How much does Harvard value International Olympiad medals in their admission process?
They do but it’s hard to tell how much. Probably very much for medalists.
But that is not the reason to participate in the Olympiads.
I think MIT values them a lot more than Harvard if you are referring to the Stem fields. If you talking about the Summer or Winter Olympics they do value those a lot
There’s a fair number of Olympiad medalists at most top universities and Harvard is no different, I know of at least half a dozen or so in my class alone from various countries (there’s likely a lot more than that) - it’s one of the most prestigious awards a high schooler can win.
However, I doubt medallists get in because of the medals alone - in my experience, more often than not, the kind of students who finish well at IMO, IOI, IPhO, etc. tend to be truly exceptional academics and the like, beyond merely being good at competition mathematics and science - the kind of students most top schools would love to have.