2020 INMO Problems/Problem 4
Let be an integer and let be real numbers such that . Prove that
For , we want to show that where and . This is equivalent to showing that , which is true.
Suppose, now, that the given inequality is true for , where . Now, consider reals with sum . Then, and , so by induction hypothesis,
This means or as desired. ~biomathematics
In general ,.
Using Tchevbycev inequality we have , .
.[Applying Induction on successive ].
Since, and , Hence , .
The RHS inequality is trivial by AM-GM inequality.
For LHS inequality I would like to use induction.
We have , and .
. Suppose , the statement is true for such that , and .
Now , consider .
Suppose , is median of the sequence, and and .
Our induction step is complete.
This two claim leads and equality for . ~trishan