CMO 1986

In the diagram line segments $AB$ and $CD$ are of length 1 while angles $ABC$ and $CBD$ are $90^\circ$ and $30^\circ$ respectively. Find $AC$ .

import geometry;
import graph;

unitsize(1.5 cm);

pair A, B, C, D;

B = (0,0);
D = (3,0);
A = 2*dir(120);
C = extension(B,dir(30),A,D);

draw(A--B--D--cycle);
draw(B--C);
draw(arc(B,0.5,0,30));

label(" $A$ ", A, NW);
label(" $B$ ", B, SW);
label(" $C$ ", C, NE);
label(" $D$ ", D, SE);
label(" $30^\circ$ ", (0.8,0.2));
label(" $90^\circ$ ", (0.1,0.5));

perpendicular(B,NE,C-B);

A Mathlon is a competition in which there are $M$ athletic events. Such a competition was held in which only $A$ , $B$ , and $C$ participated. In each event $p_1$ points were awarded for first place, $p_2$ for second and $p_3$ for third, where $p_1 > p_2 > p_3 > 0$ and $p_1$ , $p_2$ , $p_3$ are integers. The final scores for $A$ was 22, for $B$ was 9 and for $C$ was also 9. $B$ won the 100 metres. What is the value of $M$ and who was second in the high jump?

A chord $ST$ of constant length slides around a semicircle with diameter $AB$ . $M$ is the midpoint of $ST$ and $P$ is the foot of the perpendicular from $S$ to $AB$ . Prove that $\angle SPM$ is constant for all positions of $ST$ .

For all positive integers $n$ and $k$ , define $F(n,k) = \sum_{r = 1}^n r^{2k - 1}$ . Prove that $F(n,1)$ divides $F(n,k)$ .

Let $u_1$ , $u_2$ , $u_3$ , $\dots$ be a sequence of integers satisfying the recurrence relation $u_{n + 2} = u_{n + 1}^2 - u_n$ . Suppose $u_1 = 39$ and $u_2 = 45$ . Prove that 1986 divides infinitely many terms of the sequence.