International Mathematical Olympiad · 1987

Problems

Problem 1Let $p_n(k)$ be the number of permutations of the set $\{1, \ldots, n\}$ , $n \ge 1$ , that have exactly $k$ fixed points. Prove that $\sum_{k=0}^{n} k \cdot p_n(k) = n!$ . (A permutation $f$ of a set $S$ is a one-to-one mapping of $S$ onto itself; an element $i$ of $S$ is a fixed point of $f$ if $f(i) = i$ .)Solutions: 2
Problem 2In an acute-angled triangle $ABC$ , the interior bisector of angle $A$ intersects $BC$ at $L$ and intersects the circumcircle of $ABC$ again at $N$ . From point $L$ , perpendiculars are drawn to $AB$ and $AC$ , with feet $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.Solutions: 1
Problem 3Let $x_1, x_2, \ldots, x_n$ 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 $a_1, a_2, \ldots, a_n$ , not all $0$ , such that $|a_i| \le k - 1$ for all $i$ and $|a_1 x_1 + a_2 x_2 + \cdots + a_n x_n| \le \dfrac{(k-1)\sqrt{n}}{k^n - 1}$ .Solutions: 1
Problem 4Prove that there is no function $f$ from the set of non-negative integers into itself such that $f(f(n)) = n + 1987$ for every non-negative integer $n$ .Solutions: 2
Problem 5Let $n$ 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.Solutions: 1
Problem 6Let n be an integer at least 2. Prove that if $k^2+k+n$ is prime for every integer k with $0\le k\le\sqrt{n/3}$ , then it is prime for every integer k with $0\le k\le n-2$ .Solutions: 1