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This paper deals with the problem of positive observation for linear time-delay systems for which the states take nonnegative values whenever the initial conditions are nonnegative. We focus on the design of positive observers (possibly with time-delay) which guarantee nonnegative estimates of the current states. We derive necessary and sufficient conditions for the existence of a positive observer (extended Luemberger-type) and show that the solvability of the problem can be decided via standard linear programming techniques. Moreover, on the negative side, it is shown that one cannot stabilize any unstable positive time-delay system by using extended Luenberger type positive observers. In other words, the separation principle does not hold.