MathDB
Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1985 Spain Mathematical Olympiad
2
2
Part of
1985 Spain Mathematical Olympiad
Problems
(1)
exists a subset E of Z \to Z so that ... ?
Source: Spanish Mathematical Olympiad 1985 P2
8/2/2018
Determine if there exists a subset
E
E
E
of
Z
×
Z
Z \times Z
Z
×
Z
with the properties: (i)
E
E
E
is closed under addition, (ii)
E
E
E
contains
(
0
,
0
)
,
(0,0),
(
0
,
0
)
,
(iii) For every
(
a
,
b
)
≠
(
0
,
0
)
,
E
(a,b) \ne (0,0), E
(
a
,
b
)
=
(
0
,
0
)
,
E
contains exactly one of
(
a
,
b
)
(a,b)
(
a
,
b
)
and
−
(
a
,
b
)
-(a,b)
−
(
a
,
b
)
.Remark: We define
(
a
,
b
)
+
(
a
′
,
b
′
)
=
(
a
+
a
′
,
b
+
b
′
)
(a,b)+(a',b') = (a+a',b+b')
(
a
,
b
)
+
(
a
′
,
b
′
)
=
(
a
+
a
′
,
b
+
b
′
)
and
−
(
a
,
b
)
=
(
−
a
,
−
b
)
-(a,b) = (-a,-b)
−
(
a
,
b
)
=
(
−
a
,
−
b
)
.
Subset
Integers
algebra