PAMO 2005

For any positive real numbers $a,b$ and $c$ , prove: $$\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} \geq \dfrac{2}{a+b} + \dfrac{2}{b+c} + \dfrac{2}{c+a} \geq \dfrac{9}{a+b+c}$$

Let $S$ be a set of integers with the property that any integer root of any non-zero polynomial with coefficients in $S$ also belongs to $S$ . If $0$ and $1000$ are elements of $S$ , prove that $-2$ is also an element of $S$ .

Let $ABC$ be a triangle and let $P$ be a point on one of the sides of $ABC$ . Construct a line passing through $P$ that divides triangle $ABC$ into two parts of equal area.

Let $[ {x} ]$ be the greatest integer less than or equal to $x$ , and let $\{x\}=x-[x]$ . Solve the equation: $[x] \cdot \{x\} = 2005x$

Noah has to fit 8 species of animals into 4 cages of the Arc. He planes to put two species of animal in each cage. It turns out that, for each species of animal, there are at most 3 other species with which it cannot share a cage. Prove that there is a way to assign the animals to the cages so that each species shares a cage with a compatible species.

Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a function such that: For all $a$ and $b$ in $\mathbb{Z} - \{0\}$ , $f(ab) \geq f(a) + f(b)$ . Show that for all $a \in \mathbb{Z} - \{0\}$ we have $f(a^n) = nf(a)$ for all $n \in \mathbb{N}$ if and only if $f(a^2) = 2f(a)$