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News: Balkan Mathematical Olympiad problems (30 June 2013)

The Balkan Mathematical Olympiad paper was sat today; the problems and proposing countries are:

In a triangle ABC, the excircle ω_a opposite A touches AB at P and AC at Q, and the excircle ω_b opposite B touches BA at M and BC at N. Let K be the projection of C onto MN and let L be the projection of C onto PQ. Show that the quadrilateral MKLP is cyclic.

(Bulgaria)
Determine all positive integers x, y and z such that x⁵ + 4^y = 2013^z.

(Serbia)
Let S be the set of positive real numbers. Find all functions f: S³ → S such that, for all positive real numbers x, y, z and k, the following three conditions are satisfied:

(a) xf(x, y, z) = zf(z, y, x),

(b) f(x, yk, k²z) = kf(x, y, z),

(c) f(1, k, k+1) = k+1.

(United Kingdom: J. E. Smith)
In a mathematical competition, some competitors are friends; friendship is mutual, that is to say that when A is a friend of B, then B is also a friend of A. We say that n ≥ 3 different competitors A₁, A₂, …, A_n form a weakly-friendly cycle if A_i is not a friend of A_i+1 for 1 ≤ i ≤ n (A_n+1 = A₁), and there are no other pairs of non-friends among the compenents of the cycle.

The following property is satisfied:

for every competitor C and every weakly-friendly cycle S of competitors not including C, the set of competitors D in S which are not friends of C has at most one element.

Prove that all competitors of this mathematical competition can be arranged into three rooms, such that every two competitors in the same room are friends.

(Serbia)