9.1 For four pairwise distinct numbers $\mathrm{x}, \mathrm{y}, \mathrm{s}, \mathrm{t}$, the equality $\frac{\mathrm{x}+\mathrm{s}}{\mathrm{x}+\mathrm{t}}=\frac{\mathrm{y}+\mathrm{t}}{\mathrm{y}+\mathrm{s}}$ is satisfied. Find the sum of all four numbers.
Solution.

$(x+s)(y+s)=(x+t)(y+t)$

$x y+x s+y s+s^{2}=x y+x t+y t+t^{2}$,

$x s-x t+y s-y t+s^{2}-t^{2}=0$

$x(s-t)+y(s-y)+(s+t)(s-t)=0$,

$(s-t)(x+y+s+t)=0$.

Since $\mathrm{s} \neq \mathrm{t}$, then $\mathrm{x}+\mathrm{y}+\mathrm{s}+\mathrm{t}=0$.