ZA-SAMO 1997

From an initial triangle $\Delta A_0B_0C_0$ , a sequence of triangles $\Delta A_1B_1C_1$ , $A_2B_2C_2$ , ... is formed such that, at each stage, $A_{k + 1}$ , $B_{k + 1}$ and $C_{k + 1}$ are the points where the incircle of $\Delta A_kB_kC_k$ touches the sides $B_kC_k$ , $C_kA_k$ and $A_kB_k$ respectively.

(a) Express $\angle A_{k + 1}B_{k + 1}C_{k + 1}$ in terms of $\angle A_kB_kC_k$ .

(b) Deduce that, as $k$ increases, $\angle A_kB_kC_k$ tends to $60^{\circ}$ .

Find all natural numbers with the property that, when the first digit is moved to the end, the resulting number is $\dfrac{7}{2}$ times the original one.

Find all solutions $x,y \in \mathbb{Z}$ , $x,y \geq 0$ , to the equation $$1 + 3^x = 2^y.$$

Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ which satisfy $$f(m + f(n)) = f(m) + n$$ for all $m,n \in \mathbb{Z}$ .

A circle and a point $P$ higher than the circle lie in the same vertical plane. A particle moves along a straight line under gravity from $P$ to a point $Q$ on the circle. Given that the distance travelled from $P$ in time $t$ is equal to $\dfrac{1}{2}gt^2 \sin{\alpha}$ , where $\alpha$ is the angle of inclination of the line $PQ$ to the horizontal, give a geometrical characterization of the point $Q$ for which the time taken from $P$ to $Q$ is a minimum.

Six points are connected in pairs by lines, each of which is either red or blue. Every pair of points is joined. Determine whether there must be a closed path having four sides all of the same colour. (A path is closed if it begins and ends at the same point.)