AUSTRIAN-POLISH 1998

Let $x_1, x_2,y _1,y_2$ be real numbers such that $x_1^2 + x_2^2 \le 1$ . Prove the inequality $$(x_1y_1 + x_2y_2 - 1)^2 \ge (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)$$

Find all pairs of real numbers $(x, y)$ satisfying the following system of equations $2-x^{3}=y, 2-y^{3}=x$ .

For positive integers $m, n$ , denote $$S_m(n)=\sum_{1\le k \le n} \left[ \sqrt[k^2]{k^m}\right]$$ Prove that $S_m(n) \le n + m (\sqrt[4]{2^m}-1)$

Determine all pairs $(a, b)$ of positive integers for which the equation $x^3 - 17x^2 + ax - b^2 = 0$ has three integer roots (not necessarily different).

Consider all pairs $(a, b)$ of natural numbers such that the product $a^ab^b$ written in decimal system ends with exactly $98$ zeros. Find the pair $(a, b)$ for which the product $ab$ is the smallest.

In each unit square of an infinite square grid a natural number is written. The polygons of area $n$ with sides going along the gridlines are called admissible, where $n > 2$ is a given natural number. The value of an admissible polygon is defined as the sum of the numbers inside it. Prove that if the values of any two congruent admissible polygons are equal, then all the numbers written in the unit squares of the grid are equal. (We recall that a symmetric image of polygon $P$ is congruent to $P$ .)

Given a triangle $ABC$ , points $K,L,M$ are the midpoints of the sides $BC,CA,AB$ , and points $X,Y,Z$ are the midpoints of the arcs $BC,CA,AB$ of the circumcircle not containing $A,B,C$ respectively. If $R$ denotes the circumradius and $r$ the inradius of the triangle, show that $r+KX+LY+MZ=2R$ .