INEQUALITIES FOR OLYMPIAD

                            INEQUALITIES FOR OLYMPIAD:-

If you are learning inequalities for olympiads then there will be many inequalities that you will come across. Here we will discuss some important inequalities that you will most probably find while solving Mathematic Olympiad questions.

▪We will discuss here -  Basic Inequalities.

                                         -  A.M. and G.M and H.M Inequalities. 

                                         -  Exponential Inequalities.

                                         -  Logarithmic Inequalities.

                                         - Cauchu Schwartz

Inequality (*)

BASIC INEQUALITIES:-

Understand basic inequalities is quite simply. Think of a weighing scale with a simple Quadratic polynomial "x²+5" on both sides.

You will observe that since these are equal that's why the plates are at a same level.

Now think of adding 6 in any one side (lets say on the right side) then what will you observe? The right plate will lean towards right side.

So here we find a inequality and we can write it like, x²+5 < x²+11.

This is the basic idea of inequality.

RULES OF ADDITION,MULTIPLICATION AND DIVISION FOR INEQUALITIES:-

● ADDITION:- It is same as you do while solving any equation.

     

● MULTIPLICATION:- It is also same as solving any equation but not in case of cross-multiplication you can't cross multiply an inequality as it will change the degree of the polynomials .Or simply think of an inequality say,2<3, so if we multiply -1 we will end up with(-2)<(-3) which is not possible right?                        

● DIVISION:- It has cases you can't simply divide inequalities. It has cases. It will depend on the question you are solving. You will have to do such question by taking cases.

● SUBSTRACTION:- Same as solving an equation

REASONS BEHIND THESE RULES:-

The reasons are also simple. Again you have to think about a weighing scale. Take the same example of the inequality x²+5 < x²+11. Now lets first understand the logic of addition and substraction .Now answer yourself if you substract or add 11 from both sides will it affect the inequality? Similarly in the inequality 6<7 if we substract or add 2  then what will happen? Then it will look like 4<5 or 8<9 here the inequality is still same!

Apply this logic to multiplication rule as well.( not in all cases)

●A.M,G.M INEQUALITIES:-

If you have studied sequence and series then we don't need to elaborate this much but for those who have not let's give you a short introduction,

Let this be a series:- a,b,c,d,..............,n 

Then A.M of this series will be : (a+b+c+d+..............+n)/n

And G.M of this is :- ⁿ√(abcd..............n)

☆Then the inequality will be A.M≥G.M☆

▪That is (a+b+c+d+..............+n)/n≥ⁿ√(abcd..............n)

ILLUSTRATIONS:-

Q1)Prove that

Using AM-GM INEQUALITY 

ADDING THESE WE HAVE

Q2)

Q3)

Q4)

EXERCISE:-

Q) Solve this inequality 3x-8/x+7>8.( Do this only if you know wavy curve method)

Q) Statements: A > B, B ≥ C, C < D

      Conclusions:
        I. A > C
        II. A = C

which one is true?

 a) both are correct   b)either one is correct   c)I is only true   d)II is only true

Q)Prove (a³+b³+c³)/3≥ abc

Q)Prove that (a+1/a)≥ 2

Q)Prove that 2/(1/a+1/b)≤√ab

Q) Let a₂, . . . , an be n − 1 positive real numbers, where n ≥ 3, such that a₂a₃ · · · an = 1. Prove that

 (1 + a₂)² (1 + a₃)³ · · ·(1 + an)ⁿ > n^{n      (*****IMO 2012*******)(Must try)}

^{REST OF THE TOPICS WILL BE POSTED LATER.}

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Written by:- S.D

Edited by:- M.Jackson

Research:- Team MATHFREAKS

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