The assaults were the latest in a five month-long wave of Palestinian violence that shows no sign of abating.
In the Jerusalem attack, police spokeswoman Luba Samri said the 20-year-old Palestinian "walked through Damascus Gate and then turned around with a knife in his hand, ran at the officers and attacked them." One of the officers suffered knife wounds to the head, she said, the other was cut in the hand and struggled with the attacker before shooting him.
The other two attacks today took place in the West Bank during clashes with soldiers. A Palestinian rammed his car into Israeli troops during a "violent riot" near the West Bank city of Ramallah before troops opened fire and killed him, the military said.
Later, in separate clashes near Bethlehem, also in the West Bank, the military said two Palestinians shot at soldiers who returned fire, killing one and wounding the other.
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The assaults were the latest in months of near-daily Palestinian attacks against civilians and security personnel, mostly stabbings, shootings and attacks in which cars are used to ram into Israelis. Since mid-September, 28 Israelis have been killed. During that time, 165 Palestinians were killed by Israeli fire, the majority said by Israel to be attackers. The rest died in clashes with security forces.
In Washington, State Department spokesman Mark Toner also said today that Weisman was a U.S. Citizen. Toner condemned Thursday's attack, saying it underscores the need for Israelis and Palestinians to reject violence, reduce tensions and restore calm.
The Rami Levy supermarket, popular with both Israeli and Palestinian shoppers has been targeted by Palestinian attackers before over the past months of violence. It is located in the West Bank, territory Israel captured from Jordan in the 1967 war that the Palestinians demand for part of their future state, a position that has global support.