1994 USAMO Problems/Problem 4
Let be a sequence of positive real numbers satisfying for all . Prove that, for all
Since each is positive, by Muirhead's inequality, . Now we claim that
, giving works, but we set the base case , which gives . Now assume that it works for . By our assumption, now we must prove that, for case, , which is clearly true for . So we are done.
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