USAMO
Guide
Home
Contests
Problemset
Standings
Announcements
…
Back to Problems
Prev
Next
1900
Symmetric System
Editorial
algebra
systems of equations
symmetric expressions
Let
x
x
x
,
y
y
y
, and
z
z
z
be positive real numbers that satisfy the following system of equations:
x
2
+
x
y
+
y
2
=
75
x^2 + xy + y^2 = 75
x
2
+
x
y
+
y
2
=
75
y
2
+
y
z
+
z
2
=
16
y^2 + yz + z^2 = 16
y
2
+
y
z
+
z
2
=
16
z
2
+
z
x
+
x
2
=
39
z^2 + zx + x^2 = 39
z
2
+
z
x
+
x
2
=
39
What is the exact value of
3
(
x
y
+
y
z
+
z
x
)
2
3(xy + yz + zx)^2
3
(
x
y
+
y
z
+
z
x
)
2
?
Sign in to submit your answer
Sign In
