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Poland Contests
Poland - Second Round
1981 Poland - Second Round
4
4
Part of
1981 Poland - Second Round
Problems
(1)
2 sequences with quotients and remainders
Source: Polish MO Recond Round 1981 p4
9/9/2024
The given natural numbers are
k
,
n
k, n
k
,
n
. We inductively define two sequences of numbers
(
a
j
)
(a_j)
(
a
j
)
and
(
r
j
)
(r_j)
(
r
j
)
as follows: Step one: we divide
k
k
k
by
n
n
n
and get the quotient
a
1
a_1
a
1
and the remainder
r
i
r_i
r
i
, step j: we divide
k
+
r
j
−
1
k+r_{j-1}
k
+
r
j
−
1
by
n
n
n
and get the quotient
a
j
a_j
a
j
and the remainder
r
j
r_j
r
j
. Calculate the sum of
a
1
+
…
+
a
n
a_1 + \ldots + a_n
a
1
+
…
+
a
n
.
number theory
remainder