A1. Find all real x such that √(x - 1/x) + √(1 - 1/x) > (x - 1)/x. | |
A2. Show that there are 1977 non-similar triangles such that the angles A, B, C satisfy (sin A + sin B + sin C)/(cos A + cos B + cos C) = 12/7 and sin A sin B sin C = 12/25. | |
A3. Into how many regions do n circles divide the plane, if each pair of circles intersects in two points and no point lies on three circles? | |
B1. p(x) is a real polynomial of degree 3. Find necessary and sufficient conditions on its coefficients in order that p(n) is integral for every integer n. | |
B2. The real numbers a_{0}, a_{1}, ... , a_{n+1} satisfy a_{0} = a_{n+1} = 0 and | a_{k-1} - 2a_{k} + a_{k+1}| ≤ 1 for k = 1, 2, ... , n. Show that |a_{k}| ≤ k(n + 1 - k)/2 for all k. | |
B3. The planes p and p' are parallel. A polygon P on p has m sides and a polygon P' on p' has n sides. Find the largest and smallest distances between a vertex of P and a vertex of P'. |
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