CR-MO 2015

A function $f$ defined on integers such that $f (n) =n + 3$ if $n$ is odd $f (n) = \frac{n}{2}$ if $n$ is even

If $k$ is an odd integer, determine the values for which $f (f (f (k))) = k$ .

Find all functions $f: R \to R$ such that $f (f (x) f (y)) = xy$ and there is no $k \in R -\{0,1,-1\}$ such that $f (k) = k$ .

Let $\vartriangle A_1B_1C_1$ and $l_1, m_1, n_1$ be the trisectors closest to $A_1B_1$ , $B_1C_1$ , $C_1A_1$ of the angles $A_1, B_1, C_1$ respectively. Let $A_2 = l_1 \cap n_1$ , $B_2 = m_1 \cap l_1$ , $C_2 = n_1 \cap m_1$ . So on we create triangles $\vartriangle A_nB_nC_n$ . If $\vartriangle A_1B_1C_1$ is equilateral prove that exists $n \in N$ , such that all the sides of $\vartriangle A_nB_nC_n$ are parallel to the sides of $\vartriangle A_1B_1C_1$ .

Consider $\vartriangle ABC$ , right at $B$ , let $I$ be its incenter and $F,D,E$ the points where the circle inscribed on sides AB, $BC$ and $AC$ , respectively. If $M$ is the intersection point of $CI$ and $EF$ , and $N$ is the intersection point of $DM$ and $AB$ . Prove that $AN = ID$ .

Let $A, B, C, D$ be points that lie on the same circle . Let $F$ be such that the arc $AF$ is congruent with the arc $BF$ . Let $P$ be the intersection point of the segments $DF$ and $AC$ . Let $Q$ be intersection point of the $CF$ and $BD$ segments. Prove that $PQ \parallel AB$ .

Show that there are no triples $(a, b, c)$ of positive integers such that a) $a + c, b + c, a + b$ do not have common multiples in pairs. b) $\frac{c^2}{a + b},\frac{b^2}{a + c},\frac{a^2}{c + b}$ are integer numbers.