2008 Indonesia MO Problems/Problem 2
Contents
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 |