- Evaluate
∫
_{1}^{2}(ln*x*)/(2 - 2*x*+*x*^{2})*dx*. - Determine the real numbers
*k*such that ∑_{n=1}^{∞}((2*n*)!/(4^{n}*n*!*n*!))^{k}is convergent. - Let
*n*be a positive integer and let M_{n}(ℤ_{2}) denote the*n*by*n*matrices with entries from the integers mod 2. If*n*≥2, prove that the number of matrices*A*in M_{n}(ℤ_{2}) satisfying*A*^{2}= 0 (the matrix with all entries zero) is an even positive integer. - For a positive integer
*a*, let*P*(*a*) denote the largest prime divisor of*a*^{2}+ 1. Prove that there exist infinitely many triples (*a*,*b*,*c*) of distinct positive integers such that*P*(*a*) =*P*(*b*) =*P*(*c*). - Suppose that
*m*,*n*,*r*are positive integers such that1 +Prove that*m*+*n*√3 = (2 + √3)^{2r-1}.*m*is a perfect square. - Let
*A*,*B*,*P*,*Q*,*X*,*Y*be square matrices of the same size. Suppose that*A*+*B*+*AB*= *XY**AX*=*XQ**P*+*Q*+*PQ*= *YX**PY*=*YB*.

Prove that*AB*=*BA*. - Let
*q*be a real number with |*q*|≠1 and let*k*be a positive integer. Define a Laurent polynomial*f*_{k}(*X*) in the variable*X*, depending on*q*and*k*, by*f*_{k}(*X*) = ∏_{i=0}^{k-1}(1 -*q*^{i}*X*)(1 -*q*^{i+1}*X*^{-1}). (Here ∏ denotes product.) Show that the constant term of*f*_{k}(*X*), i.e. the coefficient of*X*^{0}in*f*_{k}(*X*), is equal to(1 -*q*^{k+1})(1 -*q*^{k+2})...(1 -*q*^{2k})/((1 -*q*)(1 -*q*^{2})...(1 -*q*^{k})).

Peter Linnell 2016-10-22