| A1. Do there exist tetrahedra T1, T2 such that (1) vol T1 > vol T2, and (2) every face of T2 has larger area than any face of T1? | |
| A2. Let F(n) be the number of paths P0, P1, ... , Pn of length n that go from P0 = (0,0) to a lattice point Pn on the line y = 0, such that each Pi is a lattice point and for each i < n, Pi and Pi+1 are adjacent lattice points a distance 1 apart. Show that F(n) = (2n)Cn. | |
| A3. N is a number of the form ∑k=160 ak kkk, where each ak = 1 or -1. Show that N cannot be a 5th power. | |
| B1. Let V be the set of all vectors (x,y) with integral coordinates. Find all real-valued functions f on V such that (a) f(v) = 1 for all v of length 1; (b) f(v + w) = f(v) + f(w) for all perpendicular v, w ∈ V. (The vector (0,0) is considered to be perpendicular to any vector.) | |
| B2. k1, k2 are circles with different radii and centers K1, K2. Neither lies inside the other, and they do not touch or intersect. One pair of common tangents meet at A on K1K2, the other pair meet at B on K1K2. P is any point on k1. Show that there is a diameter of K2 with one endpoint on the line PA and the other on the line PB. | |
| B3. The real numbers x, y, z satisfy x2 + y2 + z2 = 2. Show that x + y + z ≤ 2 + xyz. When do we have equality? |
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