The first powerful blast was reportedly caused by a suicide attacker, who rammed his explosives-laden vehicle into a security fence around the Cairo Security Directorate, killing four and injuring 76 others, officials at the Interior and Health Ministry said.
In a second blast hours later, at least one person was killed when a small improvised explosive device went off near a police vehicle close to a metro station in Giza. At least 11 persons were wounded in the attack.
The fourth blast at the Radobis Cinema theatre in Cairo's Haram district killed one person, state TV reported.
The deadly blasts raised concerns that an Islamist insurgency is gathering pace in the polarised Arab nation as Egyptian forces have intensified a crackdown on Islamists.
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The Islamist elements have been blamed for a series of attacks on the army and police following the ouster of president Mohamed Morsi by the powerful military on July 3, 2013.
In a statement, president Adly Mansour pledged to "severely punish" those involved in "planning, financing, inciting, participating in or executing" such attacks, saying that tampering with state security is a "red line".
An al-Qaeda-inspired militant group Ansar Beit al-Maqdis (Champions of Jerusalem) claimed responsibility for the attack on the police headquarters, BBC reported.