A better question is: "Why should country X perform well in international Olympiads"?

It is just entertainment. It doesn't have any meaningful, deep or relevant impact on culture and quality of life of a country. At best it boost narcissistic jingoism and nationalism.

Life in a country won't get any better because someone that carries their flag appeared on television winning a medal. This is just a globalized version of the Roman "panis et circensis", but without the panis.