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2023 China National Olympiad
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Part of
2023 China National Olympiad
Problems
(1)
China MO 2023 Problem 1: Brother sequences
Source: China MO 2023 Day 1 P1
12/29/2022
Define the sequences
(
a
n
)
,
(
b
n
)
(a_n),(b_n)
(
a
n
)
,
(
b
n
)
by \begin{align*} & a_n, b_n > 0, \forall n\in\mathbb{N_+} \\ & a_{n+1} = a_n - \frac{1}{1+\sum_{i=1}^n\frac{1}{a_i}} \\ & b_{n+1} = b_n + \frac{1}{1+\sum_{i=1}^n\frac{1}{b_i}} \end{align*} 1) If
a
100
b
100
=
a
101
b
101
a_{100}b_{100} = a_{101}b_{101}
a
100
b
100
=
a
101
b
101
, find the value of
a
1
−
b
1
a_1-b_1
a
1
−
b
1
; 2) If
a
100
=
b
99
a_{100} = b_{99}
a
100
=
b
99
, determine which is larger between
a
100
+
b
100
a_{100}+b_{100}
a
100
+
b
100
and
a
101
+
b
101
a_{101}+b_{101}
a
101
+
b
101
.
algebra
Sequence