We study the periodicities of a system of difference equations xn+1=max1/xn,An/yn-k,yn+1=max1/yn,Bn/xn-k, where initial values (x-k,y-k),…,(x0,y0)∈(0,+∞)×(0,+∞). We show that if An,Bn∈(0,1) are two periodic sequences, then every solution of the above system is eventually periodic with period 2. If k is even, there must be one in xn and yn converges to period two solution.