## A very tricky trig problem from China !! RMO or IITJEE Advanced mathematics training

**Problem: (China 2003)**

Let n be a fixed positive integer. Determine the smallest positive real number such that for any , , , , in the interval , if

, then

.

**Solution:**

**(by Yumin Huang)**

The answer is:

, if

, if

, if

The case is trivial.

If ,we claim that with equality if and only if . It suffices to show that

or,

Because , we get

.

By setting,

, the last inequality becomes

, or

Squaring both sides and clearing denominators, we get , that is,

. This establishes our claim.

Now, assume that . We claim that . Note that ; by setting and letting , we find that , and so the left hand side of the desired inequality approaches . It suffices to show that

.

Without loss of generality, assume that . Then,

It suffices to show that **relation ***

But, because ,

Consequently, by the **arithmetic geometric mean inequality, **

Because

we have

or $latex $

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