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Olympiad proof Problem - Day 1

From this day onwards I am going to post some proof problems.So that you can use these as practice problems for Olympiads. Let us start from basic level (not very basic though).


Find the minimum value of the expression below

\[\dfrac{a}{b+c+d} + \dfrac{b}{a+c+d} + \dfrac{c}{a+b+d} + \dfrac{d}{a+b+c}\]

Details and Assumptions:

  • \(a\), \(b\), \(c\) and \(d\) are positive real numbers.

Try to make different approaches.


Try more proof problems at Olympiad Proof Problems.

Note by Surya Prakash
2 years ago

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By using \(A.M-H.M\) we get , \[\dfrac{\displaystyle\sum_{cyc}^{a,b,c,d}\left(\frac{a}{b+c+d}+1\right)}{4}\geq\frac{4}{\sum_{cyc}^{a,b,c,d}\left(\dfrac{1}{1+\dfrac{a}{b+c+d}}\right)}\] \[\displaystyle\sum_{cyc}^{a,b,c,d}\left(\dfrac{a}{b+c+d}+1\right)\geq\dfrac{16}{3}\] \[\displaystyle\sum_{cyc}^{a,b,c,d}\left(\dfrac{a}{b+c+d}\right)\geq\dfrac{16}{3}-4\] \[\displaystyle\sum_{cyc}^{a,b,c,d}\left(\dfrac{a}{b+c+d}\right)\geq\dfrac{4}{3}\]

Shivam Jadhav - 2 years ago

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Nice solution. Do you have any more approaches?

Surya Prakash - 2 years ago

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Generalization:

\[\Large{\sum_{i=1}^n}\large{ \dfrac{a_i}{\small{\displaystyle\sum_{j=1}^{n}a_{j}-a_i}}\geq \dfrac{n}{n-1}}\]

When \(n=3\) we get Nesbitt's inequality and when \(n=4\) we get the inequality of this note.

Nihar Mahajan - 2 years ago

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Proof:

\[\sum_{cyc} \dfrac{a}{\sum_{cyc} a - a}+n -n\] \[=\sum_{cyc} (\dfrac{a}{\sum_{cyc} a - a}+1) -n\] \[=\sum_{cyc} (\dfrac{\sum_{cyc} a}{\sum_{cyc} a - a} -n\] \[=\sum_{cyc} a [\dfrac{1}{\sum_{cyc} a - a}] -n\] Titu's Lemma/Cauchy-Shwarz in engel form\[\geq \sum_{cyc} a [\dfrac{n^2}{(n-1)\sum_{cyc} a}]-n\] \[=\dfrac{n^2}{n-1} -n\] \[=\dfrac{n}{n-1}\]

Sualeh Asif - 2 years ago

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@Nihar Mahajan nice idea..... how did you reach at this generalization?

Dev Sharma - 2 years ago

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When I was proving this inequality , I felt a bit restricted , hence gave the generalization

Nihar Mahajan - 2 years ago

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So, here is my solution.

The given expression is homogeneous equation of degree \(0\). So, we can make a constraint \(a+b+c+d=1\). And this constraint gives us that \(0 < a,b,c,d <1\).

The expression transforms into \[\sum_{a,b,c,d} \dfrac{a}{1-a} \]

Let \(f(x) = \dfrac{x}{1-x}\). Then we get that \(f''(x) = -2(x-1)^{-3} > 0\) for all \(x<1\).

It implies that

\[f(a) + f(b) + f(c) + f(d) \geq 4f\left(\dfrac{a+b+c+d}{4}\right) = 4 f \left(\dfrac{1}{4}\right) = \dfrac{4}{3}\].

Surya Prakash - 2 years ago

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We can also solve it using Titu's Lemma. I am currently out of station. I would post the proof as soon as I return home.

Nihar Mahajan - 2 years ago

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1) put all 4 equal. To get minimum as 4/3. 2)put deniminator of all fractions as w,x,yand z. Now find each a,b,c and in terms of w,x,y and z. All these (w,x,y and z )are positive real no.s. now apply AM>=GM . 3) Rather finding out each a to d in terms of w,x,y and z , one can add +1 to each term take (a+b+c+d) common apply AM>=GM . Later substract 4 from answer

Aakash Khandelwal - 2 years ago

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How can you say that minimum occurs when all the four quantities are equal? You should prove that.

Surya Prakash - 2 years ago

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'put all 4 equal. To get minimum as 4/3'.. WHY?

Dev Sharma - 2 years ago

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What if we get maximum when all 4 are equal?

Nihar Mahajan - 2 years ago

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