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UKMT Mathematical Olympiad for Girls, 2013

Number theory Difficulty 2.0 Junior Prove it United Kingdom

(a) Find all positive integers $a$ and $b$ for which $a^2 - b^2 = 18$ .

(b) The diagram shows a sequence of points $P_0, P_1, P_2, P_3, P_4, \ldots$ , which spirals out around the point $O$ . For any point $P$ in the sequence, the line segment joining $P$ to the next point is perpendicular to $OP$ and has length $3$ . The distance from $P_0$ to $O$ is $29$ . What is the next value of $n$ for which the distance from $P_n$ to $O$ is an integer?

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