SE-SKOLORNAS 1986

Show that the polynomial $x^6 -x^5 +x^4 -x^3 +x^2 -x+\frac34$ has no real zeroes.

The diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ intersect at $O$ . If $S_1$ and $S_2$ are the areas of triangles $AOB$ and $COD$ and S that of $ABCD$ , show that $\sqrt{S_1} + \sqrt{S_2} \le \sqrt{S}$ . Prove that equality holds if and only if $AB$ and $CD$ are parallel.

Let $N \ge 3$ be a positive integer. For every pair $(a,b)$ of integers with $1 \le a <b \le N$ consider the quotient $q = b/a$ . Show that the pairs with $q < 2$ are equally numbered as those with $q > 2$ .

Prove that $x = y = z = 1$ is the only positive solution of the system $$\left\{ \begin{array}{l} x+y^2 +z^3 = 3 y+z^2 +x^3 = 3 z+x^2 +y^3 = 3 \end{array} \right.$$

In the arrangement of $pn$ real numbers below, the difference between the greatest and smallest numbers in each row is at most $d$ , $d > 0$ . $$\begin{array}{l} a_{11} \, a_{12} \, ... \, a_{1n} a_{21} \, a_{22} \, ... \, a_{2n} \, . \, \, \, \, . \, \, \, \, \, \, \, \, . \, . \, \, \, \, . \, \, \, \, \, \, \, \, . \, . \, \, \, \, . \, \, \, \, \, \, \, \, . a_{n1} \, a_{n2} \, ... \, a_{nn} \end{array}$$ Prove that, when the numbers in each column are rearranged in decreasing order, the difference between the greatest and smallest numbers in each row will still be at most d.

The interval $[0,1]$ is covered by a finite number of intervals. Show that one can choose a number of these intervals which are pairwise disjoint and have the total length at least $1/2$ .