Question bank IMO 1987

Let $S$ be a set of $n$ elements. We denote the number of all permutations of $S$ that have exactly $k$ fixed points by $p_n(k)$. Prove that
$$\sum_{k=0}^n kp_n(k)=n!.$$

The prolongation of the bisector $AL$ ($L\in BC$) in the acute-angled triangle $ABC$ intersects the circumscribed circle at point $N$. From point $L$ to the sides $AB$ and $AC$ are drawn the perpendiculars $LK$ and $LM$ respectively. Prove that the area of the triangle $ABC$ is equal to the area of the quadrilateral $AKNM$.

Suppose $x_1,x_2,\ldots,x_n$ are real numbers with $x_1^2+x_2^2+\cdots+x_n^2=1$. Prove that for any integer $k\gt 1$ there are integers $e_i$ not all $0$ and with $|e_i|\lt k$ such that
$$|e_1x_1+e_2x_2+\cdots+e_nx_n|\le\frac{(k-1)\sqrt n}{k^n-1}.$$

Does there exist a function $f:\mathbb N\to\mathbb N$, such that $f(f(n))=n+1987$ for every natural number $n$?

Prove that for every natural number $n\ge3$ it is possible to put $n$ points in the Euclidean plane such that the distance between each pair of points is irrational and each three points determine a nondegenerate triangle with rational area.

Let $f(x)=x^2+x+p$, $p\in\mathbb N$. Prove that if the numbers $f(0),f(1),\ldots,f([\sqrt{p/3}])$ are primes, then all the numbers $f(0),f(1),\ldots,f(p-2)$ are primes.