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.
Let a, b and c be positive real numbers. Prove that
a^{2}b(b−c)/(a+b)
+ b^{2}c(c−a)/(b+c)
+
c^{2}a(a−b)/(c+a)
≥ 0.
Let ABC be an acute triangle with orthocentre H.
Let M be the midpoint of AC. Let C_{1}
on AB be the foot of the perpendicular from C, and
let H_{1} be the reflection of H in AB.
Let the points P, Q and R be the orthogonal
projections of C_{1} onto the
lines AH_{1}, AC and CB,
respectively. Let M_{1} be the point such that the
circumcentre of triangle PQR is the midpoint of the segment
MM_{1}.
Prove that M_{1} lies
on BH_{1}.
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.
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.
