18th Vietnamese Mathematical Olympiad 1980 Problems

18th Vietnamese Mathematical Olympiad 1980 Problems

A1.  Let x₁, x₂, ... , x_n be real numbers in the interval [0, π] such that (1 + cos x₁) + (1 + cos x₂) + ... + (1 + cos x_n) is an odd integer. Show that sin x₁ + sin x₂ + ... + sin x_n ≥ 1.
A2.  Let x₁, x₂, ... , x_n be positive reals with sum s. Show that (x₁ + 1/x₁)² + (x₂ + 1/x₂)² + ... + (x_n + 1/x_n)² ≥ n(n/s + s/n)².

A3.  P is a point inside the triangle A₁A₂A₃. The ray A_iP meets the opposite side at B_i. C_i is the midpoint of A_iB_i and D_i is the midpoint of PB_i. 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 + ... + x_n-1x_n for non-negative reals x_i such that x₁ + x₂ + ... + x_n = s.