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2001 Bulgaria National Olympiad
2
System of equations!
System of equations!
Source: Bulgaria MO 2001 Day 2 Problem 2
February 4, 2015
algebra unsolved
algebra
Problem Statement
Find all real values
t
t
t
for which there exist real numbers
x
x
x
,
y
y
y
,
z
z
z
satisfying :
3
x
2
+
3
x
z
+
z
2
=
1
3x^2 + 3xz + z^2 = 1
3
x
2
+
3
x
z
+
z
2
=
1
,
3
y
2
+
3
y
z
+
z
2
=
4
3y^2 + 3yz + z^2 = 4
3
y
2
+
3
yz
+
z
2
=
4
,
x
2
ā
x
y
+
y
2
=
t
x^2 - xy + y^2 = t
x
2
ā
x
y
+
y
2
=
t
.
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