Problem 5

Two-color the regions cut out by the $n$ chords inside the circle red and blue so that adjacent regions have different colors, with the two regions incident to vertex $1$ colored red on the right and blue on the left viewed from $1$. Orient the boundary of every red region clockwise and every blue region anticlockwise; every chord segment separating a red and a blue region receives the same orientation from both sides. A present starting at $1$ moves along its chord in the direction consistent with the two adjacent regions, and at each intersection it turns onto the crossing chord while continuing to border one of those two regions, so it always follows the region orientation. Around the outer circle the region colors alternate, so the chord orientations at vertices $1,2,\ldots,2n$ alternate between outgoing at odd vertices and incoming at even vertices. Hence a present from $1$ can only exit at the $n$ even-numbered vertices $2,4,\ldots,2n$.