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Russia Contests
All-Russian Olympiad Regional Round
2023 All-Russian Olympiad Regional Round
10.10
10.10
Part of
2023 All-Russian Olympiad Regional Round
Problems
(1)
Another easy inequality with square roots
Source: ARO Regional stage 2023 10.10
2/16/2023
Prove that for all positive reals
x
,
y
,
z
x, y, z
x
,
y
,
z
, the inequality
(
x
−
y
)
3
x
2
+
y
2
+
(
y
−
z
)
3
y
2
+
z
2
+
(
z
−
x
)
3
z
2
+
x
2
≥
0
(x-y)\sqrt{3x^2+y^2}+(y-z)\sqrt{3y^2+z^2}+(z-x)\sqrt{3z^2+x^2} \geq 0
(
x
−
y
)
3
x
2
+
y
2
+
(
y
−
z
)
3
y
2
+
z
2
+
(
z
−
x
)
3
z
2
+
x
2
≥
0
is satisfied.
algebra
Russia
inequalities proposed