MathDB

Let

n

be a positive integer. Given are points

P_1,\ P_2,\ \cdots,\ P_{4n}

of which any three points are not collinear. For

i=1,\ 2,\ \cdots,\ 4n

, rotating half-line

P_iP_{i-1}

clockwise in

90^\circ

about the pivot

P_i

gives half-line

P_iP_{i+1}.

Find the maximum value of the number of the pairs of

(i,\ j)

such that line segments

P_iP_{i+1}

and

P_jP_{j+1}

intersect at except endpoints. Note that :

P_0=P_{4n},\ P_{4n+1}=P_1

and

1\leq i<j\leq 4n.

Problem Statement