1992 IMO Problems/Problem 6
Problem
For each positive integer , is defined to be the greatest integer such that, for every positive integer , can be written as the sum of positive squares.
(a) Prove that for each .
(b) Find an integer such that .
(c) Prove that there are infinitely many integers such that .
Solution
(a) Let be a positive integer. We will prove that .
Assume for the sake of contradiction that there exists a positive integer such that . Then, there exists a positive integer such that .
Consider the number . By definition of , for every positive integer , can be written as the sum of positive squares. In particular, can be written as the sum of positive squares.
However, it is a well-known result that any positive integer can be expressed as the sum of at most positive squares. Therefore, cannot be expressed as the sum of positive squares, which is a contradiction. Hence, for each .
(b) To find an integer such that , we need to show that can be expressed as the sum of positive squares.
Consider the number . We can express it as the sum of perfect squares of and perfect square of . Therefore, .
(c) To prove that there are infinitely many integers such that , note that for any integer where is a non-negative integer, we have . Since there are infinitely many non-negative integers , there are infinitely many integers such that . By M. Nazaryan.
See Also
1992 IMO (Problems) • Resources | ||
Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Last Question |
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