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Functional Equations

Functional Equations - Problem Solving


The function ff satisfies (xy)f(x+y)(x+y)f(xy)+4xy(x2y2)=0,(x-y)f(x+y)-(x+y)f(x-y)+4xy(x^2-y^2)=0,

whenever xy |x| \neq |y | .

Given that f(1)=2,f(1)=2, what is f(x)f(x)?

Function f(n)f(n) defined over all positive integers nn satisfies the following: f(1)=1,f(2)=2, andf(n+2)=f(n+1)+(f(n))2+3581 for n1.\begin{aligned} & f(1)=1, f(2)=2, \text{ and}\\ & f(n+2)=f(n+1)+\left(f(n)\right)^2+3581 \text{ for } n \geq 1. \end{aligned} How many of the 35813581 integers f(1),f(2),,f(3581)f(1), f(2), \ldots, f(3581) are multiples of 7?7?

What is the polynomial p(x)p(x) that satisfies p(x2+4)=(p(x))2+4 and p(0)=0?p(x^2+4)=\left(p(x)\right)^2+4 \text{ and } p(0)=0?

A function ff from the positive integers to the positive integers satisfies the conditions
1) f(n)=2f(n1)+3f(n) = 2 f(n-1) + 3 ,
2) f(1)=9. f(1) = 9 .

What is the value of f(9) f(9) ?

If f(x)f(x) is a polynomial satisfying 27f(x3)4f(x2)x6f(3x)+46=0 27 f(x^3) -4f(x^2) - x^6 f(3x) + 46 = 0 , what is f(10)?f(10)?


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