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