The goal of this set of notes is to improve our problem solving and proof writing skills. You are encouraged to submit a solution to any of these problems, and join in the discussion in #imo-discussion on Saturday 24 at 9:00 pm IST ,8 30 PDT. For more details, see IMO Problems Discussion Group.
Sorry Everyone!, I was really busy the last whole week. Here is the next set of problems from the 1960 IMO.
4. (HUN) Construct a triangle whose lengths of heights and (from A and B, respectively) and length of median (from A) are given.
5. (CZS) A cube ABCDA'B'C'D' is given.
(a) Find the locus of all midpoints of segments , where is any point on segment and any point on segment B'D'.
(b) Find the locus of all points on segments such that
6. (BUL) An isosceles trapezoid with bases and and height is given.
(a) On the line of symmetry construct the point P such that both (nonbase) sides are seen from P with an angle of .
(b) Find the distance of from one of the bases of the trapezoid.
(c) Under what conditions for , and can the point be constructed (analyze all possible cases)?
7. (GDR) A sphere is inscribed in a regular cone. Around the sphere a cylinder is circumscribed so that its base is in the same plane as the base of the cone. Let be the volume of the cone and the volume of the cylinder.
(a) Prove that is impossible.
(b) Find the smallest for which , and in this case construct the angle at the vertex of the cone.