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) = x^{n} - nx^{n-1} + (n^{2}-n)/2 x^{n-2} + a_{n-3}x^{n-3} + ... + a_{1}x + a_{0} (where n > 2) has n real roots. Find the values of a_{0}, a_{1}, ... , 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 x_{1}, x_{2}, x_{3}, ... 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 x^{3} + px^{2} + 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
jscholes@kalva.demon.co.uk
23 July 2002