IIT JEE , AIEEE Forum - Mathematics , Algebra

hsbhatt

You have reversed the inequality

The correct inequality would be $\sum a_kx_k \ge \sum b_kx_k$

As an example consider the sequence (1,1,1) which is majorized by (3,0,0). (Majorization is the word for conditions (ii) and (iii) in your problem). Let (x₁,x₂,x₃) be (1,2,3) which also satisfy the given condition.

The proof uses the Abel Summation formula:

$a_1b_1+a_2b_2+...+a_nb_n = \\ \\<br/>(a_1-a_2)b_1+(a_2-a_3)(b_1+b_2)+(a_3-a_4)(b_1+b_2+b_3)+...+(a_{n-1}-a_n)(b_1+b_2+...+b_{n-1})+a_n(b_1+b_2+...+b_n)$

We apply this to the LHS

$a_1x_1+a_2x_2+..+a_nx_n = \\ \\<br/>(x_1-x_2)a_1+(x_2-x_3)(a_1+a_2)+...+x_n(a_1+a_2+...+a_n)$

We are given that

$\sum_{k=1}^K a_k < \sum_{k=1}^K b_k (K=1,2,3,...,n-1)$

The coefficient of LHS in the above summation is $x_k - x_{k+1}<0$

Since multiplying by a negative quantity changes the direction of the inequality, we have

$(x_k - x_{k+1})(a_1+a_2+...+a_k)>(x_k-x_{k+1})(b_1+b_2+...+b_k)$

For the last term we have x_n(a_1+a_2+...+a_n)= x_n(b_1+b_2+...+b_n)

Hence

a_1x_1+a_2x_2+...+a_nx_n > (x_1-x_2)b_1+(x_2-x_3)(b_1+b_2)+...+x_n(b_1+b_2+b_3+...+b_n)

and the RHS again by Abel Summation formula is nothing but

b_1x_1+b_2x_2+...+b_nx_n and hence the inequality is proved

The inequality as given by you would be true if $x_1 \ge x_2 \ge x_3 \ge...\ge x_n$

hsbhatt

edit: I have been a bit lax with some statements in the previous post.

There is a sentence "the coefficient of LHS in the above.." the inequality should be $x_{k-1} - x_k \le 0$

So the inequality that follows is not strict which is as the problem statement desires.

reddevil_2009

sir........you have used Abel Summation formula in your solution..............but a normal twelth standard student would not know about it................So how should we attempt it in a simplified basic way (though lengthy) ................

And can you tell me which inequalities and formulae to mug up................I know Cauchy's,Jensons,Tchebychef's ,Weirstras inequalities................Any others??????????plzzzzzz suggest and give linnks.............

hsbhatt

I really dont know how to answer this. Although you may be in 12th, your instructors seem to be giving you problems without paying heed to that! In the past you have even brought IMO problems to this forum.

Abel Summation is just another way of writing out the LHS that can easily be adopted in this problem. No special pyrotechnics involved.This will definitely look unfamilar in a JEE forum because we dont do much of inequalities in JEE. So you ask an olympiad type of question and ask a JEE type answer, there is little that can be done. But in an olympiad forum, you give this majorization condition, it becomes easy meat as the Abel Summation technique is well known and it will be done by them without having to exercise their brain at all.( so solving this problem doesnt mean one is a genius. It only means you happened to be familiar with the technique)

Yet again I want to stress, for JEE oriented students, you dont really have to go deep into inequalities, Number Theory etc. Just remember, JEE is looking for Engineers and not mathletes

For Inequalities - practice AM-GM (most useful) and Cauchy Schwarz (very basic). Know well the inequalities concerrning bounds of sin A + sin B + sin C, tan A + tan B + tan C etc. One more is the inequality that arises in Riemann Integral [m(b-a) <= I <= M(b-a) stuff]. This one had come in JEE last year and its easy to spot too.

For Number Theory - Well, Unique Factorisation to Fermat's Little theorem a^p-a is divisible by p is about the range. Knowledge of Wilson's Theorem [(p-1)!+1 is divisible by p ] will not harm you. Get good practice in stuff concerning Euclid's lemma like a square leaves a remainder of 0 or 1 on division by 3, odd squares are always of the form 4k+1 and 8k+1 etc.. These do help in some problems. If you could learn congruences your life will be easier in problems that need you to find remainders. But be thorough with dealing with rational and irrational numbers. You will almost certainly not encounter a problem that deals with NT exclusively, but it will be woven into a problem.

A case in point: A circle is drawn around the point $(0, \sqrt 2)$ . Call a point (x,y) rational if both x and y are rational. Prove that there are at most two rational points on the circle.

(I was told this was a problem that appeared in one of the JEEs, but I cannot confirm that.).

What they will be looking for in-depth is: Coordinate Geometry (just see last year's JEE), Complex Numbers, Calculus (Diff. Equations) and Vectors. These are essential toolkits for any good engineer and thats where they will be looking to test you.

I said this so that all of you can just check your flight paths and if required effect a course correction.

PS: Please dont interpret me to mean that I am trying to dampen your enthusiasm. I am cautioning you guys because I know of enough students who made it big in the olympiad scene but didnt do so well in JEE. If you plan to specialize in a pure sciences subject it is quite alright that you study that subject more. But if you are looking for a engineering career, then be pragmatic and balance your studying so that you are well prepared in M, P and C and particularly in the fields they are looking for.

reddevil_2009

Sir this problem wasnt given by my instructor...................I was just trying some Olympiad problems so that I could be 100% ready for JEE.................I dont want to take any chances.................

And thanx Sir for helping me and making me aware of important topics for JEE..............

And plzzzz dont embarrass me by saying that you are dampening my enthusiasm...........This site has been a boon for me and you and other experts are like GOD to me....................

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