"A large explosion" targeted the military intelligence headquarters in Rafah on the border with the Gaza Strip, one official said.
Minutes later, a second explosion hit an army checkpoint nearby.
Four soldiers were killed and 13 injured in the first attack, while one soldier died and six were hurt at the checkpoint, officials said.
Witnesses said the first blast shattered the windows of buildings in Rafah's Imam Ali area.
The military intelligence headquarters is located in a high-security military zone and surrounded by checkpoints, sources in Rafah said.
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The attacks come just days after an Al-Qaeda inspired group in the Sinai peninsula claimed credit for a Cairo bombing that targeted the interior minister last week, pledging more attacks.
The military has poured troops and armour into Sinai to quell Islamist militant activity which surged after the army overthrew Islamist president Mohamed Morsi on July 3.
A subsequent crackdown on Islamists by authorities left hundreds dead.
"We call on all Muslims in Egypt to stay away from all military and interior ministry installations to preserve their lives," the statement said.