NL-NWO 1984

The circles $C_1$ and $C_2$ with radii $r_1$ and $r_2$ touch the line $p$ at the point $P$ . $C_1$ lies entirely within $C_2$ . Line $q \perp p$ intersects $p$ at $S$ and touches $C_1$ at $R$ . $q$ intersects $C_2$ at $M$ and $N$ , where $N$ is between $R$ and $S$ . a) Prove that line $PR$ bisects angle $\angle MPN$ . b) Calculate the ratio $r_1 : r_2$ if line $PN$ bisects angle $\angle RPS$ .

The circuit diagram drawn (see figure ) contains a battery $B$ , a lamp $L$ and five switches $S_1$ to $S_5$ . The probability that switch $S_3$ is closed (makes contact) is $\frac23$ , for the other four switches that probability is $\frac12$ (the probabilities are mutually independent). Calculate the probability that the light is on.

$L$ $B$ $S_1$ $S_2$ $S_3$ $S_4$ $S_5$

For $n = 1,2,3,...$ . $a_n$ is defined by: $$a_n =\frac{1 \cdot 4 \cdot 7 \cdot ... (3n-2)}{2 \cdot 5 \cdot 8 \cdot ... (3n-1)}$$ Prove that for every $n$ holds that $$\frac{1}{\sqrt{3n+1}}\le a_n \le \frac{1}{\sqrt[3]{3n+1}}$$

By placing parentheses in the expression $1:2:3$ we can get two different number values: $(1 : 2) : 3 = \frac16$ and $1 : (2 : 3) = \frac32$ . Now brackets are placed in the expression $1:2:3:4:5:6:7:8$ . Multiple bracket pairs are allowed, whether or not in nest form. (a) What is the largest numerical value we can get, and what is the smallest? (b) How many different number values can be obtained?