Problem 39

Algebra Level 5

Consider an equation x 4 + a x 3 + b x 2 + c x + d = 0 x^4+ax^3+bx^2+cx+d = 0 with roots (of x x ) x 1 , x 2 , x 3 x_1,x_2,x_3 and x 4 x_4 . If x 1 2016 + x 2 2016 + x 3 2016 + x 4 2016 = 4 x_1^{2016} + x_2^{2016} + x_3^{2016} + x_4^{2016} = 4 , find the maximum value of a a , a max a_{\text{max}} .

Submit your answer as a max + b + c + d + x 1 + x 2 + x 3 + x 4 a_{\text{max}} +b+c+d+x_1+x_2+x_3+x_4 when a max a_{\text{max}} is fulfilled.


Set .


The answer is 11.

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1 solution

Shubham Poddar
May 28, 2016

For maximum a, all the roots must be equal to -1. Hence a=4, b=6, c=4, d=1, x1=-1, x2=-1, x3=-1,x4=-1 Hence the ans is 4+6+4+1-1-1-1-1=11.

@shubham poddar How can you say this that all roots MUST be equal to -1 ??? Show your working!!!

Aaghaz Mahajan - 2 years, 8 months ago

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