- Evaluate
∫
_{0}^{π/2}(cos^{4}*x*+ sin*x*cos^{3}*x*+ sin^{2}*x*cos^{2}*x*+ sin^{3}*x*cos*x*)/(sin^{4}*x*+ cos^{4}*x*+ 2 sin*x*cos^{3}*x*+ 2 sin^{2}*x*cos^{2}*x*+ 2 sin^{3}*x*cos*x*)*dx*. - Solve in real numbers the equation
3
*x*-*x*^{3}= √(*x*+ 2). - Find nonzero complex numbers
*a*,*b*,*c*,*d*,*e*such that*a*+*b*+*c*+*d*+*e*= -1 *a*^{2}+*b*^{2}+*c*^{2}+*d*^{2}+*e*^{2}= 15 1/ *a*+ 1/*b*+ 1/*c*+ 1/*d*+ 1/*e*= -1 1/ *a*^{2}+1/*b*^{2}+1/*c*^{2}+1/*d*^{2}+1/*e*^{2}= 15 *abcde*= -1

- Define
*f*(*n*) for*n*a positive integer by*f*(1) = 3 and*f*(*n*+ 1) = 3^{f(n)}. What are the last two digits of*f*(2012)? - Determine whether the series
∑
_{n=2}^{∞}1/ln*n*- (1/ln*n*)^{(n+1)/n}is convergent. - Define a sequence (
*a*_{n}) for*n*a positive integer inductively by*a*_{1}= 1 and*a*_{n}=*n*/(∏_{d | n}*a*_{d}) | 1≤*d*<*n*. Thus*a*_{2}= 2,*a*_{3}= 3,*a*_{4}= 2 etc. Find*a*_{999000}. - Let
*A*_{1},*A*_{2},*A*_{3}be three pairwise unequal 2×2 matrices with entries in ℂ (the complex numbers). Let tr denote the trace of a matrix (sotr( ( *a**b**c**d*) ) = *a*+*d*)Suppose {

*A*_{1},*A*_{2},*A*_{3}} is closed under matrix multiplication (i.e. given*i*,*j*, there exists*k*such that*A*_{i}*A*_{j}=*A*_{k}), and tr(*A*_{1}+*A*_{2}+*A*_{3})≠3. Prove that there exists*i*such that*A*_{i}*A*_{j}=*A*_{j}*A*_{i}for all*j*(here*i*,*j*are 1, 2 or 3).

Peter Linnell 2012-10-28