TECHNICAL ANALYSIS OF MEANSREVIEWS VERY SHORT FOR EVERY inequality [DRIVE! Sos]

I / Preface.

Inequalities are always beautiful problems, and good because they always need thinking! However, solving them is the opposite, finding a solution for them is extremely hard and difficult. And for problems with 2 or more variables, things become even more difficult. After a period of learning experience and searching, we have found a technique to evaluate the simple and beautiful inequalities. Due to the difficulty of the problems, sometimes some solutions are a bit long, but in return is the beauty of the method and we do not need Maple.

Author.

II / Basis

Assume  is an expression to be proved. (We will prove )

The technical basis of [DRIVE! Sos] is very simple, based only on the determination of two factors (and ). Satisfy:

+)  ()

+)  ()

Then:

The technical basis is simple, As long as we determine the factor k satisfying as above, the problem is proved.

Please tell your readers, when combining this technique with Buffalo Way, the problem is extremely magical! In addition, when combined with the Titu’s Lemma, we found countless types of !

We come to Part III: Application.

III / Application

Problem :Math problems (Korean MO Final,2016):

For  are any three real numbers. Prove that:

Solution.

Too great right? Let's go to lesson , a very familiar problem.

Problem : Prove that  for all

Solution.

We have:

And:

Apply the formula in Part II we have:

Problem : (Schur inequality)

Let . Prove that:

Solution.

Type : (tthnew)

With . (Consider the case so that the denominator is always> 0, but this is only for  so please do not state)

Type : (DOTOANNANG)

Problem : (Combined with )

Give Prove that:

Solution.

Without loss of generality, suppose . Let then

Substitute and the inequality becomes:

We turn the opposite and obtain:

It is quite similar of Ji Chen's decomposition.

Problem : (IMO )

Let  be the lengths of sides of a triangle. Prove that:

Solution.

Type : (tthnew)

 with

Type : (DOTOANNANG)

See here: IMO 1983 - Haidangel

Type : (According to Bernhard Leeb)

 with

Please also add that [DRIVE! sos] have countless types of , not necessarily rigid to find  coefficients  as in part II. This type  proves that.

Post is ongoing update ... The next section will talk about the great combination between the Titu’s Lemma and [DRIVE! sos]

References:

*https://diendantoanhoc.net/ topic / 187455-sumlimits-cycitsumititaitbitcitgeqqit0 /? p = 721176

Author: DOTOANNANG

* VMF Mathematics Forum (https://diendantoanhoc.net/)

* AoPS forum