MathDB

A4

Part of 2020-IMOC

Problems(1)

polynomial sequence, writing numbers never stops

Source: IMOC 2020 A4

8/12/2021
One day, before his work time at Jane Street, Sunny decided to have some fun. He saw that there are some real numbers a−1,…,a−ka_{-1},\ldots,a_{-k} on a blackboard, so he decided to do the following process just for fun: if there are real numbers a−k,…,an−1a_{-k},\ldots,a_{n-1} on the blackboard, then he computes the polynomial Pn(t)=(1−a−kt)⋯(1−an−1t).P_n(t)=(1-a_{-k}t)\cdots(1-a_{n-1}t). He then writes a real number ana_n, where an=iPn(i)−iPn(−i)Pn(i)+Pn(−i).a_n=\frac{iP_n(i)-iP_n(-i)}{P_n(i)+P_n(-i)}. If ana_n is undefined (that is, Pn(i)+Pn(−i)=0P_n(i)+P_n(-i)=0), then he would stop and go to work. Show that if Sunny writes some real number on the blackboard twice (or equivalently, there exists m>n≥0m>n\ge0 such that am=anam=an), then the process never stops. Moreover, show that in this case, all the numbers Sunny writes afterwards will already be written before. (usjl)
algebrapolynomial