1981 IMO Problems

Problems of the 22nd IMO 1981 U.S.A.

Day I

Problem 1

$\displaystyle P$ is a point inside a given triangle $\displaystyle ABC$ . $\displaystyle D, E, F$ are the feet of the perpendiculars from $\displaystyle P$ to the lines $\displaystyle BC, CA, AB$ , respectively. Find all $\displaystyle P$ for which

$\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$

is least.

Solution

Problem 2

Let $\displaystyle 1 \le r \le n$ and consider all subsets of $\displaystyle r$ elements of the set $\{ 1, 2, \ldots , n \}$ . Each of these subsets has a smallest member. Let $\displaystyle F(n,r)$ denote the arithmetic mean of these smallest numbers; prove that

$F(n,r) = \frac{n+1}{r+1}.$

Solution

Problem 3

Determine the maximum value of $\displaystyle m^2 + n^2$ , where $\displaystyle m$ and $\displaystyle n$ are integers satisfying $m, n \in \{ 1,2, \ldots , 1981 \}$ and $\displaystyle ( n^2 - mn - m^2 )^2 = 1$ .

Solution

Day II

Problem 4

(a) For which values of $\displaystyle n>2$ is there a set of $\displaystyle n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $\displaystyle n-1$ numbers?

(b) For which values of $\displaystyle n>2$ is there exactly one set having the stated property?

Solution

Problem 5

Three congruent circles have a common point $\displaystyle O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle and the point $\displaystyle O$ are collinear.