## 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)**

## IMO 2014 Problem 3

Convex quadrilateral has . Point is the foot of the perpendicular from to . Points and lie on sides and , respectively, such that lies inside triangle and

Prove that the line is tangent to the circumcircle of triangle .

**IMO 2014 Problem 3 (Day 1)**

## IMO 1981 Day 1

**Problem 1.** Let be a point inside a given triangle and denote the feet of the perpendiculars from to the lines respectively. Find such that the quantity

is minimal.

**Problem 2.** Let and consider all subsets of elements of the set . Each of these subsets has a smallest member. Let denote the arithmetic mean of these smallest numbers. Prove that

**Problem 3.** Determine the maximum value of where with .