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2022 Kyiv City MO Round 1
Problem 2
Hardest algebra ever
Hardest algebra ever
Source: Kyiv City MO 2022 Round 1, Problem 9.2
January 23, 2022
algebra
inequalities
Problem Statement
For any reals
x
,
y
x, y
x
,
y
, show the following inequality:
(
x
+
4
)
2
+
(
y
+
2
)
2
+
(
x
−
5
)
2
+
(
y
+
4
)
2
≤
(
x
−
2
)
2
+
(
y
−
6
)
2
+
(
x
−
5
)
2
+
(
y
−
6
)
2
+
20
\sqrt{(x+4)^2 + (y+2)^2} + \sqrt{(x-5)^2 + (y+4)^2} \le \sqrt{(x-2)^2 + (y-6)^2} + \sqrt{(x-5)^2 + (y-6)^2} + 20
(
x
+
4
)
2
+
(
y
+
2
)
2
+
(
x
−
5
)
2
+
(
y
+
4
)
2
≤
(
x
−
2
)
2
+
(
y
−
6
)
2
+
(
x
−
5
)
2
+
(
y
−
6
)
2
+
20
(Proposed by Bogdan Rublov)
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