IMO 1987

Problem 1. Let $p_n(k)$ be the number of permutations of the set $\{1,\dots ,n\}$, $n\ge 1$, which have exactly $k$ fixed points. Prove that ${\sum }_{k=0}^{n}k\cdot p_n(k)=n!.$

(Remark: A permutation $f$ of a set $S$ is a one-to-one mapping of $S$ onto itself. An element $i$ in $S$ is called a fixed point of the permutation $f$ if $f(i)=i$.)

Problem 2. In an acute-angled triangle $ABC$ the interior bisector of the angle $A$ intersects $BC$ at $L$ and intersects the circumcircle of $ABC$ again at $N$. From point $L$ perpendiculars are drawn to $AB$ and $AC$, the feet of these perpendiculars being $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.

Problem 3. Let $x_1$, $x_2$, ..., $x_n$ be real numbers satisfying $x_1^2+x_2^2+\cdots +x_n^2=1$. Prove that for every integer $k\ge 2$ there are integers $a_1$, $a_2$, ..., $a_n$, not all 0, such that $|a_i|\le k-1$ for all $i$ and

Problem 4. Prove that there is no function $f$ from the set of non-negative integers into itself such that $f(f(n))=n+1987$ for every $n$.

Problem 5. Let $n$ be an integer greater than or equal to 3. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.

Problem 6. Let $n$ be an integer greater than or equal to 2. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le \sqrt{n/3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.