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Part of 2024 German National Olympiad

Problems(1)

Symmetric equations in five variables with many solutions

Source: Germany 2024, Problem 1

The five real numbers

v,w,x,y,s

satisfy the system of equations \begin{align*} v&=wx+ys,\\ v^2&=w^2x+y^2s,\\ v^3&=w^3x+y^3s. \end{align*} Show that at least two of them are equal.

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