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National and Regional Contests
Italy Contests
ITAMO
2020 ITAMO
2
2
Part of
2020 ITAMO
Problems
(1)
Simple Italian NT
Source: Italy National Olympiad 2020 P2
9/30/2020
Determine all the pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of positive integers, such that all the following three conditions are satisfied: 1-
b
>
a
b>a
b
>
a
and
b
ā
a
b-a
b
ā
a
is a prime number 2- The last digit of the number
a
+
b
a+b
a
+
b
is
3
3
3
3- The number
a
b
ab
ab
is a square of an integer.
number theory