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Strange number theory from 2017 Taiwan TST

Source: 2017 Taiwan TST Round 3

April 13, 2018
conicsellipsenumber theory

Problem Statement

Choose a rational point P0(xp,yp)P_0(x_p,y_p) arbitrary on ellipse C:x2+2y2=2098C:x^2+2y^2=2098. Define P1,P2,⋯P_1,P_2,\cdots recursively by the following rules:
(1)(1) Choose a lattice point Qi=(xi,yi)∉CQ_i=(x_i,y_i)\notin C such that ∣xi∣<50|x_i|<50 and ∣yi∣<50|y_i|<50. (2)(2) Line PiQiP_iQ_i intersects CC at another point Pi+1P_{i+1}.
Prove that for any point P0P_0 we can choose suitable points Q0,Q1,⋯Q_0,Q_1,\cdots such that ∃k∈N∪{0}\exists k\in\mathbb{N}\cup\{0\}, OPk‾2=2017\overline{OP_k}^2=2017.
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