SE-SKOLORNAS 1981

Let $N = 11\cdots 122 \cdots 25$ , where there are $n$ $1$ s and $n+1$ $2$ s. Show that $N$ is a perfect square.

Does $$\left\{ \begin{array}{l} x^y = z y^z = x z^x = y \end{array} \right.$$ have any solutions in positive reals apart from $x = y = z= 1$ ?

Find all polynomials $p(x)$ of degree $5$ such that $p(x) + 1$ is divisible by $(x-1)^3$ and $p(x) - 1$ is divisible by $(x+1)^3$ .

A cube side $5$ is divided into $125$ unit cubes. $N$ of the small cubes are black and the rest white. Find the smallest $N$ such that there must be a row of $5$ black cubes parallel to one of the edges of the large cube.

$ABC$ is a triangle. $X$ , $Y$ , $Z$ lie on $BC$ , $CA$ , $AB$ respectively. Show that area $XYZ$ cannot be smaller than each of area $AYZ$ , area $BZX$ , area $CXY$ .

Show that there are infinitely many triangles with side lengths $a$ , $b$ , $c$ , where $a$ is a prime, $b$ is a power of $2$ and $c$ is the square of an odd integer.