VMO 1966

Let $x, y$ and $z$ be nonnegative real numbers satisfying the following conditions: (1) $x + cy \le 36$ ,(2) $2x+ 3z \le 72$ , where $c$ is a given positive number. Prove that if $c \ge 3$ then the maximum of the sum $x + y + z$ is $36$ , while if $c < 3$ , the maximum of the sum is $24 + \frac{36}{c}$ .

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.

(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].