Is it permissible to continue shortening and combining prayers in Cairo, knowing that the distance between Cairo and Fayoum is 100 kilometers, the questioner does not have a residence in Cairo, and the work period there has extended to three years?

It is permissible to shorten prayers within the mentioned distance, and in Cairo if the stay is less than four days, unless you pray behind a resident imam. It is permissible for you to lead the residents in prayer, as the Messenger of Allah, may Allah bless him and grant him peace, did during the conquest of Mecca when he said: "Pray four rak'ahs, for we are a people on a journey." It is permissible for you to combine prayers, either by advancing or delaying them, but this is not permissible for the residents who follow you. Their combining by advancing prayers is invalid, whereas their combining by delaying prayers is valid, though they incur sin by delaying the prayer beyond its prescribed time.