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Serbia National Math Olympiad
2019 Serbia National Math Olympiad
6
2019 Serbia MO Day 2 P6
2019 Serbia MO Day 2 P6
Source: 2019 Serbia MO
April 7, 2019
algebra
Sequence
Problem Statement
Sequences
(
a
n
)
n
=
0
∞
(a_n)_{n=0}^{\infty}
(
a
n
)
n
=
0
∞
and
(
b
n
)
n
=
0
∞
(b_n)_{n=0}^{\infty}
(
b
n
)
n
=
0
∞
are defined with recurrent relations :
a
0
=
0
,
a
1
=
1
,
a
n
+
1
=
2018
n
a
n
+
a
n
−
1
for
n
≥
1
a_0=0 , \;\;\; a_1=1, \;\;\;\; a_{n+1}=\frac{2018}{n} a_n+ a_{n-1}\;\;\; \text {for }\;\;\; n\geq 1
a
0
=
0
,
a
1
=
1
,
a
n
+
1
=
n
2018
a
n
+
a
n
−
1
for
n
≥
1
and
b
0
=
0
,
b
1
=
1
,
b
n
+
1
=
2020
n
b
n
+
b
n
−
1
for
n
≥
1
b_0=0 , \;\;\; b_1=1, \;\;\;\; b_{n+1}=\frac{2020}{n} b_n+ b_{n-1}\;\;\; \text {for }\;\;\; n\geq 1
b
0
=
0
,
b
1
=
1
,
b
n
+
1
=
n
2020
b
n
+
b
n
−
1
for
n
≥
1
Prove that :
a
1010
1010
=
b
1009
1009
\frac{a_{1010}}{1010}=\frac{b_{1009}}{1009}
1010
a
1010
=
1009
b
1009
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