BG-MO 1989

In triangle $ABC$ , point $O$ is the center of the excircle touching the side $BC$ , while the other two excircles touch the sides $AB$ and $AC$ at points $M$ and $N$ respectively. A line through $O$ perpendicular to $MN$ intersects the line $BC$ at $P$ . Determine the ratio $AB/AC$ , given that the ratio of the area of $\triangle ABC$ to the area of $\triangle MNP$ is $2R/r$ , where $R$ is the circumradius and $r$ the inradius of $\triangle ABC$ .

Prove that the sequence $(a_n)$ , where $$a_n=\sum_{k=1}^n\left\{\frac{\left\lfloor2^{k-\frac12}\right\rfloor}2\right\}2^{1-k},$$ converges, and determine its limit as $n\to\infty$ .

Let $p$ be a real number and $f(x)=x^p-x+p$ . Prove that:

(a) Every root $\alpha$ of $f(x)$ satisfies $|\alpha|<p^{\frac1{p-1}}$ ; (b) If $p$ is a prime number, then $f(x)$ cannot be written as the product of two non-constant polynomials with integer coefficients.

At each of the given $n$ points on a circle, either $+1$ or $-1$ is written. The following operation is performed: between any two consecutive numbers on the circle their product is written, and the initial $n$ numbers are deleted. Suppose that, for any initial arrangement of $+1$ and $-1$ on the circle, after finitely many operations all the numbers on the circle will be equal to $+1$ . Prove that $n$ is a power of two.

Prove that the perpendiculars, drawn from the midpoints of the edges of the base of a given tetrahedron to the opposite lateral edges, have a common point if and only if the circumcenter of the tetrahedron, the centroid of the base, and the top vertex of the tetrahedron are collinear.

Let $x,y,z$ be pairwise coprime positive integers and $p\ge5$ and $q$ be prime numbers which satisfy the following conditions:

(i) $6p$ does not divide $q-1$ ; (ii) $q$ divides $x^2+xy+y^2$ ; (iii) $q$ does not divide $x+y-z$ .

Prove that $x^p+y^p\ne z^p$ .