| A1. Given R, r > 0. Two circles are drawn radius R, r which meet in two points. The line joining the two points is a distance D from the center of one circle and a distance d from the center of the other. What is the smallest possible value for D+d? | |
| A2. A is a set of n non-negative integers. We say it has property P if the set {x + y: x, y ∈ A} has n(n+1)/2 elements. We call the largest element of A minus the smallest element, the diameter of A. Let f(n) be the smallest diameter of any set A with property P. Show that n2/4 ≤ f(n) < n3. | |
| A3. Let R be the reals, show that there are no functions f, g: R → R such that g(f(x)) = x3 and f(g(x)) = x2 for all x. Let S be the set of all real numbers > 1. Show that there are functions f, g : S → S satsfying the condition above. | |
| B1. Let Fn be the Fibonacci sequence F1 = F2 = 1, Fn+2 = Fn+1 + Fn. Put Vn = √(Fn2 + Fn+22). Show that Vn, Vn+1, Vn+2 are the sides of a triangle of area ½. | |
| B2. c is a rational. Define f0(x) = x, fn+1(x) = f(fn(x)). Show that there are only finitely many x such that the sequence f0(x), f1(x), f2(x), ... takes only finitely many values. | |
| B3. f is a map on the plane such that two points a distance 1 apart are always taken to two points a distance 1 apart. Show that for any d, f takes two points a distance d apart to two points a distance d apart. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
Brasil home
© John Scholes
jscholes@kalva.demon.co.uk
12 Oct 2003
Last corrected/updated 12 Oct 03