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2024 Junior Balkan MO
1
Junior Balkan Mathematical Olympiad 2024- P1
Junior Balkan Mathematical Olympiad 2024- P1
Source: JBMO 2024
June 27, 2024
JBMO
Balkan
inequalities
algebra
Problem Statement
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers such that
a
2
+
b
2
+
c
2
=
1
4
.
a^2 + b^2 + c^2 = \frac{1}{4}.
a
2
+
b
2
+
c
2
=
4
1
.
Prove that
1
b
2
+
c
2
+
1
c
2
+
a
2
+
1
a
2
+
b
2
≤
2
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
.
\frac{1}{\sqrt{b^2 + c^2}} + \frac{1}{\sqrt{c^2 + a^2}} + \frac{1}{\sqrt{a^2 + b^2}} \le \frac{\sqrt{2}}{(a + b)(b + c)(c + a)}.
b
2
+
c
2
1
+
c
2
+
a
2
1
+
a
2
+
b
2
1
≤
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
2
.
Proposed by Petar Filipovski, Macedonia
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