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Problems
Contests
National and Regional Contests
PEN Problems
PEN A Problems
51
51
Part of
PEN A Problems
Problems
(1)
A 51
Source:
5/25/2007
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
and
d
d
d
be odd integers such that
0
<
a
<
b
<
c
<
d
0<a<b<c<d
0
<
a
<
b
<
c
<
d
and
a
d
=
b
c
ad=bc
a
d
=
b
c
. Prove that if
a
+
d
=
2
k
a+d=2^{k}
a
+
d
=
2
k
and
b
+
c
=
2
m
b+c=2^{m}
b
+
c
=
2
m
for some integers
k
k
k
and
m
m
m
, then
a
=
1
a=1
a
=
1
.
Divisibility Theory