1983 IMO Problems/Problem 6
Let , and be the lengths of the sides of a triangle. Prove that
Determine when equality occurs.
By Ravi substitution, let , , . Then, the triangle condition becomes . After some manipulation, the inequality becomes:
By Cauchy, we have:
with equality if and only if . So the inequality holds with equality if and only if x = y = z. Thus the original inequality has equality if and only if the triangle is equilateral.
Without loss of generality, let . By Muirhead or by AM-GM, we see that .
If we can show that , we are done, since then , and we can divide by .
We first see that, , so .
Factoring, this becomes . This is the same as:
Expanding and refactoring, this is equal to . (This step makes more sense going backwards.)
Expanding this out, we have
which is the desired result.
Let be the semiperimeter, , of the triangle. Then, , , and . We let , and (Note that are all positive, since all sides must be shorter than the semiperimeter.) Then, we have , , and . Note that , so Plugging this into and doing some expanding and cancellation, we get The fact that each term on the left hand side has at least two variables multiplied motivates us to divide the inequality by , which we know is positive from earlier so we can maintain the sign of the inequality. This gives We move the negative terms to the right, giving We rewrite this as where is any real number. (This works because if we evaulate the cyclic sum, then as long as the coefficients of and on the right sum to 1 the right side will be .
Thus, we need to show that there exists a real number such that for all positive . We claim that works. This becomes , and since is positive we can multiply by to yield , or , which is obviously true by the Trivial Inequality. Thus, we are done with part (a).
(To clarify how this works, we have , , and , we add these inequalities to get )
Equality occurs if and only if , , and at the same time, since if they are not all equal then the left side will be greater than the right side. From there, it is easy to see that the equality case is , which is .
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