The violence underscored the struggle of authorities to tame an Islamist insurgency which has been gaining pace since the army toppled Islamist President Mohamed Mursi in July.
In the most high-profile attack, a suicide car bomber struck a top security compound in central Cairo early in the morning and killed at least four people, security sources said.
Another blast in the Dokki district killed one person. An explosion near a cinema on the road to the Pyramids of Giza on the outskirts of Cairo also killed led to one fatality.
Clashes in the capital and several other cities between Mursi supporters and security forces which killed 11 people also raised tensions in the biggest Arab nation.
There was no immediate claim of responsibility for the suicide attack in the parking lot of the Cairo Security Directorate
Prime Minister Hazem el-Beblawi condemned the bloodshed in a statement, saying it was an attempt by "terrorist forces" to derail the army-backed government's political road map, which is meant to lead to free and fair elections.
Later in the day, a military helicopter flew back and forth over central Cairo, underscoring concerns that another attack could occur at anytime.
Authorities have been bracing for more violence during the anniversary of Mubarak's fall, when rival political groups are expected to turn out, including supporters of army chief General Abdel Fattah al-Sisi, who ousted Mursi, as well as members of the Muslim Brotherhood and liberals.
The 2011 revolt raised hopes of a stable democracy in the Arab world's biggest nation. Instead, relentless political turmoil has hit investment and tourism hard in Egypt.
The dead from the first blast in the security headquarters included three policemen, security sources said.
The attack also heavily damaged the nearby Islamic Art Museum, an official told the state news agency.
In a statement, the office of President Adly Mansour said it would "avenge the deaths of the martyrs" who died at the Security Directorate and severely punish the perpetrators.
Reuters witnesses heard gunfire right after the first blast, which twisted metal and shattered windows of nearby shops. Wood and metal debris was scattered hundreds of metres around.
One body covered in a blanket lay in a pool of blood near a scorched car engine.
State television quoted witnesses as saying gunmen on motorcycles opened fire on buildings after the explosion. The Health Ministry said 76 people were wounded.
After ousting Mursi, Sisi unveiled a political road map he said would bring elections and calm to Egypt.
Security forces have killed hundreds of Brotherhood members and jailed thousands more, including top leaders.
The army-backed government has effectively removed the Brotherhood from politics and many Egyptians turned against it after Mursi's troubled one-year rule.
Intense pressure on the movement has severely hurt its ability to stage mass protests it had hoped would reverse what it calls an army coup that has undermined Egypt's democratic credentials.
But authorities are struggling to contain Islamist militant violence. Militants based in the Sinai have stepped up attacks on security forces since Mursi's fall, killing hundreds.
Attacks in other parts of Egypt have also risen, fuelling fears the country could face an Islamist insurgency similar to one that raged in the 1990s before Mubarak stamped it out.
The Sinai-based Islamist militant group Ansar Bayt al-Maqdis said in December it was behind a car bomb attack on a Nile Delta police compound that killed 16 people and wounded about 140.
On Thursday, gunmen killed five policemen at a checkpoint south of Cairo, the Interior Ministry said.
The police headquarters assault will likely encourage the state to crack down harder on the Brotherhood, which it accuses of terrorist acts. The group says it is a peaceful movement.
Human rights groups accuse security forces of widespread human rights abuses in their crackdown on the Brotherhood. But the group has little sympathy on the street.
The mood was tense at the site of Friday's suicide bombing. "Traitors and dogs!" yelled onlookers, an apparent reference to the assailants.
People also chanted anti-Brotherhood slogans. "The people want the execution of the Brotherhood. Execution for Mursi," they yelled.
One woman, Wafaa Ahmed, was crying outside the Cairo Security Directorate.
"These people have no sense of loyalty to the nation. This is terrorism, they want to get back at us because we finished them off ...," she said.
