KMO 1993

Consider a $9 \times 9$ array of white squares. Find the largest $n \in\mathbb N$ with the property: No matter how one chooses $n$ out of 81 white squares and color in black, there always remains a $1 \times 4$ array of white squares (either vertical or horizontal).

Let be given a triangle $ABC$ with $BC = a, CA = b, AB = c$ . Find point $P$ in the plane for which $aAP^{2}+bBP^{2}+cCP^{2}$ is minimum, and compute this minimum.

Find the smallest $x \in\mathbb{N}$ for which $\frac{7x^{25}-10}{83}$ is an integer.

An integer which is the area of a right-angled triangle with integer sides is called Pythagorean. Prove that for every positive integer $n > 12$ there exists a Pythagorean number between $n$ and $2n.$

Given $n \in\mathbb{N}$ , find all continuous functions $f : \mathbb{R}\to \mathbb{R}$ such that for all $x\in\mathbb{R},$ $$\sum_{k=0}^{n}\binom{n}{k}f(x^{2^{k}})=0.$$

Consider a triangle $ABC$ with $BC = a, CA = b, AB = c.$ Let $D$ be the midpoint of $BC$ and $E$ be the intersection of the bisector of $A$ with $BC$ . The circle through $A, D, E$ meets $AC, AB$ again at $F, G$ respectively. Let $H\not = B$ be a point on $AB$ with $BG = GH$ . Prove that triangles $EBH$ and $ABC$ are similar and find the ratio of their areas.