{"schema":"vela.problem-packet.v0.1","problem":1132,"statement":"For $x_1,\\ldots,x_n\\in [-1,1]$ let\\[l_k(x)=\\frac{\\prod_{i\\neq k}(x-x_i)}{\\prod_{i\\neq k}(x_k-x_i)},\\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\\neq k$.Let $x_1,x_2,\\ldots\\in [-1,1]$ be an infinite sequence, and let\\[L_n(x) = \\sum_{1\\leq k\\leq n}\\lvert l_k(x)\\rvert,\\]where each $l_k(x)$ is defined above with respect to $x_1,\\ldots,x_n$.Must there exist $x\\in (-1,1)$ such that\\[L_n(x) >\\frac{2}{\\pi}\\log n-O(1)\\]for infinitely many $n$?Is it true that\\[\\limsup_{n\\to \\infty}\\frac{L_n(x)}{\\log n}\\geq \\frac{2}{\\pi}\\]for almost all $x\\in (-1,1)$?","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)"}