- Find
∑
_{n=2}^{n=∞}(*n*^{2}-2*n*- 4)/(*n*^{4}+4*n*^{2}+ 16). - Evaluate
∫
_{0}^{2}(16 -*x*^{2})*x*/(16 -*x*^{2}+ √((4-*x*)(4+*x*)(12+*x*^{2})))*dx*. - Find the least positive integer
*n*such that 2^{2014}divides 19^{n}- 1. - Suppose we are given a
19×19 chessboard (a table with
19
^{2}squares) and remove the central square. Is it possible to tile the remaining 19^{2}- 1 = 360 squares with 4×1 and 1×4 rectangles? (So each of the 360 squares is covered by exactly one rectangle.) Justify your answer. - Let
*n*≥1 and*r*≥2 be positive integers. Prove that there is no integer*m*such that*n*(*n*+ 1)(*n*+ 2) =*m*^{r}. - Let
*S*denote the set of 2 by 2 matrices with integer entries and determinant 1, and let*T*denote those matrices of*S*which are congruent to the identity matrix*I*mod 3 (so( a b c d ) ∈ *T*means that

*a*,*b*,*c*,*d*∈ℤ,*ad*-*bc*= 1, and 3 divides*b*,*c*,*a*- 1,*d*- 1; `` ∈" means ``is in").- (a)
- Let
*f*:*T*→ℝ (the real numbers) be a function such that for every*X*,*Y*∈*T*with*Y*≠*I*, either*f*(*XY*) >*f*(*X*) or*f*(*XY*^{-1}) >*f*(*X*) (or both). Show that given two finite nonempty subsets*A*,*B*of*T*, there are matrices*a*∈*A*and*b*∈*B*such that if*a'*∈*A*,*b'*∈*B*and*a'b'*=*ab*, then*a'*=*a*and*b'*=*b*. - (b)
- Show that there is no
*f*:*S*→ℝ such that for every*X*,*Y*∈*S*with*Y*≠±*I*, either*f*(*XY*) >*f*(*X*) or*f*(*XY*^{-1}) >*f*(*X*).

- Let
*A*,*B*be two points in the plane with integer coordinates*A*= (*x*_{1},*y*_{1}) and*B*= (*x*_{2},*y*_{2}). (Thus*x*_{i},*y*_{i}∈ℤ, for*i*= 1, 2.) A path π :*A*→*B*is a sequence of**down**and**right**steps, where each step has an integer length, and the initial step starts from*A*, the last step ending at*B*. In the figure below, we indicated a path from*A*_{1}= (4, 9) to*B*_{1}= (10, 3). The distance*d*(*A*,*B*) between*A*and*B*is the number of such paths. For example, the distance between*A*= (0, 2) and*B*= (2, 0) equals 6. Consider now two pairs of points in the plane*A*_{i}= (*x*_{i},*y*_{i}) and*B*_{i}= (*u*_{i},*z*_{i}) for*i*= 1, 2, with integer coordinates, and in the configuration shown in the picture (but with arbitrary coordinates):*x*_{2}<*x*_{1}and*y*_{1}>*y*_{2}, which means that*A*_{1}is North-East of*A*_{2};*u*_{2}<*u*_{1}and*z*_{1}>*z*_{2}, which means that*B*_{1}is North-East of*B*_{2}.- Each of the points
*A*_{i}is North-West of the points*B*_{j}, for 1≤*i*,*j*≤2. In terms of inequalities, this means that*x*_{i}< min{*u*_{1},*u*_{2}} and*y*_{i}> max{*z*_{1},*z*_{2}} for*i*= 1, 2.

- (a)
- Find the distance between two points
*A*and*B*as before, as a function of the coordinates of*A*and*B*. Assume that*A*is North-West of*B*. - (b)
- Consider the
2×2 matrix
*M*=( *d*(*A*_{1},*B*_{1})*d*(*A*_{1},*B*_{2})*d*(*A*_{2},*B*_{1})*d*(*A*_{2},*B*_{2})) Prove that for any configuration of points

*A*_{1},*A*_{2},*B*_{1},*B*_{2}as described before, det*M*> 0.

Peter Linnell 2014-10-24