Are gifts and things bought with a check considered permissible (halal), knowing that the competition was held at a mixed-gender university in France on the occasion of International Women's Day?

You must repent from participating in this innovated event, which was accompanied by abominable acts like intermingling, by seeking forgiveness, feeling remorse, and resolving not to return to it. Prizes for competitions are not permissible except for those explicitly mentioned in religious texts, such as competitions in horses, camels, and archery, and what has been analogized to them that aids in the propagation of religion, like competitions in Quran, Hadith, and Fiqh. Since the prize was acquired in fields not permitted by Sharia, it appears that it is not permissible, and you must dispose of it by giving it to the poor and needy, unless the questioner herself is poor and in need, in which case she and her family may benefit from it. As for the items purchased with the prize check, there is no harm in benefiting from them, but the value of the check must be taken out and disposed of.