Nearly 70 people have been wounded following the attacks.
The attacks began with a powerful car bomb that exploded outside the police headquarters in central Cairo, killing four people and wounding 51.
Within hours, two other blasts occurred elsewhere in the city, killing one person and injuring 15.
The attacks come on the eve of the third anniversary of the 2011 uprising.
The revolution brought about the removal of the country’s decades-long ruler Hosni Mubarak.
An al-Qaeda-inspired militant group Ansar Beit al-Maqdis (Champions of Jerusalem) has admitted carrying out the attack on the police headquarters.
The group previously claimed responsibility for a car bomb attack on a security building in the northern city of Mansoura in December that killed 16 people and injured more than 100 others.
The authorities blamed Mohamed Morsi’s Muslim Brotherhood for that attack – something the group strongly denied – and declared it a terrorist group shortly afterwards.
Egyptian Interior Minister Muhammad Ibrahim said security is being stepped up around the squares where people are expected to gather to mark Saturday’s anniversary.
“We have a plan to secure all of this for the anniversary of the 25 January revolution,” he said.
“I am telling the people not to be afraid and go down.”
The Muslim Brotherhood condemned the attack that hit police headquarters at around 06:30 local time.
The group called for an investigation and added they were part of a “peaceful revolution”.
The powerful blast was felt across the city and black smoke could be seen rising over the scene.
Gunfire was reportedly heard soon after the blast, and more than 30 ambulances raced to the scene.
There was extensive damage to the large concrete building with debris strewn across the street.
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