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National and Regional Contests
Argentina Contests
Argentina National Olympiad
1997 Argentina National Olympiad
3
3
Part of
1997 Argentina National Olympiad
Problems
(1)
max S=x_1(1-x_2)+x_2(1-x_3)+x_3(1-x_4)+\cdots +x_{99}(1-x_{100})+x_ {100}(1-x_1)
Source: Argentina 1997 OMA L3 p3
5/13/2024
Let
x
1
,
x
2
,
x
3
,
…
,
x
100
x_1,x_2,x_3,\ldots ,x_{100}
x
1
,
x
2
,
x
3
,
…
,
x
100
be one hundred real numbers greater than or equal to
0
0
0
and less than or equal to
1
1
1
. Find the maximum possible value of the sum
S
=
x
1
(
1
−
x
2
)
+
x
2
(
1
−
x
3
)
+
x
3
(
1
−
x
4
)
+
⋯
+
x
99
(
1
−
x
100
)
+
x
100
(
1
−
x
1
)
.
S=x_1(1-x_2)+x_2(1-x_3)+x_3(1-x_4)+\cdots +x_{99}(1-x_{100})+x_ {100}(1-x_1).
S
=
x
1
(
1
−
x
2
)
+
x
2
(
1
−
x
3
)
+
x
3
(
1
−
x
4
)
+
⋯
+
x
99
(
1
−
x
100
)
+
x
100
(
1
−
x
1
)
.
algebra
Sum
inequalities