Školjka

%V0 Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.

%V0 Solve the equation $|x^2 -1|+ |x^2 - 4| = mx$ as a function of the parameter $m$. Which pairs $(x,m)$ of integers satisfy this equation ?

%V0 The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.

%V0 Let $f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $f(f(n) + f(m)) = m + n$ for all positive integers $n,m.$ Find the possible value for $f(1988).$

%V0 A function $f$ defined on the positive integers (and taking positive integers values) is given by: $\begin{matrix} f(1) = 1, f(3) = 3 \\ f(2n) = f(n) \\ f(4n + 1) = 2f(2n + 1) - f(n) \\ f(4n + 3) = 3f(2n + 1) - 2f(n)\text{,} \end{matrix}$ for all positive integers $n.$ Determine with proof the number of positive integers $\leq 1988$ for which $f(n) = n.$

%V0 Let $f$ and $g$ be two integer-valued functions defined on the set of all integers such that (a) $f(m + f(f(n))) = -f(f(m+ 1) - n$ for all integers $m$ and $n;$ (b) $g$ is a polynomial function with integer coefficients and g(n) = $g(f(n))$ $\forall n \in \mathbb{Z}.$