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International Contests
EGMO
2022 EGMO
4
n \neq N
n \neq N
Source: EGMO 2022/4
April 9, 2022
EGMO
algebra
EGMO2022
Problem Statement
Given a positive integer
n
≥
2
n \ge 2
n
≥
2
, determine the largest positive integer
N
N
N
for which there exist
N
+
1
N+1
N
+
1
real numbers
a
0
,
a
1
,
…
,
a
N
a_0, a_1, \dots, a_N
a
0
,
a
1
,
…
,
a
N
such that
(
1
)
(1) \
(
1
)
a
0
+
a
1
=
−
1
n
,
a_0+a_1 = -\frac{1}{n},
a
0
+
a
1
=
−
n
1
,
and
(
2
)
(2) \
(
2
)
(
a
k
+
a
k
−
1
)
(
a
k
+
a
k
+
1
)
=
a
k
−
1
−
a
k
+
1
(a_k+a_{k-1})(a_k+a_{k+1})=a_{k-1}-a_{k+1}
(
a
k
+
a
k
−
1
)
(
a
k
+
a
k
+
1
)
=
a
k
−
1
−
a
k
+
1
for
1
≤
k
≤
N
−
1
1 \le k \le N-1
1
≤
k
≤
N
−
1
.
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