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Saudi Arabia Contests
Saudi Arabia GMO TST
2018 Saudi Arabia GMO TST
1
gcd(a_1^n + a-1a_2 ...a_n, a_2^n +a_1a_2 ...a_n, ... , a_n^n +a_1a_2 ...a_n)
gcd(a_1^n + a-1a_2 ...a_n, a_2^n +a_1a_2 ...a_n, ... , a_n^n +a_1a_2 ...a_n)
Source: 2018 Saudi Arabia GMO TST II p1
July 31, 2020
number theory
greatest common divisor
GCD
Problem Statement
Let
n
n
n
be an odd positive integer with
n
>
1
n > 1
n
>
1
and let
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2,... , a_n
a
1
,
a
2
,
...
,
a
n
be positive integers such that gcd
(
a
1
,
a
2
,
.
.
.
,
a
n
)
=
1
(a_1, a_2,... , a_n) = 1
(
a
1
,
a
2
,
...
,
a
n
)
=
1
. Let
d
d
d
= gcd
(
a
1
n
+
a
1
⋅
a
2
⋅
⋅
⋅
a
n
,
a
2
n
+
a
1
⋅
a
2
⋅
⋅
⋅
a
n
,
.
.
.
,
a
n
n
+
a
1
⋅
a
2
⋅
⋅
⋅
a
n
)
(a_1^n + a_1\cdot a_2 \cdot \cdot \cdot a_n, a_2^n + a_1\cdot a_2 \cdot \cdot \cdot a_n, ... , a_n^n + a_1\cdot a_2 \cdot \cdot \cdot a_n)
(
a
1
n
+
a
1
⋅
a
2
⋅
⋅
⋅
a
n
,
a
2
n
+
a
1
⋅
a
2
⋅
⋅
⋅
a
n
,
...
,
a
n
n
+
a
1
⋅
a
2
⋅
⋅
⋅
a
n
)
. Show that the possible values of
d
d
d
are
d
=
1
,
d
=
2
d = 1, d = 2
d
=
1
,
d
=
2
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