Supporters of deposed Islamist president Mohamed Morsi, overthrown in a July military coup, tried to converge on a central Cairo square for the anniversary celebrations, when police confronted them.
At least 15 people were killed in Cairo and elsewhere and 83 were wounded, emergency services chief Khaled Khatib told reporters.
In central Cairo, policemen fired shots and tear gas to disperse stone-throwing protesters. AFP correspondents saw several suspected demonstrators being arrested and beaten.
Hundreds of thousands of people had filled the square in February 2011 to force president Hosni Mubarak to resign, and again in July 2013 to urge the army to depose his successor Morsi.
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But today, security forces guarded entrances to the square, frisking people arriving for the celebrations.
Several thousand people, some carrying pictures of army chief Abdel Fattah al-Sisi, waved Egyptian flags as warplanes flew overhead in formation and patriotic songs blared from loudspeakers.
In Delga, an Islamist bastion south of Cairo, one person was killed when Islamists clashed with civilian opponents and police, a health ministry official and witnesses said.
Clashes also erupted in the Suez Canal city of Ismailiya, a security official said.
For weeks authorities have been drumming up national fervour in state media, amid the worst political divisions in Egypt's recent history.
Prime Minister Hazem Beblawi urged Egyptians to unite, saying the country is on the road to recovery.