All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu 'alyhi wa sallam, is His slave and Messenger.
If this competition makes the participants spend any money by paying subscription fees, the cost of communication, or the like, then it is impermissible to enter it. Also, the money that is earned from participating in this competition is forbidden to be taken and utilized.
However, if the competition does not cost one anything and the travel to that country is permissible, for being an Islamic country or a non-Muslim country for the sake of seeking treatment or knowledge that are not available elsewhere, making Da‘wah (Islamic propagation) for the sake of Allaah The Almighty, spreading Islam and for similar purposes that permit traveling to the land of the disbelievers, then it would be permissible to participate in this competition. Also, the money that is obtained from it is lawful to be taken and utilized, even if it is provided by banks or by other sources, because the Prophet, sallallaahu ‘alayhi wa sallam, would accept the gifts of the disbelievers and the Jews, who might have earned their money from unlawful sources like in usury and interest and by selling alcohol and the flesh of swine.
Allaah Knows best.