| A1. X is a set with n > 2 elements. Is there a function f : X → X such that the composition f n-1 is constant, but f n-2 is not constant? | |
| A2. Given n we define ai,j as follows. For i, j = 1, 2, ... , n, ai,j = 1 for j = i, and 0 for j ≠ i. For i = 1, 2, ... , n, j = n+1, ... , 2n, ai,j = -1/n. Show that for any permutation p of (1, 2, ... , 2n) we have ∑i=1n |∑k=1n ai,p(k) | ≥ n/2. | |
| A3. W is a regular octahedron with center O. P is a plane through the center O. K(O, r1) and K(O, r2) are circles center O and radii r1, r2 such that K(O, r1) ⊆ P∩W ⊆ K(O, r2). Show that r1/r2 ≤ (√3)/2. | |
| B1. We throw a coin n times and record the results as the sequence α1, α2, ... , αn, using 1 for head, 2 for tail. Let βj = α1 + α2 + ... + αj and let p(n) be the probability that the sequence β1, β2, ... , βn includes the value n. Find p(n) in terms of p(n-1) and p(n-2). | |
| B2. Six disks with diameter 1 are placed so that they cover the edges of a regular hexagon with side 1. Show that no vertex of the hexagon is covered by two or more disks. | |
| B3. There are 1025 cities, P1, ... , P1025 and ten airlines A1, ... , A10, which connect some of the cities. Given any two cities there is at least one airline which has a direct flight between them. Show that there is an airline which can offer a round trip with an odd number of flights. |
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 24 Mar 04