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Austrian-Polish
1986 Austrian-Polish Competition
5
x^2+y^2+u^2+v^2=4, xu+yv+xv +yu=0, xyu+yuv+uvx+vxy=-2, xyuv = -1
x^2+y^2+u^2+v^2=4, xu+yv+xv +yu=0, xyu+yuv+uvx+vxy=-2, xyuv = -1
Source: Austrian Polish 1986 APMC
April 30, 2020
system of equations
algebra
Problem Statement
Find all real solutions of the system of equations
{
x
2
+
y
2
+
u
2
+
v
2
=
4
x
u
+
y
v
+
x
v
+
y
u
=
0
x
y
u
+
y
u
v
+
u
v
x
+
v
x
y
=
−
2
x
y
u
v
=
−
1
\begin{cases} x^2 + y^2 + u^2 + v^2 = 4 \\ xu + yv + xv + yu = 0 \\ xyu + yuv + uvx + vxy = - 2 \\ xyuv = -1 \end{cases}
⎩
⎨
⎧
x
2
+
y
2
+
u
2
+
v
2
=
4
xu
+
y
v
+
xv
+
y
u
=
0
x
y
u
+
y
uv
+
uvx
+
vx
y
=
−
2
x
y
uv
=
−
1
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