PAMO 2003

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?

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

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