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 at 9:00 pm IST ,8 30 PDT. For more details, see IMO Problems Discussion Group.
Here are the problems from the second day of the 1959 International mathematical Olympiad. They range from easy to harder Geometry and Combinatorics. Try your hand at them. Don't be discouraged if you cant completely solve them. Do post your inspirations and ideas towards the problems. The discussion for theses questions will be held soon. Happy Problem Solving!
Q4. Construct a right-angled triangle whose hypotenuse \(c\) is given if it is known that the median from the right angle equals the geometric mean of the remaining two sides of the triangle. (HUN)
Q5. A segment \(AB\) is given and on it a point \(M\). On the same side of \(AB\) squares \(AMCD\) and \(BMFE\) are constructed. The circumcircles of the two squares, whose centers are \(P\) and \(Q\), intersect in \(M\) and another point \(N\).
(a) Prove that lines \(FA\) and \(BC\) intersect at \(N\).
(b) Prove that all such constructed lines \(MN\) pass through the same point \(S\), regardless of the selection of \(M\).
(c) Find the locus of the midpoints of all segments \(PQ\), as \(M\) varies along the segment \(AB\).(ROM)
Q6. Let \(\alpha\) and \(\beta\) be two planes intersecting at a line \(p\). In \(\alpha\) a point \(A\) is given and in \(\beta\) a point \(C\) is given, neither of which lies on \(p\). Construct \(B\) in \(\alpha\) and \(D\) in \(\beta\) such that \(ABCD\) is an equilateral trapezoid, \(AB || CD\), in which a circle can be inscribed. (CZS)