MathDB

5

Part of 2009 Iran MO (3rd Round)

Problems(2)

Rolling a Ball

Source: Iran 3rd round 2009 - final exam problem 5

1/2/2015
A ball is placed on a plane and a point on the ball is marked. Our goal is to roll the ball on a polygon in the plane in a way that it comes back to where it started and the marked point comes to the top of it. Note that We are not allowed to rotate without moving, but only rolling. Prove that it is possible.
Time allowed for this problem was 90 minutes.
Pythagorean TheoremgeometrycombinatoricsIran
Iran(3rd round)2009

Source: Problem 5 Geometry

9/13/2009
5-Two circles S1 S_1 and S2 S_2 with equal radius and intersecting at two points are given in the plane.A line l l intersects S1 S_1 at B,D B,D and S2 S_2 at A,C A,C(the order of the points on the line are as follows:A,B,C,D A,B,C,D).Two circles W1 W_1 and W2 W_2 are drawn such that both of them are tangent externally at S1 S_1 and internally at S2 S_2 and also tangent to l l at both sides.Suppose W1 W_1 and W2 W_2 are tangent.Then PROVE AB \equal{} CD.
geometry unsolvedgeometry