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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 [[2018 February HMMT Algebra Test/Problems|this page]]. To view a list of all HMMT problems and solutions, please visit [[HMMT Problems and Solutions|this page]].
 
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 [[2018 February HMMT Algebra Test/Problems|this page]]. To view a list of all HMMT problems and solutions, please visit [[HMMT Problems and Solutions|this page]].
 
 
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==Problem==
 
==Problem==
 
Assume that the quartic <math>x^4-ax^3+bx^2-ax+d=0</math> has four real roots <math>\tfrac{1}{2} \leq x_1, x_2, x_3, x_4 \geq 2</math>. Find the maximum possible value of <math>\tfrac{(x_1+x_2)(x_1+x_3)x_4}{(x_4+x_2)(x_4+x_3)x_1}</math> over all possible values of <math>a,b,d </math>.
 
Assume that the quartic <math>x^4-ax^3+bx^2-ax+d=0</math> has four real roots <math>\tfrac{1}{2} \leq x_1, x_2, x_3, x_4 \geq 2</math>. Find the maximum possible value of <math>\tfrac{(x_1+x_2)(x_1+x_3)x_4}{(x_4+x_2)(x_4+x_3)x_1}</math> over all possible values of <math>a,b,d </math>.

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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.

ProblemEdit

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 $.

SolutionEdit

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