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Russia Contests
All-Russian Olympiad Regional Round
2023 All-Russian Olympiad Regional Round
9.9
9.9
Part of
2023 All-Russian Olympiad Regional Round
Problems
(1)
Easy inequality with square roots
Source: ARO Regional stage 2023 9.9
2/16/2023
Find the largest real
m
m
m
, such that for all positive real
a
,
b
,
c
a, b, c
a
,
b
,
c
with sum
1
1
1
, the inequality
a
b
a
b
+
c
+
b
c
b
c
+
a
+
c
a
c
a
+
b
≥
m
\sqrt{\frac{ab} {ab+c}}+\sqrt{\frac{bc} {bc+a}}+\sqrt{\frac{ca} {ca+b}} \geq m
ab
+
c
ab
+
b
c
+
a
b
c
+
c
a
+
b
c
a
≥
m
is satisfied.
algebra
Russia
inequalities proposed
Inequality