One way can be that sports increase the overall health of the nation because people who deal with sports are often healthier than those who don't. Another can be that athletes are great cultural ambassadors when they travel to tournament and represent the country. Another can be that they gather the country around a common goal, that is, supporting your country when it competes against others. The fourth can be that sports serve as a relaxation things for people because it is important for a country to have facilities where people can have fun.
