AUSTRIAN-POLISH 1979

On sides $AB$ and $BC$ of a square $ABCD$ the respective points $E$ and $F$ have been chosen so that $BE = BF$ . Let $BN$ be the altitude in triangle $BCE$ . Prove that $\angle DNF = 90$ .

Find all polynomials of the form $$P_n(x)=n!x^n+a_{n-1}x^{n-1}+\dots+a_1x+(-1)^n(n+1)$$ with integer coefficients, having $n$ real roots $x_1,\dots,x_n$ satisfying $k \leq x_k \leq k+1$ for $k=1, \dots,n$ .

Find all positive integers $n$ such that the inequality $$\left( \sum\limits_{i=1}^n a_i^2\right) \left(\sum\limits_{i=1}^n a_i \right) -\sum\limits_{i=1}^n a_i^3 \geq 6 \prod\limits_{i=1}^n a_i$$ holds for any $n$ positive numbers $a_1, \dots, a_n$ .

Determine all functions $f : N_0 \to R$ satisfying $f (x+y)+ f (x-y)= f (3x)$ for all $x,y$ .

The circumcenter and incenter of a given tetrahedron coincide. Prove that all its faces are congruent.

A positive integer $n$ and a real number $a$ are given. Find all $n$ -tuples $(x_1, ... ,x_n)$ of real numbers that satisfy the system of equations $$\sum_{i=1}^{n} x_i^k= a^k \,\,\, for \,\,\, k = 1,2, ... ,n$$

Let $n$ and $m$ be fixed positive integers. The hexagon $ABCDEF$ with vertices $A = (0,0)$ , $B = (n,0)$ , $C = (n,m)$ , $D = (n-1,m)$ , $E = (n-1,1)$ , $F = (0,1)$ has been partitioned into $n+m-1$ unit squares. Find the number of paths from $A$ to $C$ along grid lines, passing through every grid node at most once.

Let $A,B,C,D$ be four points in space, and $M$ and $N$ be the midpoints of $AC$ and $BD$ , respectively. Show that $$AB^2+BC^2+CD^2+DA^2 = AC^2+BD^2+4MN^2$$

Find the greatest power of $2$ that divides $a_n = [(3+\sqrt{11} )^{2n+1}]$ , where $n$ is a given positive integer.