I’m looking for math problems of a specific kind. Here are the conditions I hope the problems satisfy:
A good mathematician who hasn’t seen the problem before should take anywhere from 30 minutes to 2 hours to solve it.
The solution should only involve undergraduate level maths. Difficult Putnam problems are a good benchmark for what kind of maths background should be required to solve the problems. The background required can be much less than this, but it shouldn’t be more.
For whatever reason you think the problem should be more widely known. The reason is completely up to you: the solution might include some insight which you find useful, it might be particularly elegant, it might involve some surprising elements that you wouldn’t expect to appear in the context of the problem, et cetera.
It’s fine if the problem is well known, your examples don’t have to be original or obscure.
Here are some examples:
If a polynomial with rational coefficients defines an injective map Q→Q, must it also define an injective map R→R? If yes then prove this is true, if no then find an explicit counterexample.
Prove that Hom(∏k∈NZ,Z)≅⨁k∈NZ. In words, prove that homomorphisms of abelian groups from the direct product of countably many copies of Z to Z themselves form a group that’s isomorphic to the direct sum of countably many copies of Z.
If f:R→R is a continuous function such that the sequence f(α),f(2α),f(3α),… converges to 0 for every α>0, must it be the case that limx→∞f(x)=0? If yes then prove this is true, if no then find an explicit counterexample.
They are all relatively famous but they should give a sense of the flavor of what I’m looking for.
[Question] What are the best elementary math problems you know?
I’m looking for math problems of a specific kind. Here are the conditions I hope the problems satisfy:
A good mathematician who hasn’t seen the problem before should take anywhere from 30 minutes to 2 hours to solve it.
The solution should only involve undergraduate level maths. Difficult Putnam problems are a good benchmark for what kind of maths background should be required to solve the problems. The background required can be much less than this, but it shouldn’t be more.
For whatever reason you think the problem should be more widely known. The reason is completely up to you: the solution might include some insight which you find useful, it might be particularly elegant, it might involve some surprising elements that you wouldn’t expect to appear in the context of the problem, et cetera.
It’s fine if the problem is well known, your examples don’t have to be original or obscure.
Here are some examples:
If a polynomial with rational coefficients defines an injective map Q→Q, must it also define an injective map R→R? If yes then prove this is true, if no then find an explicit counterexample.
Prove that Hom(∏k∈NZ,Z)≅⨁k∈NZ. In words, prove that homomorphisms of abelian groups from the direct product of countably many copies of Z to Z themselves form a group that’s isomorphic to the direct sum of countably many copies of Z.
If f:R→R is a continuous function such that the sequence f(α),f(2α),f(3α),… converges to 0 for every α>0, must it be the case that limx→∞f(x)=0? If yes then prove this is true, if no then find an explicit counterexample.
They are all relatively famous but they should give a sense of the flavor of what I’m looking for.