such that
such that
is self-bounded if for each 
,
there exist at least 
terms of the sequence that are less than or equal to 
for a self-bounded sequence 
Y by
Given a real number 
. Prove that for every positive real number 
with 
, the following inequality holds:
![\[ \sqrt{\frac{x^2+y^2}{2}} + \frac{C}{x+y} \geqslant 1 + \frac{C}{2}.\]](//latex.artofproblemsolving.com/a/9/6/a963b229c9ab252dde1ca99f3c4b8708c96ff688.png)
Proposed by Fajar Yuliawan, Indonesia
. Prove that for every positive real number 
with 
, the following inequality holds:![\[ \sqrt{\frac{x^2+y^2}{2}} + \frac{C}{x+y} \geqslant 1 + \frac{C}{2}.\]](http://latex.artofproblemsolving.com/a/9/6/a963b229c9ab252dde1ca99f3c4b8708c96ff688.png)
Proposed by Fajar Yuliawan, Indonesia
This post has been edited 3 times. Last edited by GreenTea2593, Jan 21, 2025, 3:51 AM
Reason: punctuation
Reason: punctuation
.
must be true. If 
then 
,
then above inequality is equivalent to 
which is true.
by AM-QM.
