26th Vietnam 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 - nxn-1 + (n2-n)/2 xn-2 + an-3xn-3 + ... + a1x + a0 (where n > 2) has n real roots. Find the values of a0, a1, ... , an-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 xn+2 <= (xn + xn+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.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.

Vietnam home
© John Scholes
23 July 2002