| 1. n > 1, and a and b are positive real numbers such that an - a - 1 = 0 and b2n - b - 3a = 0. Which is larger? |
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| 2. The diagonals of a convex quadrilateral meet at right angles at X. Show that the four points obtained by reflecting X in each of the sides are cyclic. |
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| 3. Let S be the set of functions f defined on reals in the closed interval [0, 1] with non-negative real values such that f(1) = 1 and f(x) + f(y) ≤ f(x + y) for all x, y such that x + y ≤ 1. What is the smallest k such that f(x) ≤ kx for all f in S and all x? |
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| 4. The sequence an of odd positive integers is defined as follows: a1 = r, a2 = s, and an is the greatest odd divisor of an-1 + an-2. Show that, for sufficiently large n, an is constant and find this constant (in terms of r and s). |
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| 5. A sequence xn of positive reals satisfies xn-1xn+1 ≤ xn2. Let an be the average of the terms x0, x1, ... , xn and bn be the average of the terms x1, x2, ... , xn. Show that anbn-1 ≥ an-1bn. |
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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 Aug 2002