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Iran MO (2nd Round)
2017 Iran MO (2nd Round)
1
1
Part of
2017 Iran MO (2nd Round)
Problems
(1)
Number theory - Iran
Source: Iran MO 2017 - 2nd Round - P1
4/20/2017
a) Prove that there doesn't exist sequence
a
1
,
a
2
,
a
3
,
.
.
.
∈
N
a_1,a_2,a_3,... \in \mathbb{N}
a
1
,
a
2
,
a
3
,
...
∈
N
such that:
∀
i
<
j
:
g
c
d
(
a
i
+
j
,
a
j
+
i
)
=
1
\forall i<j: gcd(a_i+j,a_j+i)=1
∀
i
<
j
:
g
c
d
(
a
i
+
j
,
a
j
+
i
)
=
1
b) Let
p
p
p
be an odd prime number. Prove that there exist sequence
a
1
,
a
2
,
a
3
,
.
.
.
∈
N
a_1,a_2,a_3,... \in \mathbb{N}
a
1
,
a
2
,
a
3
,
...
∈
N
such that:
∀
i
<
j
:
p
∤
g
c
d
(
a
i
+
j
,
a
j
+
i
)
\forall i<j: p \not | gcd(a_i+j,a_j+i)
∀
i
<
j
:
p
∣
g
c
d
(
a
i
+
j
,
a
j
+
i
)
number theory
prime numbers