Interesting Inequality

Calculus Level 4

State the correct inequality

[ ( 1 + a ) ( 1 + b ) ( 1 + c ) ( 1 + d ) ] 15 > 1 5 15 ( a b c d ) X { [(1+a)(1+b)(1+c)(1+d)] }^{ 15 }>15^{15}{ (abcd) }^{X }

Find X X .


The answer is 8.

Rajat Dwivedi by Rajat Dwivedi , 25, India

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1 solution

Rajat Dwivedi
Jul 19, 2014

Interestingly if we consider the LHS, we find that it is written 1+a+b+c+d+ab+bc+ca+ad+bd+cd+abc+bcd+acd+abd+abcd1+a+b+c+d+ab+bc+ca+ad+bd+cd+abc+bcd+acd+abd+abcd. Now applying AM\ge GM we get \left( a+b+c+d+ab+bc+ca+ad+bd+cd+abc+bcd+acd+abd+abcd \right) /15\ge \sqrt [ 15 ]{ { \left( abcd \right) }^{ 8 } } So we can conclude that 1+a+b+c+d+ab+bc+ca+ad+bd+cd+abc+bcd+acd+abd+abcd1+a+b+c+d+ab+bc+ca+ad+bd+cd+abc+bcd+acd+abd+abcd >\sqrt [ 15 ]{ { \left( abcd \right) }^{ 8 } }

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FYI, place your math code within \ ( \ ) (no spaces) to get the latex to display correctly.

Can you explain why X = 8 X = 8 is the optimal solution? E.g. how do you know that we cannot have X = 8.1 X = 8.1 , especially since you used a non-strict inequality?

Do you want X X to be the maximum possible real number?

Calvin Lin Staff - 6 years, 10 months ago

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