Difference between revisions of "1978 USAMO Problems"
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==Problem 3== | ==Problem 3== | ||
| + | You are told that all integers from <math>33</math> to <math>73</math> 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 <math>73</math>. | ||
==Problem 4== | ==Problem 4== | ||
==Problem 5== | ==Problem 5== | ||
Revision as of 19:01, 16 August 2011
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?
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 .
