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Try this problem from ISI-MSQMS 2017 which involves the concept of Inequality.

(a) Prove that $a^{5}+b^{5}+c^{5}>a b c(a b+b c+c a),$ for all positive distinct values of $a, b, c$

Algebra

Inequality

Numbers

First hint

We know if we have $n$ numbers say $a_1,a_2,.....,a_n$ then AM $\geq$ GM implies

$\frac{a_1+a_2+....+a_n}{n} \geq (a_1.a_2........a_n)^\frac{1}{n}$

I assure you that the sum the can be done just by using this simple inequality,why don't you just give it a try?

Second hint

Applying AM $\geq$ GM on $a^5,a^5,a^5,b^5,c^5$ we get

$3a^5+b^5+c^5 \geq 5a^3bc$

Similarly, $a^5+3b^5+c^5 \geq 5ab^3c$

$a^5+b^5+3c^5 \geq 5abc^3$

Adding the above three equations we get $a^5+b^5+c^5 \geq abc(a^2+b^2+c^2)$

So you have all the pieces of the jigsaw puzzle with you,and the puzzle is about to be completed,just try to place the remaining few pieces in its correct position

Third hint

Now we have to show that $a^2+b^2+c^2 > ab+bc+ca$

Applying AM $\geq$ GM on $a^2,b^2$ we get,

$a^2+b^2 \geq 2ab$

Similarly,$b^2+c^2 \geq 2bc$

$a^2+c^2 \geq 2ca$

Adding the above three equations we get $a^2+b^2+c^2 \geq ab+bc+ca$

We are almost there ,so just try the last step yourself.

Final Step

Therefore, $a^5+b^5+c^5 \geq abc(a^2+b^2+c^2) \geq abc(a^2+b^2+c^2)$

i.e, $a^{5}+b^{5}+c^{5}>a b c(a b+b c+c a)$

- https://www.cheenta.com/triple-integral-iit-jam-2016-question-15/
- https://www.youtube.com/watch?v=oUyHFKVB9IY

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