## A Trigonometry problem for IITJEE Advanced Mathematics or Mathematics Olympiad

**Ref: Titu Andreescu and Zumin Feng.**

**Problem:**

Let x, y, z be positive real numbers such that . Determine the minimum value of

**Solution:**

An application of **Cauchy**–**Schwarz inequality **makes this as a one step problem. Nevertheless, we present a proof which involves only the easier inequality for real numbers x and y by setting first and and second and .

Clearly, z is a real number in the interval . Hence, there is an angle a such that . Then, , or . For an angle b, we have . Hence, we can set , and for some b, it suffices to find the minimum value of

or

Expanding the right hand side gives

Equality holds when and , which implies that and . Because , equality holds when and , that is, , , .

More later.

Nalin Pithwa

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