parmenides51 wrote: Consider non-negative real numbers $a, b, c$ satisfying the condition $a^2 + b^2 + c^2 = 2$ . Find the maximum value of the following expression $$P=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c$$ $$P=\sum_{cyc}\left(\frac{\sqrt{2-a^2}}{3-a}+a-\frac{3}{8}(a^2-1)-\frac{3}{2}\right)+3+\frac{3}{8}(a^2+b^2-2)-2022c\leq$$$$\leq3+\frac{3}{8}(a^2+b^2+c^2-2)=3.$$The equality occurs for $a=b=1$ and $c=0$.