Problem: If A, B, C are angles of a triangle, then find the maximum value of
expressed as a reduced rational. (IITJEE 1984).
First Hint: Prove that the maximum must occur for an equilateral triangle.
Second Hint: For a fixed value of one of the three angles, show that the maximum must occur when the other two are equal.
Solution:
Here, the three angles A, B, C are constrained by the requirement that they are non-negative and add up to
. Let S be the set of all ordered triples of the form
which satisfy these constraints, that is,
and denote by
. Then, the problem amounts to finding the maximum value of the function f over the set S.
Now, suppose . We claim that unless
, we can find some
such that
. Indeed, we let
and
. Note that
and hence,
. Note that
(except in the degenerate case where ). Further,
which is greater than
. This gives,
which is less than
which equals .
By a similar reasoning, unless , we can find some
such that
. A similar assertion holds if
. It then follows that when
is maximum, A, B, C must be all equal and hence, each equals
(since ). But,
. So, the maximum value of
.
More later,
Nalin Pithwa