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Baltic Way
2007 Baltic Way
18
18
Part of
2007 Baltic Way
Problems
(1)
Must a,b,c,d have the same sign?
Source: Baltic Way 2007
11/30/2010
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be non-zero integers, such that the only quadruple of integers
(
x
,
y
,
z
,
t
)
(x, y, z, t)
(
x
,
y
,
z
,
t
)
satisfying the equation
a
x
2
+
b
y
2
+
c
z
2
+
d
t
2
=
0
ax^2+by^2+cz^2+dt^2=0
a
x
2
+
b
y
2
+
c
z
2
+
d
t
2
=
0
is
x
=
y
=
z
=
t
=
0
x=y=z=t=0
x
=
y
=
z
=
t
=
0
. Does it follow that the numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
have the same sign?
number theory proposed
number theory