IMO 2018 Day 2

Day 2 of IMO 2018. Go crazy!

Day 1

Problem 4

A site is any point \((x, y)\) in the plane such that \(x\) and \(y\) are both positive integers less than or equal to \(20\).

Initially, each of the $400$ sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone.

Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones.

Problem 5

Let $a_1,a_2,\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number $\frac{a_1}{a_2} + \frac{a_2}{a_3} + \ldots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$ is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$.

Problem 6

A convex quadrilateral $ABCD$ satisfies $AB \cdot CD = BC \cdot DA$. Point $X$ lies inside $ABCD$ so that $\angle{XAB} = \angle{XCD}$ and $\angle{XBC} = \angle{XDA}$. Prove that $\angle{BXA} + \angle{DXC} = 180^{\circ}$.