1978 USAMO Problems

Problem 1

The sum of 5 real numbers is 8 and the sum of their squares is 16. What is the largest possible value for one of the numbers?

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?