Need Recommendation for High Level proof book

Question

I know how solve geometry, combinatorics, algebra/Precalc, and number theory non-proof problems pretty well. However, I lack the ability to prove theorems, certain parts of recursive functions (ex prove ,a2020 smaller than a2019-7)(first time using mathstackexchange srry bad formatting), certain geometric ideas, etc. I'm also bad at using AM-GM and other tools to prove inequalities.
I could get multiple books, if each books goes really in-depth into it's respective area(s).
A few examples of problems I am aiming to easily prove are here: PUMaC 2020 A(https://static1.squarespace.com/static/570450471d07c094a39efaed/t/6073d63922e7506b0307e748/1618204218125/2020_Indiv_Finals_A.pdf)
PUMaC 2020 B (https://static1.squarespace.com/static/570450471d07c094a39efaed/t/6073d63fce2041659f2a3cab/1618204223311/2020_Indiv_Finals_B.pdf).
While I'm not going for PUMaC tests, I do want to be able to prove these sorts of problems. If the book does contain several approaches to proof problems and normal problems, do recommend.

Answer 1

Proof writing takes quite a lot of effort to learn how to do well. I would recommend starting out by learning a bit about formal logic, specifically 'truth functional calculus'. Mathematically proofs make use of logic (though the logic therein is not typically written formally, it is nevertheless communicated clearly enough to be understood).

With a solid grasp of basic logic, you are then ready to start learning about common proof techniques (these are work for many problems, but some of the harder ones will require other insights to successfully prove; and ofc there are plenty of known problems whose answer has still eluded us, such as the Collatz conjecture).