A1. Let x_{n} = (n+1)π/3974. Find the sum of all cos(± x_{1} ± x_{2} ± ... ± x_{1987}). | |
A2. The sequences a_{0}, a_{1}, a_{2}, ... and b_{0}, b_{1}, b_{2}, ... are defined as follows. a_{0} = 365, a_{n+1} = a_{n}(a_{n}^{1986} + 1) + 1622, b_{0} = 16, b_{n+1} = b_{n}(b_{n}^{3} + 1) - 1952. Show that there is no number in both sequences. | |
A3. There are n > 2 lines in the plane, no two parallel. The lines are not all concurrent. Show that there is a point on just two lines. | |
B1. x_{1}, x_{2}, ... , x_{n} are positive reals with sum X and n > 1. h ≤ k are two positive integers. H = 2^{h} and K = 2^{k}. Show that x_{1}^{K}/(X - x_{1})^{H-1} + x_{2}^{K}/(X - x_{2})^{H-1} + x_{3}^{K}/(X - x_{3})^{H-1} + ... + x_{n}^{K}/(X - x_{n})^{H-1} ≥ X^{K-H+1}/( (n-1)^{2H-1}n^{K-H}). When does equality hold? | |
B2. The function f(x) is defined and differentiable on the non-negative reals. It satisfies | f(x) | ≤ 5, f(x) f '(x) ≥ sin x for all x. Show that it tends to a limit as x tends to infinity. | |
B3. Given 5 rays in space from the same point, show that we can always find two with an angle between them of at most 90^{o}. |
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
23 July 2002