# 2003 Pan African MO Problems

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## Day 1

### Problem 1

Let $\mathbb{N}_0=\{0, 1, 2 \cdots \}$. Find all functions: $\mathbb{N}_0 \to \mathbb{N}_0$ such that:

(1) $f(n) < f(n+1)$, all $n \in \mathbb{N}_0$;

(2) $f(2)=2$;

(3) $f(mn)=f(m)f(n)$, all $m, n \in \mathbb{N}_0$.

### Problem 2

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.

### Problem 3

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

## Day 2

### Problem 4

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 $f^{k+1}(n)=f(f^k(n))$, $\forall k \in \mathbb{N}_0$?

### Problem 5

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

### Problem 6

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