| A1. The real numbers x1, x2, ... , xn belong to the interval (0,1) and satisfy x1 + x2 + ... + xn = m + r, where m is an integer and r ∈ [0,1). Show that x12 + x22 + ... + xn2 ≤ m + r2. | |
| A2. For a permutation P = (p1, p2, ... , pn) of (1, 2, ... , n) define X(P) as the number of j such that pi < pj for every i < j. What is the expected value of X(P) if each permutation is equally likely? | |
| A3. W is a polygon. W has a center of symmetry S such that if P belongs to W, then so does P', where S is the midpoint of PP'. Show that there is a parallelogram V containing W such that the midpoint of each side of V lies on the border of W. | |
| B1. d is a positive integer and f : [0,d] → R is a continuous function with f(0) = f(d). Show that there exists x ∈ [0,d-1] such that f(x) = f(x+1). | |
| B2. The sequence a1, a2, a3, ... is defined by a1 = a2 = a3 = 1, an+3 = an+2an+1 + an. Show that for any positive integer r we can find s such that as is a multiple of r. | |
| B3. Find the largest possible volume for a tetrahedron which lies inside a hemisphere of radius 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