Let $N_0=\{0, 1, 2 \cdots \}$. Find all functions: $N_0 \to N_0$ such that: (1) $f(n) < f(n+1)$, all $n \in N_0$; (2) $f(2)=2$; (3) $f(mn)=f(m)f(n)$, all $m, n \in N_0$.

The circumference of a circle is arbitrarily divided into four arcs. The midpoints of the arcs are connected by segments. Show that two of these segments are perpendicular.

Does there exists a base in which the numbers of the form: \[ 10101, 101010101, 1010101010101,\cdots \] are all prime numbers?

Let $\mathbb{N}_0=\{0,1,2 \cdots \}$. Does there exist a function $f: \mathbb{N}__0 \to \mathbb{N}_0$ such that: \[ f^{2003}(n)=5n, \forall n \in \mathbb{N}_0 \] where we define: $f^1(n)=f(n)$ and $f^{k+1}(n)=f(f^k(n))$, $\forall k \in \mathbb{N}_0$?

Find all positive integers $n$ such that $21$ divides $2^{2^n}+2^n+1$.

Find all functions $f: R\to R$ such that: \[ f(x^2)-f(y^2)=(x+y)(f(x)-f(y)), x,y \in R \]