IMO 1985 LL SPA75

Let ABCD be a rectangle, AB = a, BC = b. Consider the

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IMO 1985 LL SPA75

Origin: SPA

Problem

Let ABCD be a rectangle, AB = a, BC = b. Consider the family of parallel and equidistant straight lines (the distance between two consecutive lines being d) that are at an the angle \varphi, 0 \leq\varphi \leq90◦, with respect to AB. Let L be the sum of the lengths of all the segments intersecting the rectangle. Find: (a) how L varies, (b) a necessary and sufficient condition for L to be a constant, and (c) the value of this constant.