Is it permissible for a contestant to share the prize of a competition with his winning relative, knowing that he (the contestant) put in all the effort and his relative did not make any effort?

The summary of previous fatwas is that it is impermissible to take a prize from someone who did not exert effort in answering the questions, and it must be returned to the granting party if that party stipulated that the contestant answer the questions himself.

However, if they did not stipulate that, then the agreement between you and the winner should be examined. If it was a contract to divide the prize that was known to both of you, then it falls under the category of a valid ju'alah (reward/stipend), and he is obligated to fulfill it. If it was unknown, then the ju'alah is invalid, and you are entitled to a compensation equivalent to the customary wage (ajrat al-mithl). If it was a promise, then scholars differ on the obligatoriness of fulfilling a promise.