The interior ministry said a policeman was shot dead in clashes with protesters in a north Cairo neighbourhood that also killed a number of demonstrators.
It was not immediately clear whether the policeman was included in the toll of 11 people reported by the state-owned Al-Ahram newspaper on its website.
Another protester was shot dead in clashes with police in the Mediterranean city of Alexandria. Officials said he was killed after he opened fire and police responded.
In central Cairo, police fired shotguns and tear gas to disperse hundreds of protesters who tried to march on the central Tahrir Square, the epicentre of the early 2011 revolt that ousted veteran strongman Hosni Mubarak.
The anti-Mubarak revolt was fuelled by police abuses and the corruption of the strongman's three-decade rule, but the police have since regained popularity amid widespread yearning for stability.
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