1999 USAMO Problems/Problem 2
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Problem
Let be a cyclic quadrilateral. Prove that
Solution
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)
See Also
1999 USAMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
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