**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