- Evaluate
∫
_{1}^{4}(*x*- 2)/((*x*^{2}+4)√*x*)*dx*. - A sequence (
*a*_{n}) is defined by*a*_{0}= -1,*a*_{1}= 0, and*a*_{n+1}=*a*_{n}^{2}- (*n*+ 1)^{2}*a*_{n-1}- 1*n*. Find*a*_{100}. - Find
∑
_{k=1}^{∞}(*k*^{2}- 2)/((*k*+ 2)!). - Let
*m*,*n*be positive integers and let [*a*] denote the residue class mod*mn*of the integer*a*(thus {[*r*] |*r*is an integer} has exactly*mn*elements). Suppose the set {[*ar*] |*r*is an integer} has exactly*m*elements. Prove that there is a positive integer*q*such that*q*is prime to*mn*and [*nq*] = [*a*]. - Find
lim
_{x-> ∞}(2*x*)^{1+1/(2x)}-*x*^{1+1/x}-*x*. - Let
*S*be a set with an asymmetric relation <; this means that if*a*,*b*∈*S*and*a*<*b*, then we do not have*b*<*a*. Prove that there exists a set*T*containing*S*with an asymmetric relation ≺ with the property that if*a*,*b*∈*S*, then*a*<*b*if and only if*a*≺*b*, and if*x*,*y*∈*T*with*x*≺*y*, then there exists*t*∈*T*such that*x*≺*t*≺*y*(*t*∈*T*means ``*t*is an element of*T*"). - Let
*P*(*x*) =*x*^{100}+20*x*^{99}+198*x*^{98}+*a*_{97}*x*^{97}+ ... +*a*_{1}*x*+ 1 be a polynomial where the*a*_{i}( 1≤*i*≤97) are real numbers. Prove that the equation*P*(*x*) = 0 has at least one complex root (i.e. a root of the form*a*+*bi*with*a*,*b*real numbers and*b*≠ 0).

Peter Linnell 2011-10-31