1978 USAMO Problems/Problem 1
Given that are real numbers such that
Determine the maximum value of .
By Cauchy Schwarz, we can see that thus Finally, which means so the maximum value of is .
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 .
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.
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