News: Balkan Mathematical Olympiad news (5 May 2010)

The Balkan Mathematical Olympiad paper was sat yesterday. Coordination has been partly done with the remaining problems to be coordinated tomorrow. The problems were as follows.

1. Let a, b and c be positive real numbers. Prove that

   a²b(b−c)/(a+b) + b²c(c−a)/(b+c) + c²a(a−b)/(c+a) ≥ 0.

2. Let ABC be an acute triangle with orthocentre H. Let M be the midpoint of AC. Let C₁ on AB be the foot of the perpendicular from C, and let H₁ be the reflection of H in AB. Let the points P, Q and R be the orthogonal projections of C₁ onto the lines AH₁, AC and CB, respectively. Let M₁ be the point such that the circumcentre of triangle PQR is the midpoint of the segment MM₁.

   Prove that M₁ lies on BH₁.

3. A strip of width w is a set of points in the plane which are on, or between, two parallel lines distance w apart. Let S be a finite set of n (n ≥ 3) points in the plane, such that any three different points from S can be covered by a strip of width 1.

   Prove that S can be covered by a strip of width 2.

4. For each positive integer n (n ≥ 2), let f(n) denote the sum of all positive integers which are at most n and are not relatively prime to n. Show that f(n+p) ≠ f(n) for each such n and for every prime p.

The involvement of the UK team in this competition is sponsored by Winton Capital Management.

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