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National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2019 Moldova Team Selection Test
6
Schur inequality
Schur inequality
Source: Moldova tst 2019
March 9, 2019
Inequality
inequalities
Moldova
Problem Statement
Let
a
,
b
,
c
≥
0
a,b,c \ge 0
a
,
b
,
c
≥
0
such that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
and
s
≥
5
s \ge 5
s
≥
5
. Prove that
s
(
a
2
+
b
2
+
c
2
)
≤
3
(
s
−
3
)
(
a
3
+
b
3
+
c
3
)
+
1
s(a^2+b^2+c^2) \le 3(s-3)(a^3+b^3+c^3)+1
s
(
a
2
+
b
2
+
c
2
)
≤
3
(
s
−
3
)
(
a
3
+
b
3
+
c
3
)
+
1
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