Now that you are going to be the Spartan contender in the Olympics, you can make your way to the Valley of Olympia to introduce yourself.
Note: Be careful not to commit any illegal actions along the way. The punishment is much more severe while the Olympics are on.
When you are close to the Temple of Zeus, use Ikaros to identify the person that you need to speak with; Paios. Head toward the stadium and speak with him. When you are ready, begin the competition. You will now have to take on a series of fighters in hand to hand combat. Note that you will not be able to use any skills. You should be careful to avoid any incoming attacks by using evasions or blocks. Quickly retaliate with a combination of light and heavy attacks to inflict the most amount of damage. When you have defeated two opponents there will be a break in the challenge and a cut scene will initiate. It is thought that Alkibiades may have been poisoned, and you must now find the judges, one of which, Kallias, is suspected of doing it. When the cut scene ends the cultist menu will automatically open. Track Kallias and find him nearby at the stadium. You can choose to kill him now or later. Either way you will progress onto the next quest.