3.

Part of 2019 Polish Junior MO Finals

Problems(1)

Algebra, fractions and numbers

Source: Polish Junior Math Olympiad 2020 Finals

x

y

z

be non-zero real numbers, such that

x + y + z = 0

and the numbers \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \text{and} \frac{x}{z} + \frac{z}{y} + \frac{y}{x} + 1 are equal. Determine their common value.