USAMO 2008 — Days 1 and 2 Question 1

Prove that for each positive integer $n$, there are pairwise relatively prime integers $k_0,\ldots,k_n$, all strictly greater than $1$, such that $k_0k_1 \dots k_n - 1$ is the product of two consecutive integers.

Proposed by Titu Andreescu; source discussion: https://aops.com/community/p1116186