MHB Function's Fun f(x^2−2016x)=f(x)⋅x+2016

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The discussion revolves around the functional equation f(x^2 - 2016x) = f(x)·x + 2016, with a focus on finding f(2017). It is established that x^2 - 2016x = 2017 leads to two possible values for x: -1 and 2017. Substituting these values results in two equations for f(2017), one involving f(-1) and the other a direct calculation. The latter simplifies to f(2017) = -1, concluding that f(2017) equals -1. The problem highlights the importance of correctly interpreting the functional equation to derive the solution.
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f(x^2*2016x) = f(x)x+2016
Then f(2017) = ?
 
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Isn't "x^2*2016x" the same as "2016x^3"?
 
Thanks, skeeter. That sounds more reasonable.

f(x^2- 2016x)= f(2017) if x^2- 2016x= 2017. That is the same as x^2- 2016x- 2017= (x+ 1)(x- 2017)= 0 so either x= -1 or x= 2017.

So either f(2017)= -1(f(-1))+ 2016 or f(2017)= 2017 f(2017)+ 2016.

For the latter, -2016= 2016 f(2017) so f(2017)= -1.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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