2007 Indonesia MO Problems/Problem 3
Problem
Let be positive real numbers which satisfy
. Prove that these three inequalities hold:
,
,
.
Solution (credit to vedran6)
Assume that , so
. Let
, making
. Therefore,
.
By substitution, we have
Note that because
, we must have
. Additionally, by the Trivial Inequality,
. Thus, we must have
, is a contradiction. Therefore, we must have
.
Because of symmetry in the equation, we can use similar steps to prove that and that
.
See Also
2007 Indonesia MO (Problems) | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 4 |
All Indonesia MO Problems and Solutions |