OME 2006

Let $P(x)$ be a polynomial with integer coefficients. Prove that if there is an integer $k$ such that none of the integers $P(1),P(2), ..., P(k)$ is divisible by $k$ , then $P(x)$ does not have integer roots.

The dimensions of a wooden octahedron are natural numbers. We painted all its surface (the six faces), cut it by planes parallel to the cubed faces of an edge unit and observed that exactly half of the cubes did not have any painted faces. Prove that the number of octahedra with such property is finite.

(It may be useful to keep in mind that $\sqrt[3]{\frac{1}{2}}=1,79 ... <1,8$ ).

$ABC$ is an isosceles triangle with $AB = AC$ . Let $P$ be any point of a circle tangent to the sides $AB$ in $B$ and to AC in C. Denote $a$ , $b$ and $c$ to the distances from $P$ to the sides $BC, AC$ and $AB$ respectively. Prove that: $a^2=bc$

Find all the functions $f:(0,+\infty) \to R$ that satisfy the equation $$f(x)f(y)+f\big(\frac{\lambda}{x})f(\frac{\lambda}{y})=2f(xy)$$ for all pairs of $x,y$ real and positive numbers, where $\lambda$ is a positive real number such that $f(\lambda )=1$

Prove that the product of four consecutive natural numbers can not be neither square nor perfect cube.

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect at $E$ . Denotes by $S_1,S_2$ and $S$ the areas of the triangles $ABE$ , $CDE$ and the quadrilateral $ABCD$ respectively. Prove that $\sqrt{S_1}+\sqrt{S_2}\le \sqrt{S}$ . When equality is reached?