All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
There are two possibilities for the case in question:
First possibility: If the prize-giving authority does not stipulate that the winner should himself be the one who solved the questions, then, the person who submitted the answers in his name is considered the owner of the prize and he has the right to dispose of it according to his will, even if he will give all of it to somebody else. However, this should happen in a genuine way, not as a sham.
We draw your attention in this regard to an important matter: solving the questions and writing the answers in the name of another person in return for giving him part of the prize is either done as a promise or a contract. If it is intended to be a promise, such an agreement is permissible. If it is intended to be a contract, then it is impermissible unless the wage is known. The contract should not include any ambiguity or uncertain elements. There is no doubt that the possibility of winning or losing and the wage being part of the prize are both clear elements of uncertainty.
Second possibility: If the prize-giving authority stipulates that the participant should himself solve the questions. In such a case, the answerer should be the same person whose name is registered in the participation form in the competition. Otherwise, the prize will be an unlawful gain which must be given back to the prize-giving authority.
Allah Knows best.