1987 IMO Problems

Problems of the 1987 IMO Cuba.

Day I

Problem 1

Let $\displaystyle p_n (k)$ be the number of permutations of the set $\displaystyle \{ 1, \ldots , n \} , \; n \ge 1$ , which have exactly $\displaystyle k$ fixed points. Prove that

$\sum_{k=0}^{n} k \cdot p_n (k) = n!$ .

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

Solution

Problem 2

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

Solution

Problem 3

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

$|a_1x_1 + a_2x_2 + \cdots + a_nx_n| \le \frac{ (k-1) \sqrt{n} }{ k^n - 1 }$ .

Solution

Day 2

Problem 4

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

Solution

Problem 5

Let $\displaystyle n$ be an integer greater than or equal to 3. Prove that there is a set of $\displaystyle n$ 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.

Solution

Problem 6

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

Solution

Page

Toolbox

Search

1987 IMO Problems

Contents

Day I

Problem 1

Problem 2

Problem 3

Day 2

Problem 4

Problem 5

Problem 6

Resources