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What are some good resources(online, books) that teach you how to tackle difficult and ugly problems in higher math arranged by subjects(analysis, topology, ODEs, groups etc) or topics(polynomials, vector spaces and such like)?

I am obsessed with solving problems and while at it I'd like to develop my skills methodically.

Also, what are some subjects(combinatorics, number theory, graph theory etc) that have a greater carryover to other subjects? As in studying what single subject gives you the most amount of tools to deal with problems in just about any math discipline?

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Problem solving strategies by Engel might be suitable. I am sure you can easily find a pdf copy by googling it.

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