## Task 6 - 231236

Let $\angle P_{0} S Q$ be an angle of any, but fixed, given size $\alpha$.

II. Determine all the initial values of $\alpha$ for which there are

(C) exactly one, (D) exactly two, (E) exactly $n$

values of $\varphi_{0}$ such that for the number $m$ found in (B) (a point $P_{m}$ exists and) the equation $S P_{m}=S P_{m-1}$ holds. In (E), $n>2$ is a given natural number.
I. The size of the angle that the light ray emanating from $P_{k}$ forms with the half-line leading from $P-k$ to $S$ is called $\varphi_{k}$.

This light ray emanating from $P_{k}$ reaches the other leg of the angle $\angle P_{0} S Q$ if and only if $\varphi_{k}1$ the existence of $P_{0}, \ldots, P_{m-1}$ due to $\alpha1$, so there is always at least one natural number $m \geq 1$ that satisfies this condition.

(C) The condition is satisfied exactly by the number $m=1$ and no other natural number $m \geq 1$ if $\frac{1}{2}+\frac{90^{\circ}}{\alpha} \leq 2$ holds. This is true for all $\alpha \geq 60^{\circ}$.

(D) The condition is satisfied exactly by the numbers $m=1$ and $m=2$ and no other natural number $m \geq 1$ if $2<\frac{1}{2}+\frac{90^{\circ}}{\alpha} \leq 3$ holds. This is true for all $\alpha$ with $36^{\circ} \leq \alpha<60^{\circ}$.

(E) The condition is satisfied exactly by the $N$ numbers $m=1, m=2, \ldots, m=N$ and no other natural number $m \geq 1$ if $N<\frac{1}{2}+\frac{90^{\circ}}{\alpha} \leq N+1$ holds.

This is true for all $\alpha$ with

$$
\frac{180^{\circ}}{2 N+1} \leq \alpha<\frac{180^{\circ}}{2 N-1}
$$

## Taken from $[5]$

### 9.25.4 Round 1983, Class 12