International Math Olympiad Questions
IMO Problems 2016
Day 1
IMO 2016 Problem 1
Triangle BCF has a right angle at B. Let A be the point on line CF such that FA = FB and F lies between A and C. Point is chosen such that so that DA = DC and AC is bisector of angle(DAB). Point E is chosen so that EA = ED and AD is the bisector of angle(EAC). Let M be the midpoint of CF. Let X be the points that AMXE is a parallelogram. Prove that BD, FX, and ME are concurrent.
IMO 2016 Problem 2
Find all integers n for which each cell of n×n table can be filled with one of the letters I, M, and O in such a way that:
(1) In each row and each column one third of the entries are I, one third are M and one-third are O.
(2) In any diagonal, if the number of entries on the diagonal is a multiple of 3, then one-third of the entries are I, one third are M, and one third are O.
Note: the rows and columns of n×n table are each labelled 1 to n in a natural order. Thus each cell corresponds to a pair of positive integer (i, j) with 1 ≤ i, j ≤ n. For n > 1, the table has 4n – 2 diagonals of two types. A diagonal of first type consists of all cells (i, j) for which i + J is a constant, and the diagonal of the second type consists of cells (i, j) for which i – J is a constant.
IMO 2016 Problem 3
Let P = A₁A₂…Aₙ be a convex polygon in the plane. The vertices A₁, A₂, …, Aₙ have integral coordinates and lie on a circle. Let S be the area of P. An odd positive integer n is given such that the squares of the side length of P are integers divisible by n. Prove that 2S is an integer divisible by n.
Day 2
IMO 2016 Problem 4
A set of positive integers is called fragrant, if it is 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) = n² + n + 1. What is the least possible positive integer value of b such that there exist a non negative integer a for which the set {P(a + 1), P(a + 2), …, P(a + b)} is fragrant?
IMO 2016 Problem 5
The equation (x – 1)(x – 2)…(x – 2016) = (x – 1)(x – 2)…(x – 2016) is written on the board, with 2016 linear factors on each side. What is the least possible value of n for which it is possible to erase exactly n of these 4032 linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?
IMO 2016 Problem 6
There are n ≥ 2 line segments in the plane such that every two segments cross and no 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. Every time he claps, each frog will 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 in such a way that no two of them will ever occupy the same intersection point at the same time.
(A) Prove that Geoff can always fulfill his wish if n is odd.
(B) Prove that Geoff can never fulfill his wish if n is even.