One student has been killed and dozens injured after police in Cairo opened fire on students protesting against the military government.

Buildings at Al-Azhar University, the world's most important Islamic school, were set on fire by students who have been protesting against the coup that ousted President Mohamed Morsi and the Muslim Brotherhood from government.

Khaled el-Haddad was shot outside the faculty of commerce on Saturday after protests which were enflamed by the government's decision on Wednesday to declare the Muslim Brotherhood a terrorist group.

The state-run newspaper Al-Ahram said the clashes began when security forces fired teargas to disperse pro-Brotherhood students who were preventing their classmates from entering university buildings to take exams. Protesters threw rocks at police and set alight tyres to counter the teargas.

State TV broadcast footage of black smoke billowing from the faculty of commerce building and said "terrorist students" had also set the agriculture faculty building on fire.

Shaimaa Mounir, a member of the "Students Against the Coup" movement, said Khaled el-Haddad died of a wound, though it was not clear whether he had been hit by birdshot or another kind of ammunition.

The violence followed a day of clashes across the country that left five people dead. Supporters of the Brotherhood took to the streets on Friday after the Islamist group was designated a terrorist organisation – a move that increases the penalties for dissent against the government, which was installed after Morsi's ousting in July following mass protests against his rule.

Human Rights Watch said on Saturday that the designation was politically driven and was intended to end the movement's activities.

"By rushing to point the finger at the Brotherhood without investigations or evidence, the government seems motivated solely by its desire to crush a major opposition movement," said Sarah Leah Whitson of the New York-based rights group.