1994 USAMO Problems/Problem 4

Since each $a_{i}$ is positive, by Muirhead's inequality, $2(\sum a_{i}^2) \ge (\sum a)^2 \ge n$ . Now we claim that $\frac{n}{2}> frac{1}{4}(1+...\frac{1}{n)}$

$n=1$ , giving $1/2>1/4$ works, but we set the base case $n=2$ , which gives $1>3/8$ . Now assume that it works for $n$ . By our assumption, now we must prove that, for $n+1$ case, $1/2>\frac{1}{n+1}$ , which is clearly true for $n>1$ . So we are done.

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1994 USAMO Problems/Problem 4