Virginia Tech Regional Mathematics Contest

- A path zig-zags from (1, 0) to (0, 0) along line segments
, where
*P*_{0}is (1, 0) and*P*_{n}is (2^{-n},(- 2)^{-n}), for*n*> 0. Find the length of the path. - A triangle with sides of lengths
*a*,*b*, and*c*is partitioned into two smaller triangles by the line which is perpendicular to the side of length*c*and passes through the vertex opposite that side. Find*integers**a*<*b*<*c*such that each of the two smaller triangles is similar to the original triangle and has sides of integer lengths. - Let
*a*_{1},*a*_{2},...,*a*_{n}be an arbitrary rearrangement of 1, 2,...,*n*. Prove that if*n*is odd, then (*a*_{1}-1)(*a*_{2}-2)...(*a*_{n}-*n*) is even. - Let
*p*(*x*) be given by*p*(*x*) =*a*_{0}+*a*_{1}*x*+*a*_{2}*x*^{2}+ ... +*a*_{n}*x*^{n}and let |*p*(*x*)|≤|*x*| on [- 1, 1].(a) Evaluate

*a*_{0}. (b) Prove that |*a*_{1}|≤1. - A sequence of integers
{
*n*_{1},*n*_{2},...} is defined as follows:*n*_{1}is assigned arbitrarily and, for*k*> 1,*n*_{k}= ∑_{j=1}^{j=k-1}*z*(*n*_{j}),*z*(*n*) is the number of 0's in the binary representation of*n*(each representation should have a leading digit of 1 except for zero which has the representation 0). An example, with*n*_{1}= 9, is {9, 2, 3, 3, 3,...}, or in binary, {1001, 10, 11, 11, 11,...}.- (a)
- Find
*n*_{1}so hat lim_{k-> ∞}*n*_{k}= 31, and calculate*n*_{2},*n*_{3},...,*n*_{10}. - (b)
- Prove that, for every choice of
*n*_{1}, the sequence {*n*_{k}} converges.

- A sequence of polynomials is given by
*p*_{n}(*x*) =*a*_{n+2}*x*^{2}+*a*_{n+1}*x*-*a*_{n}, for*n*≥ 0, where*a*_{0}=*a*_{1}= 1 and, for*n*≥ 0,*a*_{n+2}=*a*_{n+1}+*a*_{n}. Denote by*r*_{n}and*s*_{n}the roots of*p*_{n}(*x*) = 0, with*r*_{n}≤*s*_{n}. Find lim_{n-> ∞}*r*_{n}and lim_{n-> ∞}*s*_{n}. - Let
*A*= {*a*_{ij}} and*B*= {*b*_{ij}} be*n*×*n*matrices such that*A*^{-1}exists. Define*A*(*t*) = {*a*_{ij}(*t*)} and*B*(*t*) = {*b*_{ij}(*t*)} by*a*_{ij}(*t*) =*a*_{ij}for*i*<*n*,*a*_{nj}(*t*) =*ta*_{nj},*b*_{ij}(*t*) =*b*_{ij}for*i*<*n*, and*b*_{nj}(*t*) =*tb*_{nj}. For example, if*A*=[ 1 2 3 4 ] then

*A*(*t*) =[ 1 2 3 *t*4 *t*] Prove that

*A*(*t*)^{-1}*B*(*t*) =*A*^{-1}*B*for*t*> 0 and any*n*. (Partial credit will be given for verifying the result for*n*= 3.) - On Halloween, a black cat and a witch encounter each other near a
large mirror positioned along the
*y*-axis. The witch is*invisible except by reflection*in the mirror. At*t*= 0, the cat is at (10, 10) and the witch is at (10, 0). For*t*≥ 0, the witch moves toward the cat at a speed numerically equal to their distance of separation and the cat moves toward the apparent position of the witch, as seen by reflection, at a speed numerically equal to their reflected distance of separation. Denote by (*u*(*t*),*v*(*t*)) the position of the cat and by (*x*(*t*),*y*(*t*)) the position of the witch.- (a)
- Set up the equations of motion of the cat and the witch for
*t*≥ 0. - (b)
- Solve for
*x*(*t*) and*u*(*t*) and find the time when the cat strikes the mirror. (Recall that the mirror is a perpendicular bisector of the line joining an object with its apparent position as seen by reflection.)

Peter Linnell 2011-06-27