The file is 561K, which is much smaller than the MS word's product (1.5MB for 22pages, I'll die if I make 86 pages of it)

The isn't "v5.0" since I planned to make the v5 as a temporary end of this set of notes, but I have to publicize this one since a huge amount of new content has been included,

Comparing with v4.0, the following chapters are new:

7.4 A updated list of exercises about finite differences.

7.5 Convergences (A standard pure maths content?)

8.1 System of equation (This is trash. I'll have to amend this later.)

8.2 Roots of unity (Half of pure math syb.)

8.3-8.5 Cubic, quartic equation and discriminants

8.6 Special types of equation (Reference of , maybe need more explaination)

8.7 Application and Riemann's hypothesis -- The most interesting part for me! Linking a problem in integer function with the zeta function, and a brief introduction on the Riemann's hypothesis. However my proof may have some problems. At least we know my f(x) grows same as zeta function.

9.1 Nature of inequality Despite the inequality appeared before, we start from definition.

9.2 Solving inequality, as well as introducing the set notation. Maybe a chance for introducing NZQRC?

9.3 Proving inequality -- staring the olympiad stuffs

9.4 AM-GM-HM inequality

9.5 Cauchy-Schwarz inequality (unfinished)

**A new exmaple in old section**

**P.50 example 6.5.9-11 An extremely hard problem from crazy pure math (AL 1981)**

For 9.3 and afterwards I think the difficulty may be harder than problems in HKALE, I will try to find the examples from HKALE, APMO and different MOs. If the examples are too difficult just tell me.

A list of inequality planned to discuss:

1)AM-GM-HM --- and the generalized verion, Power Mean Inequality

2)Cauchy-Schwarz --- and the extended verion (vector and complex numbers)

3)Rearrangement

4)Jensen

5)Holders?...

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