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1981 IMO Problems

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

Contents

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.

Solution

Problem 6

The function \displaystyle f(x,y) satisfies

(1) \displaystyle f(0,y)=y+1,

(2) \displaystyle f(x+1,0)=f(x,1),

(3) \displaystyle f(x+1,y+1)=f(x,f(x+1,y)),

for all non-negative integers \displaystyle x,y. Determine \displaystyle f(4,1981).

Solution

Resources

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