| A1. Segments AC and BD meet at P, and |PA| = |PD|, |PB| = |PC|. O is the circumcenter of the triangle PAB. Show that OP and CD are perpendicular. | |
| A2. Find all functions f : Q+ → Q+, where Q+ is the positive rationals, such that f(x+1) = f(x) + 1 and f(x3) = f(x)3 for all x. | |
| A3. Show that for real numbers x1, x2, ... , xn we have ∑i=1m (∑j=1n xixj/(i+j) ) ≥ 0. When do we have equality? | |
| B1. The functions f0, f1, f2, ... are defined on the reals by f0(x) = 8 for all x, fn+1(x) = √(x2 + 6fn(x)). For all n solve the equation fn(x) = 2x. | |
| B2. The base of a regular pyramid is a regular 2n-gon A1A2...A2n. A sphere passes through the apex S of the pyramid and cuts the edge SAi at Bi (for i = 1, 2, ... , 2n). Show that ∑ SB2i-1 = ∑ SB2i. | |
| B3. Show that k3! is divisible by (k!)k2+k+1. |
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
20 March 2004
Last corrected/updated 25 Mar 04