1999 USAMO Problems/Problem 2
Let be a cyclic quadrilateral. Prove that
Let arc of the circumscribed circle (which we assume WLOG has radius 0.5) have value , have , have , and have . Then our inequality reduces to, for :
This is equivalent to by sum-to-product and use of :
Clearly . As sine is increasing over , .
Similarly, . The result now follows after multiplying the first inequality by , the second by , and adding. (Equality holds if and only if and , ie. is a parallelogram.)
--Suli 11:23, 5 October 2014 (EDT)
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