If you take a look at many of the sports in the winter olympics, they require quite expensive infrastructure that can only be used by a small elite of athletes. Ski Jump and bob are the best examples - your government has to be willing to invest multiple millions in a ski jump ramp that can only be used by a selected few athletes. And your bob / skeleton track is also expensive as fuck and cannot be used by the general public.

Other disciplines are also about the willingness of your state to pour money into winning gold: Take a look at ice dancing. There the Chinese and Russians have those inhuman selection processes where athletes are selected as children and are trained in inhuman conditions for years. If you want gold as a country, you have to invest money here to finance the athletes' livelihoods and provide them with the necessary ice rink, etc., so that they can train full-time. But of course, there are good reasons for a country not to do that. There are certainly better ways to spend taxpayers' money than financing individual athletes who may then win a medal every four years.