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QEDMO
2005 QEDMO 1st
11 (Z3)
11 (Z3)
Part of
2005 QEDMO 1st
Problems
(1)
A+b+c divides a²+b²+c²
Source: QEDMO 2005
11/8/2005
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive integers such that
a
2
+
b
2
+
c
2
a^2+b^2+c^2
a
2
+
b
2
+
c
2
is divisble by
a
+
b
+
c
a+b+c
a
+
b
+
c
. Prove that at least two of the numbers
a
3
,
b
3
,
c
3
a^3,b^3,c^3
a
3
,
b
3
,
c
3
leave the same remainder by division through
a
+
b
+
c
a+b+c
a
+
b
+
c
.
modular arithmetic
number theory proposed
number theory