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International Contests
International Olympiad of Metropolises
2021 IOM
3
A tricky #3 algebra
A tricky #3 algebra
Source: IOM 2021 #3
December 8, 2021
algebra
Problem Statement
Let
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
(
n
≥
2
n\geq 2
n
≥
2
) be nonnegative real numbers whose sum is
n
2
\frac{n}{2}
2
n
. For every
i
=
1
,
…
,
n
i=1,\ldots,n
i
=
1
,
…
,
n
define
b
i
=
a
i
+
a
i
a
i
+
1
+
a
i
a
i
+
1
a
i
+
2
+
⋯
+
a
i
a
i
+
1
⋯
a
i
+
n
−
2
+
2
a
i
a
i
+
1
⋯
a
i
+
n
−
1
b_i=a_i+a_ia_{i+1}+a_ia_{i+1}a_{i+2}+\cdots+ a_ia_{i+1}\cdots a_{i+n-2}+2a_ia_{i+1}\cdots a_{i+n-1}
b
i
=
a
i
+
a
i
a
i
+
1
+
a
i
a
i
+
1
a
i
+
2
+
⋯
+
a
i
a
i
+
1
⋯
a
i
+
n
−
2
+
2
a
i
a
i
+
1
⋯
a
i
+
n
−
1
where
a
j
+
n
=
a
j
a_{j+n}=a_j
a
j
+
n
=
a
j
for every
j
j
j
. Prove that
b
i
≥
1
b_i\geq 1
b
i
≥
1
holds for at least one index
i
i
i
.
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