5th Vietnam 1966 problems

1.  [missing]
2.  a, b are two fixed lines through O. Variable lines x, y are parallel. x intersects a at A and b at C, y intersects a at B and b at D. The lines AD and BC meet at M. The line through M parallel to x meets a at L and b at N. What can you say about L, M, N? Find the locus M.
3.  (1) ABCD is a rhombus. A tangent to the inscribed circle meets AB, DA, BC, CD at M, N, P, Q respectively. Find a relationship between BM and DN.
(2) ABCD is a rhombus and P a point inside. The circles through P with centers A, B, C, D meet the four sides AB, BC, CD, DA in eight points. Find a property of the resulting octagon. Use it to construct a regular octagon.
(3) [unclear].

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.

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© John Scholes
16 July 2002