1978 USAMO Problems/Problem 1
Problem
Given that are real numbers such that
,
.
Determine the maximum value of .
Solution 1
By Cauchy Schwarz, we can see that thus Finally, which means so the maximum value of is .
from: Image from Gon Mathcenter.net
Solution 2
Seeing as we have an inequality with constraints, we can use Lagrange multipliers to solve this problem. We get the following equations:
If , then according to and according to , so . Setting the right sides of and equal yields . Similar steps yield that . Thus, becomes and becomes . Solving the system yields , so the maximum possible value of is .
Solution 3
A re-writing of Solution 1 to avoid the use of Cauchy Schwarz. We have and The second equation times 4, then minus the first equation, The rest follows.
J.Z.
See Also
1978 USAMO (Problems • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.