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A cyclic system of equations with two possible outcomes

Source: Germany 2019, Problem 6

June 20, 2019

algebrasystem of equations

Problem Statement

Suppose that real numbers

x,y

and

z

satisfy the following equations:
\begin{align*} x+\frac{y}{z} &=2,\\ y+\frac{z}{x} &=2,\\ z+\frac{x}{y} &=2. \end{align*}
Show that

s=x+y+z

must be equal to

3

7

.
Note: It is not required to show the existence of such numbers

x,y,z