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Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Kyiv City MO
2021 Kyiv City MO
2021 Kyiv City MO Round 1
10.4
10.4
Part of
2021 Kyiv City MO Round 1
Problems
(1)
Beautiful inequality
Source: Kyiv City MO 2021 Round 1, Problem 10.4
12/21/2023
Positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy
a
2
+
b
2
+
c
2
+
a
+
b
+
c
=
6
a^2 + b^2 + c^2 + a + b + c = 6
a
2
+
b
2
+
c
2
+
a
+
b
+
c
=
6
. Prove the following inequality:
2
(
1
a
2
+
1
b
2
+
1
c
2
)
≥
3
+
1
a
+
1
b
+
1
c
2(\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}) \geq 3 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}
2
(
a
2
1
+
b
2
1
+
c
2
1
)
≥
3
+
a
1
+
b
1
+
c
1
Proposed by Oleksii Masalitin
inequalities