**Problem:**

The positive numbers a, b, c, A, B, C satisfy

.

Prove that .

**All Soviet Union Olympiad, 1987.**

**Solution:**

Proving that is equivalent to showing that

But this is equivalent to

If , then the above is obviously true. If not, then we have

and thus, our inequality holds.

More later,

Nalin Pithwa

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Neat. I’m seeing an alternate solution where you start with the identity describing the inequality of arithmetic and geometric means in the equivalent form

$xy \leq (x + y)^2 / 4$

which applied to yields

and then summing these we get the slightly stronger inequality

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Thanks for participating !!

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Oh, and there, 3s later I see I misread the identity…Nevermind.

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Yes, many thanks.

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