OME 2007

Let $a_0, a_1, a_2, a_3, a_4$ be five positive numbers in the arithmetic progression with a difference $d$ . Prove that $a^3_2 \leq \frac{1}{10}(a^3_0 + 4a^3_1 + 4a^3_3 + a^3_4).$

Determine all the possible non-negative integer values that are able to satisfy the expression: $\frac{(m^2+mn+n^2)}{(mn-1)}$ if $m$ and $n$ are non-negative integers such that $mn \neq 1$ .

$O$ is the circumcenter of triangle $ABC$ . The bisector from $A$ intersects the opposite side in point $P$ . Prove that the following is satisfied: $$AP^2 + OA^2 - OP^2 = bc.$$

What are the positive integer numbers that we are able to obtain in $2007$ distinct ways, when the sum is at least out of two positive consecutive integers? What is the smallest of all of them? Example: the number 9 is written in exactly two such distinct ways: $9 = 4 + 5$ $9 = 2 + 3 + 4.$

Let $a \neq 1$ and be a real positive number and $n$ be an integer greater than $1.$ Demonstrate that $n^2 < \frac{(a^n + a^{-n} -2)}{(a + a^{-1} -2)}.$

Given a halfcircle of diameter $AB = 2R$ , consider a chord $CD$ of length $c$ . Let $E$ be the intersection of $AC$ with $BD$ and $F$ the inersection of $AD$ with $BC$ . Prove that the segment $EF$ has a constant length and direction when varying the chord $CD$ about the halfcircle.