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1987 IMO Problems

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Problems of the 1987 IMO Cuba.

Contents

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

Resources

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