2008 Indonesia MO Problems/Problem 2
Contents
[hide]Problem
Prove that for every positive reals and
,
Solution 1
By the Cauchy-Schwarz Inequality, and
, with equality happening in the earlier inequality when
and equality happening in the latter inequality when
. Because
,
By the AM-GM Inequality, we know that
. For the equality case,
, so
. Additionally, by the AM-GM Inequality,
. For the equality case,
, so
. Because
,
Therefore, since
and
and
, we must have
, with equality happening when
.
Solution 2
Let
Since this function is concave up, according to Jensen's inequality, we can get
which means
.
In this problem, it turns into
.The conclusion we try to find is that
So we can see that
.
Take reciprocal for both sides we can get
.
Take RHS,
.
Now we have to prove that
.
which turns to
. It is always correct according to
inequality, it happens when
.
~bluesoul
See Also
2008 Indonesia MO (Problems) | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 3 |
All Indonesia MO Problems and Solutions |