- Let
*d*be a positive integer and let*A*be a*d*×*d*matrix with integer entries. Suppose*I*+*A*+*A*^{2}+ ... +*A*^{100}= 0 (where*I*denotes the identity*d*×*d*matrix, so*I*has 1's on the main diagonal, and 0 denotes the zero matrix, which has all entries 0). Determine the positive integers*n*≤100 for which*A*^{n}+*A*^{n+1}+ ... +*A*^{100}has determinant ±1. - For
*n*a positive integer, define*f*_{1}(*n*) =*n*and then for*i*a positive integer, define*f*_{i+1}(*n*) =*f*_{i}(*n*)^{fi(n)}. Determine*f*_{100}(75)mod 17 (i.e. determine the remainder after dividing*f*_{100}(75) by 17, an integer between 0 and 16). Justify your answer. - Prove that
cos(π/7) is a root of the equation
8
*x*^{3}-4*x*^{2}- 4*x*+ 1 = 0, and find the other two roots. - Let
Δ
*ABC*be a triangle with sides*a*,*b*,*c*and corresponding angles*A*,*B*,*C*(so*a*=*BC*and*A*= ∠*BAC*etc.). Suppose that 4*A*+ 3*C*= 540^{o}. Prove that (*a*-*b*)^{2}(*a*+*b*) =*bc*^{2}. - Let
*A*,*B*be two circles in the plane with*B*inside*A*. Assume that*A*has radius 3,*B*has radius 1,*P*is a point on*A*,*Q*is a point on*B*, and*A*and*B*touch so that*P*and*Q*are the same point. Suppose that*A*is kept fixed and*B*is rolled once round the inside of*A*so that*Q*traces out a curve starting and finishing at*P*. What is the area enclosed by this curve?

- Define a sequence by
*a*_{1}= 1,*a*_{2}= 1/2, and*a*_{n+2}=*a*_{n+1}-*a*_{n}*a*_{n+1}/2 for*n*a positive integer. Find lim_{n-> ∞}*na*_{n}. - Let
∑
_{n=1}^{∞}*a*_{n}be a convergent series of positive terms (so*a*_{i}> 0 for all*i*) and set*b*_{n}= 1/(*na*_{n}^{2}) for*n*≥1. Prove that ∑_{n=1}^{∞}*n*/(*b*_{1}+*b*_{2}+ ... +*b*_{n}) is convergent.

Peter Linnell 2010-10-31