Problem

Source: Romanian TST 1979 day 2 P3

Tags: inequalities, simultaneous equation, algebra, system of equations



Let $a,b,c\in \mathbb{R}$ with $a^2+b^2+c^2=1$ and $\lambda\in \mathbb{R}_{>0}\setminus\{1\}$. Then for each solution $(x,y,z)$ of the system of equations: \[ \begin{cases} x-\lambda y=a,\\ y-\lambda z=b,\\ z-\lambda x=c. \end{cases} \]we have $\displaystyle x^2+y^2+z^2\leqslant \frac1{(\lambda-1)^2}$. Radu Gologan