1998 IMO Shortlist Problems/A3
Let be positive real numbers such that . Prove that
Solution1
Using Titu's Lemma, we rewrite the equation, getting that . which means that . Applying AM-GM inequality, we can get that Which means the left side is greater than or equal to 36. Now let's observe the right side. Using C-S inequality we can easily get that . So In the end, we can get that Since , we can get that Now consider the right side , the left side so left side is always larger or equal to right sides and we are done ~bluesoul
Solution2
WLOG, we assume that , so . Now we can apply Chebyshev's inequality, getting that . Now we can observe that all three terms, get their minimum value while . Which means the minimum value of the original expression holds while getting and we are done ~bluesoul