2002 USA TST Problems/Problem 1
Problem
(Titu Andreescu)
Let be a triangle. Prove that
Solution
We first note that
.
Hence the inequality becomes
,
or
.
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.