show that $$[BEGF]+[CFGD]\geq \frac{4}{9} [ABC].$$When does the equality hold? (Here $[WXYZ]$ is the area of the polygon $WXYZ,$ etc.)

Let $P(x)$ be a polynomial with integer coefficients. Define a sequence ${a_n}$ by $a_0 = 0$ and $a_{n} = P(a_{n-1})$ for all $n \geq 1.$ Prove that if there exists a positive integer $m$ for which $a_m = 0,$ then $a_1 = 0$ or $a_2 = 0.$

Let $S$ be the set of all integers of the form $x^2+3xy+8y^2.$ where $x$ and $y$ are integers. (a) Show that if $u$ and $v$ are in $S,$ then so is $uv.$ (b) Can an integer of the form $23k + 7,$ with $k$ an integer, belong to $S$?

In a chess tournament there are $100$ players. On each day of the tournament, each player is designated to be ‘white’, ‘black’ or ‘idle’, and each ‘white’ player will play a game against every ‘black’ player. (You may assume that all games fixed for the day can be finished within that day.) At the end of the tournament, it was found that any two players have met exactly once. What is the minimum duration of days that the tournament lasts?

Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. Proposed by Dominik Burek and Tomasz Ciesla, Poland

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ neighbors - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. Proposed by Gurgen Asatryan, Armenia