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South Africa National Olympiad
2002 South africa National Olympiad
6
WEIRD equation with rationals
WEIRD equation with rationals
Source: South Africa 2002
October 13, 2005
number theory unsolved
number theory
Problem Statement
Find all rational numbers
a
a
a
,
b
b
b
,
c
c
c
and
d
d
d
such that
8
a
2
−
3
b
2
+
5
c
2
+
16
d
2
−
10
a
b
+
42
c
d
+
18
a
+
22
b
−
2
c
−
54
d
=
42
,
8a^2 - 3b^2 + 5c^2 + 16d^2 - 10ab + 42cd + 18a + 22b - 2c - 54d = 42,
8
a
2
−
3
b
2
+
5
c
2
+
16
d
2
−
10
ab
+
42
c
d
+
18
a
+
22
b
−
2
c
−
54
d
=
42
,
15
a
2
−
3
b
2
+
21
c
2
−
5
d
2
+
4
a
b
+
32
c
d
−
28
a
+
14
b
−
54
c
−
52
d
=
−
22.
15a^2 - 3b^2 + 21c^2 - 5d^2 + 4ab +32cd - 28a + 14b - 54c - 52d = -22.
15
a
2
−
3
b
2
+
21
c
2
−
5
d
2
+
4
ab
+
32
c
d
−
28
a
+
14
b
−
54
c
−
52
d
=
−
22.
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