{"schema":"vela.problem-packet.v0.1","problem":1131,"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 $i\\neq k$.What is the minimal value of\\[I(x_1,\\ldots,x_n)=\\int_{-1}^1 \\sum_k \\lvert l_k(x)\\rvert^2\\mathrm{d}x?\\]In particular, is it true that\\[\\min I =2-(1+o(1))\\frac{1}{n}?\\]","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)"}