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 x^{5} + 4^{y} = 2013^{z}.
(Serbia)
Let S be the set of positive real numbers. Find all
functions f: S^{3} → 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, k^{2}z) = 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 A_{1}, A_{2},
…, A_{n} form a weaklyfriendly
cycle if A_{i} is not a friend
of A_{i+1} for
1 ≤ i ≤ n
(A_{n+1} = A_{1}), and
there are no other pairs of nonfriends among the compenents of the
cycle.
The following property is satisfied:
for every competitor C and every weaklyfriendly 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)
