OIM 2005

Determine all triples of real numbers $(a,b,c)$ such that \begin{eqnarray*} xyz &=& 8 x^2y + y^2z + z^2x &=& 73 x(y-z)^2 + y(z-x)^2 + z(x-y)^2 &=& 98 . \end{eqnarray*}

A flea jumps in a straight numbered line. It jumps first from point $0$ to point $1$ . Afterwards, if its last jump was from $A$ to $B$ , then the next jump is from $B$ to one of the points $B + (B - A) - 1$ , $B + (B - A)$ , $B + (B-A) + 1$ .

Prove that if the flea arrived twice at the point $n$ , $n$ positive integer, then it performed at least $\lceil 2\sqrt n\rceil$ jumps.

Denote by $a \bmod b$ the remainder of the euclidean division of $a$ by $b$ . Determine all pairs of positive integers $(a,p)$ such that $p$ is prime and $$a \bmod p + a\bmod 2p + a\bmod 3p + a\bmod 4p = a + p.$$

Let $O$ be the circumcenter of acutangle triangle $ABC$ and let $A_1$ be some point in the smallest arc $BC$ of the circumcircle of $ABC$ . Let $A_2$ and $A_3$ points on sides $AB$ and $AC$ , respectively, such that $\angle BA_1A_2 = \angle OAC$ and $\angle CA_1A_3 = \angle OAB$ .

Prove that the line $A_2A_3$ passes through the orthocenter of $ABC$ .

Let $n$ be a fixed positive integer. The points $A_1$ , $A_2$ , $\ldots$ , $A_{2n}$ are on a straight line. Color each point blue or red according to the following procedure: draw $n$ pairwise disjoint circumferences, each with diameter $A_iA_j$ for some $i \neq j$ and such that every point $A_k$ belongs to exactly one circumference. Points in the same circumference must be of the same color.

Determine the number of ways of coloring these $2n$ points when we vary the $n$ circumferences and the distribution of the colors.