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The 25th Vietnam Mathematical Olympiad
1987年第25届越南奥林匹克数学竞赛

A1. Let xn = (n+1)π/3974. Find the sum of all cos(± x1 ± x2 ± ... ± x1987).

A2. The sequences a0, a1, a2, ... and b0, b1, b2, ... are defined as follows. a0 = 365, an+1 = an(an1986 + 1) + 1622, b0 = 16, bn+1 = bn(bn3 + 1) - 1952. Show that there is no number in both sequences.

A3. There are n > 2 lines in the plane, no two parallel. The lines are not all concurrent. Show that there is a point on just two lines.

B1. x1, x2, ... , xn are positive reals with sum X and n > 1. h ≤ k are two positive integers. H = 2h and K = 2k. Show that x1K/(X - x1)H-1 + x2K/(X - x2)H-1 + x3K/(X - x3)H-1 + ... + xnK/(X - xn)H-1 ≥ XK-H+1/( (n-1)2H-1nK-H). When does equality hold?

B2. The function f(x) is defined and differentiable on the non-negative reals. It satisfies | f(x) | ≤ 5, f(x) f '(x) ≥ sin x for all x. Show that it tends to a limit as x tends to infinity.

B3. Given 5 rays in space from the same point, show that we can always find two with an angle between them of at most 90o.

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