News: Balkan Mathematical Olympiad news (5 May 2010)
The Balkan Mathematical Olympiad paper was sat yesterday.
done with the remaining problems to be coordinated tomorrow. The
problems were as follows.
Let a, b and c be positive real numbers. Prove that
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
Prove that M1 lies
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
Prove that S can be covered by a strip of width 2.
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.