2020 INMO Problems/Problem 4
Problem
Let be an integer and let
be
real numbers such that
. Prove that
Solution(1)
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
Solution(2)
Define,.
,
In general ,.
.
.
Using Tchevbycev inequality we have ,
.
.
.
.[Applying Induction on successive
].
.
.
.
.[using GM-AM]
.
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
.
and
.
Our induction step is complete.
This two claim leads and equality for
.