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National and Regional Contests
The Philippines Contests
Philippine MO
2023 Philippine MO
1
Diophantine muahahaha
Diophantine muahahaha
Source: Philippine MO 2023/1
March 19, 2023
PMO
algebra
number theory
Diophantine equation
Problem Statement
Find all ordered pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of positive integers such that
a
2
+
b
2
+
25
=
15
a
b
a^2 + b^2 + 25 = 15ab
a
2
+
b
2
+
25
=
15
ab
and
a
2
+
a
b
+
b
2
a^2 + ab + b^2
a
2
+
ab
+
b
2
is prime.
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