CMO 2001 Problem 5

Let $P_0$ , $P_1$ , $P_2$ be three points on the circumference of a circle with radius $1$ , where $P_1P_2 = t < 2$ . For each $i \ge 3$ , define $P_i$ to be the centre of the circumcircle of $\triangle P_{i-1} P_{i-2} P_{i-3}$ .

(1) Prove that the points $P_1, P_5, P_9, P_{13},\cdots$ are collinear.

(2) Let $x$ be the distance from $P_1$ to $P_{1001}$ , and let $y$ be the distance from $P_{1001}$ to $P_{2001}$ . Determine all values of $t$ for which $\sqrt[500]{ \frac xy}$ is an integer.