2014 IMO Shortlist C9

There are $n$ circles drawn on a piece of paper in such a way that any two circles intersect in two points, and no three circles pass through the same point. Turbo the snail slides along the circles in the following fashion. Initially he moves on one of the circles in clockwise direction. Turbo always keeps sliding along the current circle until he reaches an intersection with another circle. Then he continues his journey on this new circle and also changes the direction of moving, i.e. from clockwise to anticlockwise or vice versa. Suppose that Turbo's path entirely covers all circles. Prove that $n$ must be odd.

在一张纸上画$n$个圆，任意两个圆相交于两点，并且没有三个圆经过同一点。涡轮蜗牛按照以下方式沿着圆圈滑动。最初，他沿顺时针方向在其中一个圆圈上移动。 Turbo 总是沿着当前圆不断滑动，直到到达与另一个圆的交点。然后他在这个新的圆圈上继续他的旅程，并改变移动方向，即从顺时针到逆时针，反之亦然。假设 Turbo 的路径完全覆盖所有圆。证明 $n$ 一定是奇数。