Translation:Sự cạnh tranh
He is competitive: anh ấy có tính cạnh tranh [he has the quality of competitiveness]
The football competition: giải bóng đá (giải đá banh in colloquial Southern Vietnamese). [competitions are normally "giải"]. Cuộc thi can also mean "competition".
It was their competitiveness that helped them win: (là) sự/tính cạnh tranh (đã) giúp họ chiến thắng [It is/was the quality of competitiveness that help(ed) them win]
There are some other ways to say it but I would use those ^