Difference between revisions of "1981 IMO Problems/Problem 3"
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== Problem == | == Problem == | ||
− | Determine the maximum value of <math> | + | Determine the maximum value of <math>m^2 + n^2 </math>, where <math>m </math> and <math>n </math> are integers satisfying <math> m, n \in \{ 1,2, \ldots , 1981 \} </math> and <math>( n^2 - mn - m^2 )^2 = 1 </math>. |
== Solution == | == Solution == | ||
− | We first observe that since <math> | + | We first observe that since <math>\gcd(m,n)=1 </math>, <math>m</math> and <math>n</math> are [[relatively prime]]. In addition, we note that <math>n \ge m</math>, since if we had <math>n < m</math>, then <math>n^2 -nm -m^2 = n(n-m) - m^2 </math> would be the sum of two negative integers and therefore less than <math>-1</math>. We now observe |
− | < | + | <cmath> |
− | + | (m+k)^2 -(m+k)m - m^2 = -(m^2 - km - k^2) | |
− | + | </cmath>, | |
− | </ | ||
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− | i.e., <math> | + | i.e., <math>(m,n) = (a,b) </math> is a solution [[iff]]. <math>(b, a+b) </math> is also a solution. Therefore, for a solution <math>(m, n)</math>, we can perform the [[Euclidean algorithm]] to reduce it eventually to a solution <math>(1,n) </math>. It is easy to verify that if <math>n </math> is a positive integer, it must be either 2 or 1. Thus by trivial induction, all the positive integer solutions are of the form <math>(F_{n}, F_{n+1})</math>, where the <math>F_i</math> are the [[Fibonacci numbers]]. Simple calculation reveals <math>987</math> and <math>1597</math> to be the greatest Fibonacci numbers less than <math>1981</math>, giving <math>987^2 + 1597^2=3524578</math> as the maximal value. |
{{alternate solutions}} | {{alternate solutions}} | ||
− | == | + | {{IMO box|num-b=2|num-a=4|year=1981}} |
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[[Category:Olympiad Number Theory Problems]] | [[Category:Olympiad Number Theory Problems]] |
Latest revision as of 09:23, 13 May 2019
Problem
Determine the maximum value of , where and are integers satisfying and .
Solution
We first observe that since , and are relatively prime. In addition, we note that , since if we had , then would be the sum of two negative integers and therefore less than . We now observe
,
i.e., is a solution iff. is also a solution. Therefore, for a solution , we can perform the Euclidean algorithm to reduce it eventually to a solution . It is easy to verify that if is a positive integer, it must be either 2 or 1. Thus by trivial induction, all the positive integer solutions are of the form , where the are the Fibonacci numbers. Simple calculation reveals and to be the greatest Fibonacci numbers less than , giving as the maximal value.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
1981 IMO (Problems) • Resources | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
All IMO Problems and Solutions |