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This paper deals with the first-order delayed differential systems {u′+a(t)u=h(t)v+f(t,u(t−τ(t))),v′+b(t)v=g(t,u(t−τ(t))), where a, b, τ, h are continuous ω-periodic functions with ∫0ωa(t)dt=0 and ∫0ωb(t)dt>0; f∈C(R×[0,∞),R) and g∈C(R×[0,∞),[0,∞)) are ω-periodic with respect to t. By means of the fixed point theorem in cones, several new existence theorems on positive periodic solutions are established. Our main results enrich and complement those available in the literature. [ABSTRACT FROM AUTHOR]