1978 USAMO Problems

Problems from the 1978 USAMO.

Problem 1

Given that are real numbers such that

,

.

Determine the maximum value of .

Solution

Problem 2

Two square maps cover exactly the same area of terrain on different scales. The smaller map is placed on top of the larger map and inside its borders. Show that there is a unique point on the top map which lies exactly above the corresponding point on the lower map. How can this point be constructed?

Solution

Problem 3

You are told that all integers from to inclusive can be expressed as a sum of positive integers whose reciprocals sum to 1. Show that the same is true for all integers greater than .

Solution

Problem 4

Show that if the angle between each pair of faces of a tetrahedron is equal, then the tetrahedron is regular. Does a tetrahedron have to be regular if five of the angles are equal?

Solution

Problem 5

There are 9 delegates at a conference, each speaking at most three languages. Given any three delegates, at least 2 speak a common language. Show that there are three delegates with a common language.

Solution

See Also

1978 USAMO (Problems • Resources)
Preceded by 1977 USAMO	Followed by 1979 USAMO
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions