AMO 2020

Determine all pairs of $(a,b)$ of non negative integers such that: $$\dfrac{a+b}{2} - \sqrt{ab}~=~1$$

Amy and Bee play the following game. Initially, there are three piles, each containing 2020 stones. The players take turns to make a move, with Amy going first. Each move consists of choosing one of the piles available, removing the unchosen pile(s) from the game, and then dividing the chosen pile into 2 or 3 non-empty piles. A player looses the game if they are unable to make a move.

Prove that Bee can always win the game,no matter how Amy plays

Let $ABC$ be a triangle with right angle at $\angle C$ . Suppose that the tangent line at C to the circle passing through $A,B,C$ intersects the line $AB$ at $D$ . Let $E$ be the midpoint of $CD$ and let $F$ be the point on the line $EB$ such that $AF$ is parallel to $CD$ . Prove that the lines $AB$ and $CF$ are perpendicular.

Each term of an infinite sequene $a_1,a_2, \cdots$ is equal to 0 or 1. For each positive integer $n$ , [list] [] $a_n+a_{n+1} \neq a_{n+2} +a_{n+3}$ and [] $a_n + a_{n+1}+a_{n+2} \neq a_{n+3} +a_{n+4} + a_{n+5}$ Prove that if $a_1~=~0$ , then $a_{2020}~=~1$

Let $ABCD$ be a square. For a point $P$ inside it, a windmill centred at $P$ consists of 2 perpendicular lines $l_1$ and $l_2$ passing through $P$ , such that $\bullet ~$ $l_1$ intersects the sides AB and CD at W and Y, respectively, and $\bullet ~$ $l_2$ intersects the side BC and DA at X and Z, respectively.

A windmill is called round if the quadrilateral WXYZ is cyclic. Determine all points P inside the square such that every windmill centred at P is round.

A tetramino tile is a tile that can be formed by gluing together four unit square tiles, edge to edge. For each natural number n, consider a bathroom whose floor is in the shape of a $2*2n$ rectangle. Let $T_n$ be the number of ways to tile this bathroom floor with tetramino tiles. Prove that, each of the numbers $T_1 , T_2 , T_3 , \cdots$ is a perfect square.