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for some points P_i it is not possible with a total length less than 25

Source: 1972 Hungary - Kürschák Competition p3

October 15, 2022
combinatoricscombinatorial geometry

Problem Statement

ABCDABCD is a square side 1010. There are four points P1,P2,P3,P4P_1, P_2, P_3, P_4 inside the square. Show that we can always construct line segments parallel to the sides of the square of total length 2525 or less, so that each PiP_i is linked by the segments to both of the sides ABAB and CDCD. Show that for some points PiP_i it is not possible with a total length less than 2525.
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