I'm a little perplexed by the dismissiveness of your response (perhaps I'm misreading?). I'm not saying IMO problems will never be algorithmically approachable,

I'm a little perplexed by the dismissiveness of your response (perhaps I'm misreading?). I'm not saying IMO problems will never be algorithmically approachable, just saying expressing these problems in terms of Grobner bases may be harder than is suggested here. Buzzard also expresses skepticism about using Grobner bases for these sorts of problems in the comments.