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Contests
National and Regional Contests
China Contests
China Northern MO
2019 China Northern MO
3
2019 CNMO P3
2019 CNMO P3
Source: China North MO 2019, Problem 3
February 21, 2020
algebra
Problem Statement
n
(
n
≥
2
)
n(n\geq2)
n
(
n
≥
2
)
is a given intenger, and
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
are real numbers. For any
i
=
1
,
2
,
⋯
,
n
i=1,2,\cdots ,n
i
=
1
,
2
,
⋯
,
n
,
a
i
≠
−
1
,
a
i
+
2
=
a
i
2
+
a
i
a
i
+
1
+
1
.
a_i\neq -1,a_{i+2}=\frac{a_i^2+a_i}{a_{i+1}+1}.
a
i
=
−
1
,
a
i
+
2
=
a
i
+
1
+
1
a
i
2
+
a
i
.
Prove:
a
1
=
a
2
=
⋯
=
a
n
a_1=a_2=\cdots=a_n
a
1
=
a
2
=
⋯
=
a
n
. (Note:
a
n
+
1
=
a
1
,
a
n
+
2
=
a
2
a_{n+1}=a_1,a_{n+2}=a_2
a
n
+
1
=
a
1
,
a
n
+
2
=
a
2
.)
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