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Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2020 Vietnam National Olympiad
2
Inequation
Inequation
Source: VMO 2020 - Day 1-P2
December 27, 2019
inequation
inequalities
vmo
algebra
Problem Statement
a)Let
a
,
b
,
c
∈
R
a,b,c\in\mathbb{R}
a
,
b
,
c
∈
R
and
a
2
+
b
2
+
c
2
=
1
a^2+b^2+c^2=1
a
2
+
b
2
+
c
2
=
1
.Prove that:
∣
a
−
b
∣
+
∣
b
−
c
∣
+
∣
c
−
a
∣
≤
2
2
|a-b|+|b-c|+|c-a|\le2\sqrt{2}
∣
a
−
b
∣
+
∣
b
−
c
∣
+
∣
c
−
a
∣
≤
2
2
b) Let
a
1
,
a
2
,
.
.
a
2019
∈
R
a_1,a_2,..a_{2019}\in\mathbb{R}
a
1
,
a
2
,
..
a
2019
∈
R
and
∑
i
=
1
2019
a
i
2
=
1
\sum_{i=1}^{2019}a_i^2=1
∑
i
=
1
2019
a
i
2
=
1
.Find the maximum of:
S
=
∣
a
1
−
a
2
∣
+
∣
a
2
−
a
3
∣
+
.
.
.
+
∣
a
2019
−
a
1
∣
S=|a_1-a_2|+|a_2-a_3|+...+|a_{2019}-a_1|
S
=
∣
a
1
−
a
2
∣
+
∣
a
2
−
a
3
∣
+
...
+
∣
a
2019
−
a
1
∣
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