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MAA AMC
AMC 12/AHSME
1980 AMC 12/AHSME
29
System of Equations
System of Equations
Source:
January 2, 2006
algebra
system of equations
Problem Statement
How many ordered triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of integers satisfy the system of equations below?
x
2
−
3
x
y
+
2
y
z
−
z
2
=
31
−
x
2
+
6
y
z
+
2
z
2
=
44
x
2
+
x
y
+
8
z
2
=
100
\begin{array}{l} x^2-3xy+2yz-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \end{array}
x
2
−
3
x
y
+
2
yz
−
z
2
=
31
−
x
2
+
6
yz
+
2
z
2
=
44
x
2
+
x
y
+
8
z
2
=
100
(A)
0
(B)
1
(C)
2
(D) a finite number greater than 2
(E) infinately many
\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{a finite number greater than 2} \qquad \text{(E)} \ \text{infinately many}
(A)
0
(B)
1
(C)
2
(D)
a finite number greater than 2
(E)
infinately many
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