British Mathematical Olympiad

News: Balkan Mathematical Olympiad problems (30 June 2013)

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

  1. In a triangle ABC, the excircle ωa opposite A touches AB at P and AC at Q, and the excircle ωb 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.


  2. Determine all positive integers x, y and z such that x5 + 4y = 2013z.


  3. Let S be the set of positive real numbers. Find all functions fS3 → S such that, for all positive real numbers x, y, z and k, the following three conditions are satisfied:

    (a) xf(xyz) = zf(zyx),

    (b) f(xykk2z) = kf(xyz),

    (c) f(1, kk+1) = k+1.

    (United Kingdom: J. E. Smith)

  4. 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, …, An form a weakly-friendly cycle if Ai is not a friend of Ai+1 for 1 ≤ i ≤ n (An+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.


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