2018 February HMMT Algebra Test/Problem 9

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Problem 9
Statistics
Type	February
Appears on Test	Algebra Test
Proposed By	Allen Liu
Question Weight	7.70
Preceeded By	Problem 8
Followed By	Problem 10

This is Problem 9 on the 2018 February HMMT Algebra Test. To view a list of all of the 2018 February HMMT Algebra Problems, please visit this page. To view a list of all HMMT problems and solutions, please visit this page.

Problem

Assume that the quartic $x^4-ax^3+bx^2-ax+d=0$ has four real roots $\tfrac{1}{2} \leq x_1, x_2, x_3, x_4 \geq 2$. Find the maximum possible value of $\tfrac{(x_1+x_2)(x_1+x_3)x_4}{(x_4+x_2)(x_4+x_3)x_1}$ over all possible values of $a,b,d$.

Solution

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