By the Golod-Shafarevich theorem, an associative algebra R given by n generators and d <n(2)/3 homogeneous quadratic relations is not 5-step nilpotent. We prove that this estimate is optimal. Namely, we show that for every positive integer n, there is an algebra R given by n generators and inverted right perpendicularn(2)/3inverted left perpendicular homogeneous quadratic relations such that R is 5-step nilpotent. (C) 2014 Published by Elsevier Inc.