Question 160
For a cubic polynomial $P(x)$ with complex roots $z_{1}, z_{2}, z_{3}$, let
$$
M(P)=\frac{\max \left(\left|z_{1}-z_{2}\right|,\left|z_{1}-z_{3}\right|,\left|z_{2}-z_{3}\right|\right)}{\min \left(\left|z_{1}-z_{2}\right|,\left|z_{1}-z_{3}\right|,\left|z_{2}-z_{3}\right|\right)}
$$
Over all polynomials $P(x)=x^{3}+a x^{2}+b x+c$, where $a, b, c$ are nonnegative integers at most 100 and $P(x)$ has no repeated roots, the twentieth largest possible value of $M(P)$ is $m$. Estimate $A=\lfloor m\rfloor$. An estimate of $E$ earns $\max \left(0,\left\lfloor 20-20|3 \ln (A / E)|^{1 / 2}\right\rfloor\right)$ points.
$$
M(P)=\frac{\max \left(\left|z_{1}-z_{2}\right|,\left|z_{1}-z_{3}\right|,\left|z_{2}-z_{3}\right|\right)}{\min \left(\left|z_{1}-z_{2}\right|,\left|z_{1}-z_{3}\right|,\left|z_{2}-z_{3}\right|\right)}
$$
Over all polynomials $P(x)=x^{3}+a x^{2}+b x+c$, where $a, b, c$ are nonnegative integers at most 100 and $P(x)$ has no repeated roots, the twentieth largest possible value of $M(P)$ is $m$. Estimate $A=\lfloor m\rfloor$. An estimate of $E$ earns $\max \left(0,\left\lfloor 20-20|3 \ln (A / E)|^{1 / 2}\right\rfloor\right)$ points.