International Mathematical Olympiad · 2016

Problems

Problem 1In convex pentagon $ABCDE$ with $\angle B>90^\circ$ , let $F$ be a point on segment $AC$ such that $\angle FBC=90^\circ$ . It is given that $FA=FB$ , $DA=DC$ , $EA=ED$ , and rays $AC$ and $AD$ trisect angle $\angle BAE$ . Let $M$ be the midpoint of $CF$ . Let $X$ be the point such that $AMXE$ is a parallelogram. Show that lines $FX$ , $EM$ , $BD$ are concurrent.Solutions: 1
Problem 2Find all integers $n$ for which each cell of an $n\times n$ table can be filled with one of the letters $I$ , $M$ , and $O$ in such a way that: in each row and each column, one third of the entries are $I$ , one third are $M$ , and one third are $O$ ; and in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries on that diagonal are $I$ , one third are $M$ , and one third are $O$ . (Note that an $n\times n$ table has $4n-2$ diagonals in total, in both directions.)Solutions: 1
Problem 3Let $P=A_1A_2\cdots A_k$ be a convex polygon in the plane. The vertices $A_1,A_2,\dots,A_k$ have integer coordinates and lie on a circle. Let $S$ be the area of $P$ . An odd positive integer $n$ is given such that the square of the length of each side of $P$ is an integer divisible by $n$ . Prove that $2S$ is an integer divisible by $n$ .Solutions: 1
Problem 4A set of positive integers is called fragrant if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$ . What is the smallest possible value of a positive integer $b$ such that there exists a non-negative integer $a$ for which the set $\{P(a+1),P(a+2),\dots,P(a+b)\}$ is fragrant?Solutions: 1
Problem 5The equation $(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots(x-2016)$ is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?Solutions: 1
Problem 6There are $n\ge2$ line segments in the plane such that every two segments cross, and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it, facing the other endpoint. Then he will clap his hands $n-1$ times; each time he claps, every frog immediately jumps forward to the next intersection point on its segment (frogs never change the direction of their jumps). Geoff wishes to place the frogs so that no two of them ever occupy the same intersection point at the same time. (a) Prove that Geoff can always fulfil his wish if $n$ is odd. (b) Prove that Geoff can never fulfil his wish if $n$ is even.Solutions: 1