The author considers the following differential delay equation
where and are relatively prime, are positive constants and denotes the integer part. Assuming that is an odd function with positive derivative and converges as tends to some existence and multiplicity results for periodic solutions are proved. As a corollary these results yield to a proof of a conjecture due to J. L. Kaplan and J. A. Yorke [J. Math. Anal. Appl. 48, 317-324 (1974; Zbl 0293.34102)].