BALKAN 2010

The left-hand side is equal to

$$\frac{a^{3} b^{3}+b^{3} c^{3}+c^{3} a^{3}-a^{3} b^{2} c-b^{3} c^{2} a-c^{3} a^{2} b}{(a+b)(b+c)(c+a)}$$

so it is enough to show that $a^{3} b^{3}+b^{3} c^{3}+c^{3} a^{3} \geq a^{3} b^{2} c+b^{3} c^{2} a+c^{3} a^{2} b$. The AM-GM inequality gives us $a^{3} b^{3}+a^{3} b^{3}+a^{3} c^{3} \geq 3 \sqrt[3]{a^{3} b^{3} \cdot a^{3} b^{3} \cdot a^{3} c^{3}}=3 a^{3} b^{2} c$; summing this inequality and its cyclic analogs yields the desired inequality. Equality holds if and only if $a=b=c$.

Let $A B C$ be an acute-angled triangle with orthocenter $H$ and let $M$ be the midpoint of $A C$. The foot of the altitude from $C$ is $C_{1}$. Point $H_{1}$ is symmetric to $H$ in $A B$. The projections of $C_{1}$ on lines $A H_{1}, A C$ and $B C$ are $P, Q$ and $R$ respectively. If $M_{1}$ is the circumcenter of triangle $P Q R$, prove that the point symmetric to $M$ with respect to $M_{1}$ lies on line $B H_{1}$. (Serbia)

We define a $w$-strip as the set of all points in the plane that are between or on two parallel lines on a mutual distance $w$. Let $S$ be a set of $n$ points in the plane such that any three points from $S$ can be covered by a 1 -strip. Show that the entire set $S$ can be covered by a 2 -strip. (Romania)

For every integer $n \geq 2$, denote by $f(n)$ the sum of positive integers not exceeding $n$ that are not coprime to $n$. Prove that $f(n+p) \neq f(n)$ for any such $n$ and any prime number $p$. (Turkey)

Time allowed: 270 minutes. Each problem is worth 10 points.