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Problems
Contests
National and Regional Contests
PEN Problems
PEN A Problems
56
56
Part of
PEN A Problems
Problems
(1)
A 56
Source:
5/25/2007
Let
a
,
b
a, b
a
,
b
, and
c
c
c
be integers such that
a
+
b
+
c
a+b+c
a
+
b
+
c
divides
a
2
+
b
2
+
c
2
a^2 +b^2 +c^2
a
2
+
b
2
+
c
2
. Prove that there are infinitely many positive integers
n
n
n
such that
a
+
b
+
c
a+b+c
a
+
b
+
c
divides
a
n
+
b
n
+
c
n
a^n +b^n +c^n
a
n
+
b
n
+
c
n
.
modular arithmetic
induction
Divisibility Theory