- Let
*I*= 3√2∫_{0}^{x}√(1 + cos*t*)/(17 - 8 cos*t*)*dt*. If 0 <*x*< π and tan*I*= 2/√3, what is*x*? - Let
*ABC*be a right-angled triangle with ∠*ABC*= 90^{o}, and let*D*on*AB*such that*AD*= 2*DB*. What is the maximum possible value of ∠*ACD*? - Define a sequence (
*a*_{n}) for*n*≥1 by*a*_{1}= 2 and*a*_{n+1}=*a*_{n}^{1+n-3/2}. Is (*a*_{n}) convergent (i.e. lim_{n→∞}*a*_{n}< ∞)? - A positive integer
*n*is called*special*if it can be represented in the form*n*= (*x*^{2}+*y*^{2})/(*u*^{2}+*v*^{2}), for some positive integers*x*,*y*,*u*,*v*. Prove that- (a)
- 25 is special;
- (b)
- 2013 is not special;
- (c)
- 2014 is not special.

- Prove that
*x*/√(1 +*x*^{2}) +*y*/√(1 +*y*^{2}) +*z*/√(1 +*z*^{2})≤(3√3)/2 for any positive real numbers*x*,*y*,*z*such that*x*+*y*+*z*=*xyz*. - Let
*X*=( 7 8 9 8 -9 -7 -7 -7 9 ) *Y*=( 9 8 -9 8 -7 7 7 9 8 ) let

*A*=*Y*^{-1}-*X*and let*B*be the inverse of*X*^{-1}+*A*^{-1}. Find a matrix*M*such that*M*^{2}=*XY*-*BY*(you may assume that*A*and*X*^{-1}+*A*^{-1}are invertible). - Find
∑
_{n=1}^{∞}*n*/(2^{n}+2^{-n})^{2}+ (- 1)^{n}*n*/(2^{n}-2^{-n})^{2}.

Peter Linnell 2013-11-06