15th Vietnamese Mathematical Olympiad 1977 Problems

15th Vietnamese Mathematical Olympiad 1977 Problems

A1.  Find all real x such that √(x - 1/x) + √(1 - 1/x) > (x - 1)/x.
A2.  Show that there are 1977 non-similar triangles such that the angles A, B, C satisfy (sin A + sin B + sin C)/(cos A + cos B + cos C) = 12/7 and sin A sin B sin C = 12/25.

A3.  Into how many regions do n circles divide the plane, if each pair of circles intersects in two points and no point lies on three circles?
B1.  p(x) is a real polynomial of degree 3. Find necessary and sufficient conditions on its coefficients in order that p(n) is integral for every integer n.
B2.  The real numbers a₀, a₁, ... , a_n+1 satisfy a₀ = a_n+1 = 0 and | a_k-1 - 2a_k + a_k+1| ≤ 1 for k = 1, 2, ... , n. Show that |a_k| ≤ k(n + 1 - k)/2 for all k.
B3.  The planes p and p' are parallel. A polygon P on p has m sides and a polygon P' on p' has n sides. Find the largest and smallest distances between a vertex of P and a vertex of P'.