2002 USA TST Problems/Problem 1
(Titu Andreescu) Let be a triangle. Prove that
We first note that
Hence the inequality becomes
We now note that for , is decreasing and is increasing. It follows that and are sorted in opposing order, so by the rearrangement inequality,
Now, for , increases, and for , decreases; hence if is an acute or right triangle, then and are oppositely sorted. But if one angle, say , is obtuse, then we must have , and and will still be oppositely sorted, and again by the rearrangement inequality,
Adding these two conditions yields the desired inequality.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.