WMC 2018 Senior Problem 38

Level 2

A polynomial f ( x ) f(x) satisfies f ( f ( x ) + 1 ) + 1 = x 4 f\left(f\left(x\right)+1\right)+1=x^4 . Find f ( 9 ) f(9) .

81 1 18 80 9

Tiger Ang by Tiger Ang , 20, Malaysia

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1 solution

X X
Oct 20, 2018

Let f ( x ) + 1 = g ( x ) f(x)+1=g(x)

f ( f ( x ) + 1 ) + 1 = f ( g ( x ) ) + 1 = g ( g ( x ) ) = x 4 , g ( x ) = x 2 , f ( x ) = x 2 − 1 f(f(x)+1)+1=f(g(x))+1=g(g(x))=x^4,g(x)=x^2,f(x)=x^2-1

So f ( 9 ) = 81 − 1 = 80 f(9)=81-1=80

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