- Find, and give a proof of your answer, all positive integers
*n*such that neither*n*nor*n*^{2}contain a 1 when written in base 3. - Let
*S*(*n*) denote the number of sequences of length*n*formed by the three letters A,B,C with the restriction that the C's (if any) all occur in a single block immediately following the first B (if any). For example ABCCAA, AAABAA, and ABCCCC are counted in, but ACACCB and CAAAAA are not. Derive a simple formula for*S*(*n*) and use it to calculate*S*(10). - Recall that the Fibonacci numbers
*F*(*n*) are defined by*F*(0) = 0,*F*(1) = 1, and*F*(*n*) =*F*(*n*- 1) +*F*(*n*- 2) for*n*≥2. Determine the last digit of*F*(2006) (e.g. the last digit of 2006 is 6). - We want to find functions
*p*(*t*),*q*(*t*),*f*(*t*) such that- (a)
*p*and*q*are continuous functions on the open interval (0,π).- (b)
*f*is an infinitely differentiable nonzero function on the whole real line (- ∞,∞) such that*f*(0) =*f'*(0) =*f''*(0).- (c)
*y*= sin*t*and*y*=*f*(*t*) are solutions of the differential equation*y''*+*p*(*t*)*y'*+*q*(*t*)*y*= 0 on (0,π).

*f*,*p*,*q*. - Let {
*a*_{n}} be a monotonic decreasing sequence of positive real numbers with limit 0 (so*a*_{1}≥*a*_{2}≥...≥ 0). Let {*b*_{n}} be a rearrangement of the sequence such that for every non-negative integer*m*, the terms*b*_{3m+1},*b*_{3m+2},*b*_{3m+3}are a rearrangement of the terms*a*_{3m+1},*a*_{3m+2},*a*_{3m+3}(thus, for example, the first 6 terms of the sequence {*b*_{n}} could be*a*_{3},*a*_{2},*a*_{1},*a*_{4},*a*_{6},*a*_{5}). Prove or give a counterexample to the following statement: the series ∑_{n=1}^{∞}(- 1)^{n}*b*_{n}is convergent. - In the diagram below
*BP*bisects ∠*ABC*,*CP*bisects ∠*BCA*, and*PQ*is perpendicular to*BC*. If*BQ*.*QC*= 2*PQ*^{2}, prove that*AB*+*AC*= 3*BC*.

- Three spheres each of unit radius have centers
*P*,*Q*,*R*with the property that the center of each sphere lies on the surface of the other two spheres. Let*C*denote the cylinder with cross-section*PQR*(the triangular lamina with vertices*P*,*Q*,*R*) and axis perpendicular to*PQR*. Let*M*denote the space which is common to the three spheres and the cylinder*C*, and suppose the mass density of*M*at a given point is the distance of the point from*PQR*. Determine the mass of*M*.

Peter Linnell 2006-10-29