{"schema":"vela.problem-packet.v0.1","problem":1130,"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 $x_0=-1$ and $x_{n+1}=1$ and\\[\\Upsilon(x_1,\\ldots,x_n)=\\min_{0\\leq i\\leq n}\\max_{x\\in[x_i,x_{i+1}]} \\sum_k \\lvert l_k(x)\\rvert.\\]Is it true that\\[\\Upsilon(x_1,\\ldots,x_n)\\ll \\log n?\\]Describe which choice of $x_i$ maximise $\\Upsilon(x_1,\\ldots,x_n)$.","status":"solved","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)"}