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Greece National Olympiad
2023 Greece National Olympiad
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2023 Greece National Olympiad
Problems
(1)
A nice system of equations
Source: Greece National Olympiad 2023, Problem 1
2/19/2023
Find all quadruplets (x, y, z, w) of positive real numbers that satisfy the following system:
{
x
y
z
+
1
x
+
1
=
y
z
w
+
1
y
+
1
=
z
w
x
+
1
z
+
1
=
w
x
y
+
1
w
+
1
x
+
y
+
z
+
w
=
48
\begin{cases} \frac{xyz+1}{x+1}= \frac{yzw+1}{y+1}= \frac{zwx+1}{z+1}= \frac{wxy+1}{w+1}\\ x+y+z+w= 48 \end{cases}
{
x
+
1
x
yz
+
1
=
y
+
1
yz
w
+
1
=
z
+
1
z
w
x
+
1
=
w
+
1
w
x
y
+
1
x
+
y
+
z
+
w
=
48
algebra
system of equations