OME 2005

Let $a$ and $b$ be integers. Demonstrate that the equation $$(x-a)(x-b)(x-3) +1 = 0$$ has an integer solution.

Is it possible to color points in the Cartesian Plane $(x,y)$ with integer coordinates with three colors, such that each color appears infinitely many times on infinitely many lines parallel to the $x$ -axis and that any three points, each of a different color, are not in a line? Justify your answer.

We will say that a triangle is multiplicative if the product of the heights of two of its sides is equal to the length of the third side. Given $ABC \dots XYZ$ is a regular polygon with $n$ sides of length $1$ . The $n-3$ diagonals that go out from vertex $A$ divide the triangle $ZAB$ in $n-2$ smaller triangles. Prove that each one of these triangles is multiplicative.

Prove that for every positive integer $n$ , the decimal expression of $\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}$ is periodic .

Let $r,s,u,v$ be real numbers. Prove that: $$min\{r-s^2,s-u^2, u-v^2,v-r^2\}\le \frac{1}{4}$$

In a triangle with sides $a, b, c$ the side $a$ is the arithmetic mean of $b$ and $c$ . Prove that: a) $0^o \le A \le 60^o$ . b) The height relative to side $a$ is three times the inradius $r$ . c) The distance from the circumcenter to side $a$ is $R - r$ , where $R$ is the circumradius.