## 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

## Balkan Mathematical Olympiad – 2016 Problems

**Problem 1.** Find all injective functions such that for every real number and every positive integer ,

**Problem 2.** Let be a cyclic quadrilateral with . The diagonals intersect at the point and lines and intersect at the point . Let and be the orthogonal projections of onto lines and respectively, and let , and be the midpoints of , and respectively. Prove that the second intersection point of the circumcircles of triangles and lies on the segment .

**Problem 3.** Find all monic polynomials with integer coefficients satisfying the following condition: there exists a positive integer such that divides for every prime for which is a positive integer.

**Problem 4.** The plane is divided into squares by two sets of parallel lines, forming an infinite grid. Each unit square is coloured with one of colours so that no rectangle with perimeter contains two squares of the same colour. Show that no rectangle of size or contains two squares of the same colour.

## Problems of the Miklos Schweitzer Competition 2014

**Problem 1.** Let be a positive integer. Let be a familiy of sets that contains more than half of all subsets of an -element set . Prove that from we can select sets that form a separating family of , i.e., for any two distinct elements of there is a selected set containing exactly one of the two elements.

**Problem 2.** let and let be non-degenerate subintervals of the interval . Prove that

where the summation is over all pairs of indices such that and are not disjoint.

**Problem 3.** We have points in the plane, no three of them collinear. The points are colored with two colors. Prove that from the points we can form empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occuring as vertices of the triangles have the same color.

**Problem 4.** For a positive integer , let be the number of sequences of positive integers such that and for . We make the convention . Let be the unique real number greater than such that . Prove that

- (i) .
- (ii) There exists no number such that .

**Problem 5.** Let be a non-real algebraic integer of degree two, and let be the set of irreducible elements of the ring . Prove that

**Problem 6.** Let be a representation of a finite -group over a field of characteristic . Prove that if the restriction of the linear map to a finite dimensional subspace of is injective, then the subspace spanned by the subspaces () is the direct sum of these subspaces.

**Problem 7.** Lef be a continuous function and let be arbitrary. Suppose that the Minkowski sum of the graph of and the graph of (i.e. the set has Lebesgue measure zero. Does it follow then that the function is of the form , with suitable constants ?

**Problem 8.** Let be a fixed integer. Calculate the distance

where runs over polynomials of degree less than with real coefficients and runs over functions of the form

defined on the closed interval , where and .

**Problem 9.** Let , and let be a convex body, i.e. a compact convex set with nonempty interior. Define the barycenter of the body with respect to the weight function by the usual formula

Prove that the translates of the body have pairwise distinct barycenters with respect to .

**Problem 10.** To each vertex of a given triangulation of the two dimensional sphere, we assign a convex subset of the plane. Assume that the three convex sets corresponding to the three vertices of any two dimensional face of the triangulation have at least one point in common. Show that there exist four vertices such that the corresponding convex sets have at least one point in common.

**Problem 11.** Let be a random variable that is uniformly distributed on the interval , and let

Show that, as , the limit distribution of is the Cauchy distribution with density function .

## IMC 2014 Day 2 Problem 5

For every positive integer , denote by the number of permutations of such that for every . For , denote by the number of permutations of such that for every and for every . Prove that

**IMC 2014 Day 2 Problem 5**

## IMC 2014 Day 2 Problem 4

We say that a subset of is –*almost contained* by a hyperplane if there are less than points in that set which do not belong to the hyperplane. We call a finite set of points –*generic* if there is no hyperplane that -almost contains the set. For each pair of positive integers and , find the minimal number such that every finite -generic set in contains a -generic subset with at most elements.

**IMC 2014 Day 2 Problem 4**

## IMC 2014 Day 1 Problem 5

Let be a close broken line consisting of line segments in the Euclidean plane. Suppose that no three of its vertices are collinear and for each index , the triangle has counterclockwise orientation and , using the notation modulo . Prove that the number of self-intersections of the broken line is at most .

**IMC 2014 Day 1 Problem 5**

## 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)**