Do you think that being proud of someone just because he's your compatriot is rational or irrational?
After all, you just share the same nationality by random chance, therefore isn't it the same as being proud that you were born on the same day of the week as Einstein or that your hair is the same colour as George Washington's?
To the extent that people make snap judgments about other people based on race or nationality, it would seem that you'd want people that share your ethnicity or nationality to do well. This likely has a minor increase in people's perception of you.
Likewise, you probably feel shame or unhappiness when the opposite happens. Just look at what being Muslim is like in the US now.
I somehow feel proud that I made a good choice of place to live for now, if that makes any sense.
PS: I noticed they are almost all "outsiders", foreigners that immigrated to Western countries, or locals outside the traditional elite of the Western countries. The point, which is a depressing one, is that mathematics and, more generally, pure science, is an "outsider" activity nowadays.
I've never understood the swellings of national pride that occur within a country when (for example) one of their citizens wins an Olympic medal; only a handful of people played any part in that achievement, to insinuate yourself amongst them by virtue of these random relationships is almost parasitic.
Likewise, though less on-topic, I have no time for patriotism either. George Bernard Shaw put it best: "Patriotism is your conviction that this country is superior to all others because you were born in it."
If someone feels proud, he/she naturally just got a good moment in life. I guess your question was: Is it rational to identify yourself with an external entity in order to feel good/bad when something happens to this entity. The answer is: In many cases it is perfectly rational. E.g. when people start supporting an old soccer sclub that got a breakout success, old supporters refer to new ones as "gloryhunters".
By the way, I am a gloryhunter here as well. One of the four Fields medalists this year (Stas Smirnov) graduated from the same high school and college as I did.
There is another reason though: it shows that your country/culture is a good one in terms of innovation or whatever. This directly reflects on the culture that your are a part of, both in the sense that you're affected by it, and also that you are part of shaping it. So in a way, a win for your culture is a win for you.
In the widest sense, a nation is like a company. You could argue all it's members contribute to making it the place it is. If it brings forth a field medal winner, it might be a result of the favourable conditions you helped creating.