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Canada Contests
Canadian Mathematical Olympiad Qualification Repechage
2022 Canadian Mathematical Olympiad Qualification
3
\sum^{n-1}_{i=0} x_i(3x_i - 4x_{i+1} + x_{i+2}) >= 0
\sum^{n-1}_{i=0} x_i(3x_i - 4x_{i+1} + x_{i+2}) >= 0
Source: Canada Repêchage 2022/3 CMOQR
October 13, 2022
algebra
inequalities
Problem Statement
Consider n real numbers
x
0
,
x
1
,
.
.
.
,
x
n
−
1
x_0, x_1, . . . , x_{n-1}
x
0
,
x
1
,
...
,
x
n
−
1
for an integer
n
≥
2
n \ge 2
n
≥
2
. Moreover, suppose that for any integer
i
i
i
,
x
i
+
n
=
x
i
x_{i+n} = x_i
x
i
+
n
=
x
i
. Prove that
∑
i
=
0
n
−
1
x
i
(
3
x
i
−
4
x
i
+
1
+
x
i
+
2
)
≥
0.
\sum^{n-1}_{i=0} x_i(3x_i - 4x_{i+1} + x_{i+2}) \ge 0.
i
=
0
∑
n
−
1
x
i
(
3
x
i
−
4
x
i
+
1
+
x
i
+
2
)
≥
0.
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