IMO 1969 LL GBR26

Origin: GBR

Problem

A smooth solid consists of a right circular cylinder of height h and base-radius r, surmounted by a hemisphere of radius r and center O. The solid stands on a horizontal table. One end of a string is attached to a point on the base. The string is stretched (initially being kept in the vertical plane) over the highest point of the solid and held down at the point P on the hemisphere such that OP makes an angle \alpha with the horizontal. Show that if \alpha is small enough, the string will slacken if slightly displaced and no longer remain in a vertical plane. If then pulled tight through P, show that it will cross the common circular section of the hemisphere and cylinder at a point Q such that \angleSOQ = \varphi, S being where it initially crossed this section, and sin\varphi = r tan \alpha h .