NL-BXMO-TST 2016

For a positive integer $n$ that is not a power of two, we define $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$ . Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$ .

Determine all triples (x, y, z) of non-negative real numbers that satisfy the following system of equations $\begin{cases} x^2 - y = (z - 1)^2 y^2 - z = (x - 1)^2 z^2 - x = (y -1)^2 \end{cases}$ .

Let $\vartriangle ABC$ be a right-angled triangle with $\angle A = 90^o$ and circumcircle $\Gamma$ . The inscribed circle is tangent to $BC$ in point $D$ . Let $E$ be the midpoint of the arc $AB$ of $\Gamma$ not containing $C$ and let $F$ be the midpoint of the arc $AC$ of $\Gamma$ not containing $B$ . (a) Prove that $\vartriangle ABC \sim \vartriangle DEF$ . (b) Prove that $EF$ goes through the points of tangency of the incircle to $AB$ and $AC$ .

The Facebook group Olympiad training has at least five members. There is a certain integer $k$ with following property: for each $k$ -tuple of members there is at least one member of this $k$ -tuple friends with each of the other $k - 1$ . (Friendship is mutual: if $A$ is friends with $B$ , then also $B$ is friends with $A$ .) (a) Suppose $k = 4$ . Can you say with certainty that the Facebook group has a member that is friends with each of the other members? (b) Suppose $k = 5$ . Can you say with certainty that the Facebook group has a member that is friends with each of the other members?

Determine all pairs $(m, n)$ of positive integers for which $(m + n)^3 / 2n (3m^2 + n^2) + 8$