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Bulgaria Contests
Bulgaria National Olympiad
1972 Bulgaria National Olympiad
Problem 2
system in R+, four equations/variables
system in R+, four equations/variables
Source: Bulgaria 1972 P2
June 21, 2021
algebra
system of equations
equation
Problem Statement
Solve the system of equations:
{
y
(
t
−
y
)
t
−
x
−
4
x
+
z
(
t
−
z
)
t
−
x
−
4
x
=
x
z
(
t
−
z
)
t
−
y
−
4
y
+
x
(
t
−
x
)
t
−
y
−
4
y
=
y
x
(
t
−
x
)
t
−
z
−
4
z
+
y
(
t
−
y
)
t
−
z
−
4
z
=
z
x
+
y
+
z
=
2
t
\begin{cases}\sqrt{\frac{y(t-y)}{t-x}-\frac4x}+\sqrt{\frac{z(t-z)}{t-x}-\frac4x}=\sqrt x\\\sqrt{\frac{z(t-z)}{t-y}-\frac4y}+\sqrt{\frac{x(t-x)}{t-y}-\frac4y}=\sqrt y\\\sqrt{\frac{x(t-x)}{t-z}-\frac4z}+\sqrt{\frac{y(t-y)}{t-z}-\frac4z}=\sqrt z\\x+y+z=2t\end{cases}
⎩
⎨
⎧
t
−
x
y
(
t
−
y
)
−
x
4
+
t
−
x
z
(
t
−
z
)
−
x
4
=
x
t
−
y
z
(
t
−
z
)
−
y
4
+
t
−
y
x
(
t
−
x
)
−
y
4
=
y
t
−
z
x
(
t
−
x
)
−
z
4
+
t
−
z
y
(
t
−
y
)
−
z
4
=
z
x
+
y
+
z
=
2
t
if the following conditions are satisfied:
0
<
x
<
t
0<x<t
0
<
x
<
t
,
0
<
y
<
t
0<y<t
0
<
y
<
t
,
0
<
z
<
t
0<z<t
0
<
z
<
t
.H. Lesov
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