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IMC
2005 IMC
6
IMC 2005 day 2 pb 6
IMC 2005 day 2 pb 6
Source: Peter
July 26, 2005
linear algebra
matrix
IMC
college contests
Problem Statement
6. If
p
,
q
p,q
p
,
q
are rationals,
r
=
p
+
7
q
r=p+\sqrt{7}q
r
=
p
+
7
q
, then prove there exists a matrix
(
a
b
c
d
)
∈
M
2
(
Z
)
−
(
±
I
2
)
\left(\begin{array}{cc}a&b\\c&d\end{array}\right) \in M_{2}(Z)- ( \pm I_{2})
(
a
c
b
d
)
∈
M
2
(
Z
)
−
(
±
I
2
)
for which
a
r
+
b
c
r
+
d
=
r
\frac{ar+b}{cr+d}=r
cr
+
d
a
r
+
b
=
r
and
d
e
t
(
A
)
=
1
det(A)=1
d
e
t
(
A
)
=
1
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