- Find the maximum value of
*xy*^{3}+*yz*^{3}+*zx*^{3}-*x*^{3}*y*-*y*^{3}*z*-*z*^{3}*x*where 0≤*x*≤1, 0≤*y*≤1, 0≤*z*≤1. - How many sequences of 1's and 3's sum to 16? (Examples of such
sequences are
{1, 3, 3, 3, 3, 3} and
{1, 3, 1, 3, 1, 3, 1, 3}.)
- Find the area of the region of points (
*x*,*y*) in the*xy*-plane such that*x*^{4}+*y*^{4}≤*x*^{2}-*x*^{2}*y*^{2}+*y*^{2}. - Let
*ABC*be a triangle, let*M*be the midpoint of*BC*, and let*X*be a point on*AM*. Let*BX*meet*AC*at*N*, and let*CX*meet*AB*at*P*. If ∠*MAC*= ∠*BCP*, prove that ∠*BNC*= ∠*CPA*.

- Let
*a*_{1},*a*_{2},... be a sequence of nonnegative real numbers and let π,ρ be permutations of the positive integers**N**(thus π,ρ :**N**-->**N**are one-to-one and onto maps). Suppose that ∑_{n=1}^{∞}*a*_{n}= 1 and ε is a real number such that ∑_{n=1}^{∞}|*a*_{n}-*a*_{πn}| + ∑_{n=1}^{∞}|*a*_{n}-*a*_{ρn}| < ε. Prove that there exists a finite subset*X*of**N**such that |*X*∩π*X*|,|*X*∩ρ*X*| > (1 - ε)|*X*| (here |*X*| indicates the number of elements in*X*; also the inequalities < , > are strict). - Find all pairs of positive (nonzero) integers
*a*,*b*such that*ab*- 1 divides*a*^{4}-3*a*^{2}+ 1. - Let
*f*_{1}(*x*) =*x*and*f*_{n+1}(*x*) =*x*^{fn(x)}for*n*a positive integer. Thus*f*_{2}(*x*) =*x*^{x}and*f*_{3}(*x*) =*x*^{(xx)}. Now define*g*(*x*) = lim_{n--> ∞}1/*f*_{n}(*x*) for*x*> 1. Is*g*continuous on the open interval (1,∞)? Justify your answer.

Peter Linnell 2008-11-06