## IMO 2018 Problems – Day 2

**Problem 4.** A *site* is any point in the plane such that and 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 . 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 such that Amy can ensure that she places at least red stones, no matter how Ben places his blue stones.

**Problem 5.** Let be an infinite sequence of positive integers. Suppose that there is an integer such that, for each , the number

is an integer. Prove that there is a positive integer such that for all .

**Problem 6.** A convex quadrilateral satisfies . Point lies inside so that and . Prove that .

Source: AoPS

## IMO 2018 Problems – Day 1

**Problem 1.** Let be the circumcircle of acute triangle . Points and are on segments and respectively such that . The perpendicular bisectors of and intersect minor arcs and of at points and respectively. Prove that lines and are either parallel or they are the same line.

**Problem 2.** Find all integers for which there exist real numbers satisfying , and

For .

**Problem 3.** An *anti-Pascal* triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from to

Does there exist an anti-Pascal triangle with rows which contains every integer from to ?

Source: AoPS.

## IMO 2016 – Problem 1

**IMO 2016, Problem 1.** Triangle has a right angle at . Let be the point on line such that and lies between and . Point is chosen such that and is the bisector of . Point is chosen such that and is the bisector of . Let be the midpoint of . Let be the point such that is a parallelogram (where and ). Prove that the lines and are concurrent.

## IMO 2015 Problem 1

**Problem 1.** We say that a finite set of points in the plane is *balanced* if, for any two different points and in , there is a point in such that . We say that is *center-free* if for any three different points , and in , there is no points in such that .

(a) Show that for all integers , there exists a balanced set having points.

(b) Determine all integers for which there exists a balanced center-free set having points.

**Problem 2.** Find all triples of positive integers such that are all powers of 2.

**Problem 3.** Let be an acute triangle with . Let be its cirumcircle., its orthocenter, and the foot of the altitude from . Let be the midpoint of . Let be the point on such that and let be the point on such that . Assume that the points and are all different and lie on in this order.

Prove that the circumcircles of triangles and are tangent to each other.

*Source: AoPS*

## IMO 2014 Problem 6

A set of lines in the plane is in *general position* if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its *finite regions*. Prove that for all sufficiently large , in any set of lines in general position it is possible to colour at least lines blue in such a way that none of its finite regions has a completely blue boundary.

*Note:* Results with replaced by will be awarded points depending on the value of the constant .

**IMO 2014 Problem 6 (Day 2)**

## IMO 2014 Problem 5

For every positive integer , Cape Town Bank issues some coins that has value. Let a collection of such finite coins (coins does not neccesarily have different values) which sum of their value is less than . Prove that we can divide the collection into at most 100 groups such that sum of all coins’ value does not exceed 1.

**IMO 2014 Problem 5 (Day 2)**

## IMO 2014 Problem 4

Let and be on segment of an acute triangle such that and . Let and be the points on and , respectively, such that is the midpoint of and is the midpoint of . Prove that the intersection of and is on the circumference of triangle .

**IMO 2014 Problem 4 (Day 2)**