Difference between revisions of "1992 IMO Problems/Problem 6"
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(c) To prove that there are infinitely many integers <math>n</math> such that <math>S(n) = n^2 - 14</math>, note that for any integer <math>n = 4 + 15k</math> where <math>k</math> is a non-negative integer, we have <math>S(n) = n^2 - 14</math>. Since there are infinitely many non-negative integers <math>k</math>, there are infinitely many integers <math>n</math> such that <math>S(n) = n^2 - 14</math>. | (c) To prove that there are infinitely many integers <math>n</math> such that <math>S(n) = n^2 - 14</math>, note that for any integer <math>n = 4 + 15k</math> where <math>k</math> is a non-negative integer, we have <math>S(n) = n^2 - 14</math>. Since there are infinitely many non-negative integers <math>k</math>, there are infinitely many integers <math>n</math> such that <math>S(n) = n^2 - 14</math>. | ||
+ | By M. Nazaryan. | ||
==See Also== | ==See Also== |
Latest revision as of 12:27, 7 April 2024
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 |
All IMO Problems and Solutions |