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Let's say that you've added some reactants and allowed the reaction to go to equilibrium. At equilibrium, there will be some reactants and some products, so the concentrations of each will be greater than zero. After the reaction reaches equilibrium, you add some more reactants. The reaction is now in the initial phase and has to reach equilibrium, but there are still some products from the first equilibrium reaction, so the initial products will not equal zero. So yes, this is possible.
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