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2000 AIME Problems
7
System of Equations
System of Equations
Source:
January 5, 2007
number theory
relatively prime
Problem Statement
Suppose that
x
,
x,
x
,
y
,
y,
y
,
and
z
z
z
are three positive numbers that satisfy the equations
x
y
z
=
1
,
xyz=1,
x
yz
=
1
,
x
+
1
z
=
5
,
x+\frac{1}{z}=5,
x
+
z
1
=
5
,
and
y
+
1
x
=
29.
y+\frac{1}{x}=29.
y
+
x
1
=
29.
Then
z
+
1
y
=
m
n
,
z+\frac{1}{y}=\frac{m}{n},
z
+
y
1
=
n
m
,
where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
.
m+n.
m
+
n
.
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