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Indonesia Regional
2024 Indonesia Regional
1
1
Part of
2024 Indonesia Regional
Problems
(1)
Indonesia Regional Inequality
Source: 2024 Indonesia Regional MO Essay Problem 1
5/6/2024
Given a real number
C
⩽
2
C\leqslant 2
C
⩽
2
. Prove that for every positive real number
x
,
y
x,y
x
,
y
with
x
y
=
1
xy=1
x
y
=
1
, the following inequality holds:
x
2
+
y
2
2
+
C
x
+
y
⩾
1
+
C
2
\sqrt{\frac{x^2+y^2}{2}} + \frac{C}{x+y} \geqslant 1 + \frac{C}{2}
2
x
2
+
y
2
+
x
+
y
C
⩾
1
+
2
C
Proposed by Fajar Yuliawan, Indonesia
Inequality
inequalities
algebra
Indonesia
RMO