Set $P$ as a polynomial function by $p_n(x)=\sum_{k=0}^{n-1} x^k$. a) Prove that for $m,n\in N$, when dividing $p_n(x)$ by $p_m(x)$, the remainder is $$p_i(x),\forall i=0,1,...,m-1.$$b) Find all the positive integers $i,j,k$ that make $$p_i(x)+p_j(x^2)+p_k(x^4)=p_{100}(x).$$(square1zoa)
Problem
Source: Mathcenter Contest / Oly - Thai Forum 2008 R3 p9 https://artofproblemsolving.com/community/c3196914_mathcenter_contest
Tags: polynomial, algebra