What is the ruling on my winning a competition that I solved, and another person applied for it who does not meet its conditions and does not know its solution, and we agreed that I would solve it in his name in exchange for a sum of money that he would take upon our winning and my being handed the prize?

What you have asked about has two scenarios:

1. If the awarding body does not stipulate that the contestant must answer the questions himself: The prize winner is the one whose name appears on the answer, and he may give whatever he wishes to anyone who helped him. However, it must be noted that if the answering of the questions is on the basis of a promise of a portion of the prize, then it is permissible. But if it is on the basis of a contract, it is not permissible unless the payment is known and fixed, because basing the payment on a percentage of the prize involves ignorance and gharar (excessive uncertainty).

2. If the awarding body stipulates that the contestant must answer the questions himself: It is not permissible for anyone other than the person whose name is on the entry form to answer the questions. If money is taken on this basis, it is haram and must be returned to the awarding body.