2005 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

Balkan 2005-3

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?

1

balkan 2005-4

n \geq 2

be an integer. Let

S

\{1,2,\dots,n\}

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

1

Balkan 2005-2

Find all primes

p

p^2-p+1

is a perfect cube.

1

Balkan 2005-1

ABC

be an acute-angled triangle whose inscribed circle touches

AB

AC

D

E

respectively. Let

X

Y

be the points of intersection of the bisectors of the angles

\angle ACB

\angle ABC

DE

Z

be the midpoint of

BC

. Prove that the triangle

XYZ

is equilateral if and only if

\angle A = 60^\circ