Should one who catches up with the Imam prostrating in the first rak'ah repeat this rak'ah?

The latecomer is required to join the Imam in whatever state he finds him. The rak'ah is not counted for him unless he catches the bowing (ruku') with the Imam. If he does not catch the bowing of that rak'ah, it is not counted for him, and he must complete his prayer after the Imam's salutation, building upon what he caught thereafter. If it was the last rak'ah (he caught), he must perform the entire prayer.