MathDB

i.) Let

g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.

What is the remainder when the polynomial

g(x^{12}

is divided by the polynomial

g(x)

? ii.) If

k

is a positive number and

f

is a function such that, for every positive number

x, f(x^2 + 1 )^{\sqrt{x}} = k.

Find the value of

f( \frac{9 +y^2}{y^2})^{\sqrt{ \frac{12}{y} }}

for every positive number

y.

iii.) The function

f

satisfies the functional equation

f(x) + f(y) = f(x+y) - x \cdot y - 1

for every pair

x,y

of real numbers. If

f(1) = 1,

then find the numbers of integers

n,

for which

f(n) = n.

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