The math problems notebook


How much do I want to read more? 7/10

I'm glad I picked up this book. It hints at other good books too.
Made by and for passionate. It explains the two worlds, mathematic enthusiasts and real mathematician.
We dive in a fascinating world.

After seeing the first exercice, I realize I need a solid refresh of my knowledge. Good to know I have other really good books for that.


Preface

The authors met on a Sunday morning about 25 years ago in Room 113. One of us was a college student and the other was leading the Sunday Math Circle.
Most of them were participating in the mathematical competitions in vogue at that time, namely the Olympiads. Others just wanted to have good time.

The fundamental texts were the celebrated book of Richard Courant and Herbert Robbins with the mysterious title What is mathematics?
and the book of David Hilbert and Stefan Cohn–Vossen entitled Geometry and the Imagination.

Competition problems have to be solved in a short amount of time.
they have the advantage that one knows in advance that they have a solution.
In the real mathematical world, problems often had to wait not merely years but sometimes centuries to be solved.
This intellectual adventure is filled with suspense and frustration, since one does not know for sure what one is trying to prove or whether it is indeed true.

The history of mathematics abounds in examples in which a fresh mind was able to find an unexpected solution that specialists had been unable to find.
Eventually, once a solution has been found, mathematicians are then willing to try to understand it even better, and other solutions follow in time, each one simpler and clearer than the previous one.
In some sense, once solved, even the hardest problems start losing slowly their aura of difficulty and eventually become just problems. Problems that are today part of the curriculum of the average high-school student were difficult research problems three hundred years ago, solved only by brilliant mathematicians.

We wanted to have 25% easy problems concerning basic tools and methods and consisting mainly of instructional exercises.
The largest chunk contains about 50% problems of medium difficulty, which could be useful in training for mathematical competitions from local to international levels.
The remaining 25% might be considered difficult problems even for the experienced problem-solver.

the congruent numbers conjecture and the Riemann hypothesis, which are among seven Millennium Prize Problems that the Clay Mathematics Insti- tute recorded as some of the most difficult issues with which mathematicians were struggling at the turn of the second millennium and offered a reward of one million dollars for a solution to each one.

Part I - Problems

1- Number Theory