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2022 Poland - Second Round
3
3
Part of
2022 Poland - Second Round
Problems
(1)
Polish MO 2022 P3
Source: Polish MO 2022 P3
2/12/2022
Positive integers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfying the equation
a
3
+
4
b
+
c
=
a
b
c
,
a^3+4b+c = abc,
a
3
+
4
b
+
c
=
ab
c
,
where
a
≥
c
a \geq c
a
≥
c
and the number
p
=
a
2
+
2
a
+
2
p = a^2+2a+2
p
=
a
2
+
2
a
+
2
is a prime. Prove that
p
p
p
divides
a
+
2
b
+
2
a+2b+2
a
+
2
b
+
2
.
number theory
prime numbers
modular congruences
Divisibility