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Algebra Level 4

Given that $\displaystyle\sum _{ n=1 }^{ k }{ { x }_{ n }^{ 2 } } =9$ and that $\displaystyle\sum _{ n=1 }^{ k }{ { y }_{ n }^{ 2 } } =5329$

Find the maximum value of $\displaystyle\sum _{ n=1 }^{ k }{ { ({ x }_{ n }+{ y }_{ n }) }^{ 2 } }$ .

Note : ${x}_{n},{y}_{n}\ge0$ .

2 solutions

Natanael Flores
Feb 20, 2016

We have that $x^2_1+\cdots+x^2_k=9=3^2$ and $y^2_1+\cdots+y^2_k=5329=73^2$

Let $N=2x_1y_1+\cdots+2x_ky_k$

Adding the three equations....

$(x_1+y_1)^2+\cdots+(x_k+y_k)^2=5338+N$

The max possible value of $x$ is $3$ because of $(x^2=9)$ and the max possible value of $y$ is $73$ because of $(y^2=5329)$

So the max possible value of $N=2xy$ is $2*3*73=438$

Therefore the max possible value of the sum is $5338+438=5776$

upvoted!

the intended solution was to use Minkowski's inequality,which you can prove using the same logic you were using :)

Hamza A - 5 years, 3 months ago

Thnx :D, can you write the proof with minkowski?

Natanael Flores - 5 years, 3 months ago

i think i can,i'll attempt it and then i'll post it as a note

Hamza A - 5 years, 3 months ago

yay!

i proved it

here's the link

https://brilliant.org/discussions/thread/proof-of-minkowskis-inequality/

Hamza A - 5 years, 3 months ago

Soumava Pal
Mar 22, 2016

I dont think we need to use Minkowski's Inequality, as it is a far generalization of Cauchy-Schwartz Inequality, because

$\sum_{n=1}^{n=k}{(x_n+y_n)}^2=\sum_{n=1}^{n=k}{x_n}^2+\sum_{n=1}^{n=k}{y_n}^2+2\sum_{n=1}^{n=k}({x_n}{y_n})$

$\le 9+5329+2(\sum_{n=1}^{n=k}{x_n}^2\sum_{n=1}^{n=k}{y_n}^2)^{\frac{1}{2}}$ (by Cauchy-Schwartz Inequality

$=5338+2.3.73=5338+438=5776$ .