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Home » Olympiad Resources » International Competitions » Balkan MO » 2005
 
Balkan MO 2005
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06 May 2005

1 Let ABC be an acute-angled triangle whose inscribed circle touches AB and AC at D and E respectively. Let X and Y be the points of intersection of the bisectors of the angles \angle ACB and \angle ABC with the line DE and let Z be the midpoint of BC. Prove that the triangle XYZ is equilateral if and only if \angle A = 60^\circ.
2 Find all primes p such that p^2-p+1 is a perfect cube. S
3 Let a,b,c be positive real numbers. Prove the inequality
\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq a+b+c+\frac{4(a-b)^2}{a+b+c}.
When does equality occur? 		S
4 Let n \geq 2 be an integer. Let S be a subset of \{1,2,\dots,n\} such that S neither contains two elements one of which divides the other, nor contains two elements which are coprime. What is the maximal possible number of elements of such a set S? S
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