{"schema":"vela.problem-packet.v0.1","problem":119,"statement":"Let $z_i$ be an infinite sequence of complex numbers such that $\\lvert z_i\\rvert=1$ for all $i\\geq 1$, and for $n\\geq 1$ let\\[p_n(z)=\\prod_{i\\leq n} (z-z_i).\\]Let $M_n=\\max_{\\lvert z\\rvert=1}\\lvert p_n(z)\\rvert$. Is it true that $\\limsup M_n=\\infty$?Is it true that there exists $c>0$ such that for infinitely many $n$ we have $M_n > n^c$?Is it true that there exists $c>0$ such that, for all large $n$,\\[\\sum_{k\\leq n}M_k > n^{1+c}?\\]","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}