**Ref: Titu Andreescu**

**Problem:**

Let be the set of nth roots of unity. Prove that the following statements are equivalent:

a) there is such that .

b) there is such that .

**(Romanian Mathematical Olympiad — Second Round, 1990).**

**Solution:**

Assume that there exists such that . Setting we have , hence, . On the other hand, and , hence, , as desired.

Conversely, if , , set . Since , and , we have , and , as desired.

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More later,

Nalin Pithwa