2008 Indonesia MO Problems/Problem 2
Revision as of 19:55, 22 March 2020 by Rockmanex3 (talk | contribs) (Solution to Problem 2 -- oh baby a triple!)
Problem
Prove that for every positive reals and
,
Solution
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
.
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 |