Oops, is it inequalities ?

Algebra Level 4

For positive real numbers $a,b,c,d$ , find the remainder when the minimum possible integer value of the following expression is divided by $11$ $\sqrt{(a^3+b^3+c^3+d^3)\Bigl( \dfrac{2.25}{a^3}+\dfrac{20.25}{b^3} + \dfrac{42.25}{c^3}+\dfrac{400}{d^3}\Bigr)}$

This is a part of the set 11≡ awesome (mod remainders) .

0 4 8 10

1 solution

Aditya Raut
Sep 15, 2014

By Cauchy Schwartz inequality, for 2 sets $\{a_1,a_2,...,a_n\}$ and $\{ b_1,b_2,...,b_n\}$ , we have $\left| \sum_{a_i b_i} \right|^2 \leq \Bigl(\sum a_i^2 \Bigr) \Bigl(\sum b_i^2 \Bigr)$

For given equation, consider the sets

$\{ \sqrt{a^3}, \sqrt{b^3},\sqrt{c^3},\sqrt{d^3} \}$ and $\{\dfrac{1.5}{\sqrt{a^3}},\dfrac{4.5}{\sqrt{b^3}},\dfrac{6.5}{\sqrt{c^3}},\dfrac{20}{\sqrt{d^3}}\}$

Then by above inequality, given expression's least value is $1.5+4.5+6.5+20=32.5$ Thus, it's minimum integer value will be $33$ and remainder when $33$ is divided by $11$ is $\boxed{0}$

Cauchy Schwarz was staring at my face!!

Shabarish Ch - 6 years, 8 months ago

Oops, is it inequalities ?

This is a part of the set 11≡ awesome (mod remainders) .

1 solution

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