A1. Find all real solutions to x^{3} - 3x^{2} - 8x + 40 = 8(4x + 4)^{1/4} = 0. | |
A2. The sequence a_{0}, a_{1}, a_{2}, ... is defined by a_{0} = 1, a_{1} = 3, a_{n+2} = a_{n+1} + 9a_{n} for n even, 9a_{n+1} + 5a_{n} for n odd. Show that a_{1995}^{2} + a_{1996}^{2} + a_{1997}^{2} + a_{1998}^{2} + a_{1999}^{2} + a_{2000}^{2} is divisible by 20, and that no a_{2n+1} is a square. | |
A3. ABC is a triangle with altitudes AD, BE, CF. A', B', C' are points on AD, BE, CF such that AA'/AD = BB'/BE = CC'/CF = k. Find all k such that A'B'C' is similar to ABC for all triangles ABC. | |
B1. ABCD is a tetrahedron. A' is the circumcenter of the face opposite A. B', C', D' are defined similarly. p_{A} is the plane through A perpendicular to C'D', p_{B} is the plane through B perpendicular to D'A', p_{C} is the plane through C perpendicular to A'B', and p_{D} is the plane through D perpendicular to B'C'. Show that the four planes have a common point. If this point is the circumcenter of ABCD, must ABCD be regular? | |
B2. Find all real polynomials p(x) such that p(x) = a has more than 1995 real roots, all greater than 1995, for any a > 1995. Multiple roots are counted according to their multiplicities. | |
B3. How many ways are there of coloring the vertices of a regular 2n-gon with n colors, such that each vertex is given one color, and every color is used for two non-adjacent vertices? Colorings are regarded as the same if one is obtained from the other by rotation. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
22 July 2002