26th Vietnamese Mathematical Olympiad 1988 Problems

26th Vietnamese Mathematical Olympiad 1988 Problems

A1.  994 cages each contain 2 chickens. Each day we rearrange the chickens so that the same pair of chickens are never together twice. What is the maximum number of days we can do this?

A2.  The real polynomial p(x) = xⁿ - nx^n-1 + (n²-n)/2 x^n-2 + a_n-3x^n-3 + ... + a₁x + a₀ (where n > 2) has n real roots. Find the values of a₀, a₁, ... , a_n-3.

A3.  The plane is dissected into equilateral triangles of side 1 by three sets of equally spaced parallel lines. Does there exist a circle such that just 1988 vertices lie inside it?

B1.  The sequence of reals x₁, x₂, x₃, ... satisfies x_n+2 <= (x_n + x_n+1)/2. Show that it converges to a finite limit.

B2.  ABC is an acute-angled triangle. Tan A, tan B, tan C are the roots of the equation x³ + px² + qx + p = 0, where q is not 1. Show that p ≤ √27 and q > 1.

B3.  For a line L in space let R(L) be the operation of rotation through 180 deg about L. Show that three skew lines L, M, N have a common perpendicular iff R(L) R(M) R(N) has the form R(K) for some line K.