# 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 $$\angle{BXA} + \angle{DXC} = 180^{\circ}$$.

Note by Sharky Kesa
6 months, 1 week ago

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