MathDB

3

Part of 2020 Czech and Slovak Olympiad III A

Problems(1)

x^2 - 3y + p = z, y^2 - 3z + p = x, z^2 - 3x + p = y , 3x3 parameter system

Source: 2020 Czech and Slovak Olympiad III A p3

11/24/2020
Consider the system of equations {x2−3y+p=z,y2−3z+p=x,z2−3x+p=y\begin{cases} x^2 - 3y + p = z, \\ y^2 - 3z + p = x, \\ z^2 - 3x + p = y \end{cases} with real parameter pp. a) For p≥4p \ge 4, solve the considered system in the field of real numbers. b) Prove that for p∈(1,4)p \in (1, 4) every real solution of the system satisfies x=y=zx = y = z.
(Jaroslav Svrcek)
system of equationsalgebraParametric Equationsparameter