A1. a_{1}, a_{2}, ... , a_{n} are real numbers such that 1/2 ≤ a_{i} ≤ 5 for each i. The real numbers x_{1}, x_{2}, ... , x_{n} satisfy 4x_{i}^{2} - 4a_{i}x_{i} + (a_{i} - 1)^{2} = 0. Let S = (x_{1} + x_{2} + ... + x_{n})/n, S_{2} = (x_{1}^{2} + x_{2}^{2} + ... + x_{n}^{2})/n. Show that √S_{2} ≤ S + 1. | |
A2. P is a pyramid whose base is a regular 1986-gon, and whose sloping sides are all equal. Its inradius is r and its circumradius is R. Show that R/r ≥ 1 + 1/cos(π/1986). Find the total area of the pyramid's faces when equality occurs. | |
A3. The polynomial p(x) has degree n and p(1) = 2, p(2) = 4, p(3) = 8, ... , p(n+1) = 2^{n+1}. Find p(n+2). | |
B1. ABCD is a square. ABM is an equilateral triangle in the plane perpendicular to ABCD. E is the midpoint of AB. O is the midpoint of CM. The variable point X on the side AB is a distance x from B. P is the foot of the perpendicular from M to the line CX. Find the locus of P. Find the maximum and minimum values of XO. | |
B2. Find all n > 1 such that (x_{1}^{2} + x_{2}^{2} + ... + x_{n}^{2}) ≥ x_{n}(x_{1} + x_{2} + ... + x_{n-1}) for all real x_{i}. | |
B3. A sequence of positive integers is constructed as follows. The first term is 1. Then we take the next two even numbers: 2, 4. Then we take the next three odd numbers: 5, 7, 9. Then we take the next four even numbers: 10, 12, 14, 16. And so on. Find the nth term of the sequence. |
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