Riddles, brain puzzles and mathematical problems

[BAndrew]

1)Fun fact:

Consider the following function:

f(n) = sqrt(9/121*100^n +(112-44n)/121)

Find for fun (with the help of a computer) f(95). Notice something strange?

Calculate 10000 or so digits to see something cool.


2)Math riddle:
Find the positive integers x,y,z so that
x^2 + y^2 + z^2 = N^3
x^3 + y^3+ z^3 = M^2

(at least 1 solution required + how you found it)
Are there infinite solutions?

3)IMPOSSIBLE(?) PROBLEM

Find the smallest number with the following property:

When devided with AB, ABA, ABAB the remainder is A, AB, ABA respectively (Letters A and B symbolize the digits of the number). You have AB days to solve the problem, otherwise ABA million humans will be killed and you will have to pay AB^10 $.