### 18th Vietnam 1980 problems A1.  Let x1, x2, ... , xn be real numbers in the interval [0, π] such that (1 + cos x1) + (1 + cos x2) + ... + (1 + cos xn) is an odd integer. Show that sin x1 + sin x2 + ... + sin xn ≥ 1. A2.  Let x1, x2, ... , xn be positive reals with sum s. Show that (x1 + 1/x1)2 + (x2 + 1/x2)2 + ... + (xn + 1/xn)2 ≥ n(n/s + s/n)2. A3.  P is a point inside the triangle A1A2A3. The ray AiP meets the opposite side at Bi. Ci is the midpoint of AiBi and Di is the midpoint of PBi. Show that area C1C2C3 = area D1D2D3. B1.  Show that for any tetrahedron it is possible to find two perpendicular planes such that if the projection of the tetrahedron onto the two planes has areas A and A', then A'/A > √2. B2.  Does there exist real m such that the equation x3 - 2x2 - 2x + m has three different rational roots? B3.  Given n > 1 and real s > 0, find the maximum of x1x2 + x2x3 + x3x4 + ... + xn-1xn for non-negative reals xi such that x1 + x2 + ... + xn = s. To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.

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© John Scholes
jscholes@kalva.demon.co.uk
23 July 2002