OBM 1996

Show that there exists infinite triples $(x,y,z) \in N^3$ such that $x^2+y^2+z^2=3xyz$ .

Does there exist a set of $n > 2, n < \infty$ points in the plane such that no three are collinear and the circumcenter of any three points of the set is also in the set?

Let $f(n)$ be the smallest number of 1s needed to represent the positive integer $n$ using only 1s, $+$ signs, $\times$ signs and brackets $(,)$ . For example, you could represent 80 with 13 1s as follows: $(1+1+1+1+1)(1+1+1+1)(1+1+1+1)$ . Show that $3 \log(n) \leq \log(3)f(n) \leq 5 \log(n)$ for $n > 1$ .

$ABC$ is acute-angled. $D$ s a variable point on the side BC. $O_1$ is the circumcenter of $ABD$ , $O_2$ is the circumcenter of $ACD$ , and $O$ is the circumcenter of $AO_1O_2$ . Find the locus of $O$ .

There are $n$ boys $B_1, B_2, ... , B_n$ and $n$ girls $G_1, G_2, ... , G_n$ . Each boy ranks the girls in order of preference, and each girl ranks the boys in order of preference. Show that we can arrange the boys and girls into n pairs so that we cannot find a boy and a girl who prefer each other to their partners.

For example if $(B_1, G_3)$ and $(B_4, G_7)$ are two of the pairs, then it must not be the case that $B_4$ prefers $G_3$ to $G_7$ and $G_3$ prefers $B_4$ to $B_1$ .

Let p(x) be the polynomial $x^3 + 14x^2 - 2x + 1$ . Let $p^n(x)$ denote $p(p^(n-1)(x))$ . Show that there is an integer N such that $p^N(x) - x$ is divisible by 101 for all integers x.