{"schema":"vela.problem-packet.v0.1","problem":481,"statement":"Let $a_1,\\ldots,a_r,b_1,\\ldots,b_r\\in \\mathbb{N}$ such that $\\sum_{i}\\frac{1}{a_i}>1$. For any finite sequence of $n$ (not necessarily distinct) integers $A=(x_1,\\ldots,x_n)$ let $T(A)$ denote the sequence of length $rn$ given by\\[(a_ix_j+b_i)_{1\\leq j\\leq n, 1\\leq i\\leq r}.\\]Prove that, if $A_1=(1)$ and $A_{i+1}=T(A_i)$, then there must be some $A_k$ with repeated elements.","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)"}