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Iran MO (2nd Round)
2018 Iran MO (2nd Round)
3
3
Part of
2018 Iran MO (2nd Round)
Problems
(1)
Iran MO Number theory
Source: Iran second round 2018 day 1 problem 3
4/26/2018
Let
a
>
k
a>k
a
>
k
be natural numbers and
r
1
<
r
2
<
…
r
n
,
s
1
<
s
2
<
⋯
<
s
n
r_1<r_2<\dots r_n,s_1<s_2<\dots <s_n
r
1
<
r
2
<
…
r
n
,
s
1
<
s
2
<
⋯
<
s
n
be sequences of natural numbers such that:
(
a
r
1
+
k
)
(
a
r
2
+
k
)
…
(
a
r
n
+
k
)
=
(
a
s
1
+
k
)
(
a
s
2
+
k
)
…
(
a
s
n
+
k
)
(a^{r_1}+k)(a^{r_2}+k)\dots (a^{r_n}+k)=(a^{s_1}+k)(a^{s_2}+k)\dots (a^{s_n}+k)
(
a
r
1
+
k
)
(
a
r
2
+
k
)
…
(
a
r
n
+
k
)
=
(
a
s
1
+
k
)
(
a
s
2
+
k
)
…
(
a
s
n
+
k
)
Prove that these sequences are equal.
number theory