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The 29th Vietnam Mathematical Olympiad
1991年第29届越南数学奥林匹克

A1. Find all real-valued functions f(x) on the reals such that f(xy)/2 + f(xz)/2 - f(x) f(yz) ≥ 1/4 for all x, y, z.

A2. For each positive integer n and odd k > 1, find the largest number N such that 2N divides kn - 1.

A3. The lines L, M, N in space are mutually perpendicular. A variable sphere passes through three fixed points A on L, B on M, C on N and meets the lines again at A', B', C'. Find the locus of the midpoint of the line joining the centroids of ABC and A'B'C'.

B1. 1991 students sit in a circle. Starting from student A and counting clockwise round the remaining students, every second and third student is asked to leave the circle until only one remains. (So if the students clockwise from A are A, B, C, D, E, F, ... , then B, C, E, F are the first students to leave.) Where was the surviving student originally sitting relative to A ?

B2. The triangle ABC has centroid G. The lines GA, GB, GC meet the circumcircle again at D, E, F. Show that 3/R ≤ 1/GD + 1/GE + 1/GF ≤ √3 (1/AB + 1/BC + 1/CA), where R is the circumradius.

B3. 对不起,我没有找到这个题目。

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