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.
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