**Problem:**

Let a and b satisfy and .

- Prove that if m and n are positive integers with , then .
- For each positive integer n, consider a quadratic function: .

Show that has two roots that are in between -1 and 1.

**Solution:**

Let . Consider with . Since , we have . Hence, . Call this relationship I.

On the other hand, notice that , , , , which implies

….call this relationship II.

From relationships I and II, it follows that

which can be written as

, or equivalently, . That is, .

It remains to prove that . Indeed, as .

The equality occurs if and only if .

2) Since discriminant , has two distinct real roots . Also, note that if , then the following holds:

,

,

.

We conclude that .

*Cheers,*

*Nalin Pithwa.*

**Reference: Selected Problems of the Vietnamese Mathematical Olympiad (1962-2009), Le Hai Chau, Le Hai Khoi. **

Amazon India link: