CONO-SUR 2013

Four distinct points are marked in a line. For each point, the sum of the distances from said point to the other three is calculated; getting in total 4 numbers.

Decide whether these 4 numbers can be, in some order: a) $29,29,35,37$ b) $28,29,35,37$ c) $28,34,34,37$

In a triangle $ABC$ , let $M$ be the midpoint of $BC$ and $I$ the incenter of $ABC$ . If $IM$ = $IA$ , find the least possible measure of $\angle{AIM}$ .

Nocycleland is a country with $500$ cities and $2013$ two-way roads, each one of them connecting two cities. A city $A$ neighbors $B$ if there is one road that connects them, and a city $A$ quasi-neighbors $B$ if there is a city $C$ such that $A$ neighbors $C$ and $C$ neighbors $B$ . It is known that in Nocycleland, there are no pair of cities connected directly with more than one road, and there are no four cities $A$ , $B$ , $C$ and $D$ such that $A$ neighbors $B$ , $B$ neighbors $C$ , $C$ neighbors $D$ , and $D$ neighbors $A$ . Show that there is at least one city that quasi-neighbors at least $57$ other cities.

Let $M$ be the set of all integers from $1$ to $2013$ . Each subset of $M$ is given one of $k$ available colors, with the only condition that if the union of two different subsets $A$ and $B$ is $M$ , then $A$ and $B$ are given different colors. What is the least possible value of $k$ ?

Let $d(k)$ be the number of positive divisors of integer $k$ . A number $n$ is called balanced if $d(n-1) \leq d(n) \leq d(n+1)$ or $d(n-1) \geq d(n) \geq d(n+1)$ . Show that there are infinitely many balanced numbers.

Let $ABCD$ be a convex quadrilateral. Let $n \geq 2$ be a whole number. Prove that there are $n$ triangles with the same area that satisfy all of the following properties:

a) Their interiors are disjoint, that is, the triangles do not overlap. b) Each triangle lies either in $ABCD$ or inside of it. c) The sum of the areas of all of these triangles is at least $\frac{4n}{4n+1}$ the area of $ABCD$ .