Virginia Tech Regional Mathematics Contest

- Three infinitely long circular cylinders each with unit radius have
their axes along the
*x*,*y*and*z*-axes. Determine the volume of the region common to all three cylinders. (Thus one needs the volume common to {*y*^{2}+*z*^{2}__<__1}, {*z*^{2}+*x*^{2}__<__1}, {*x*^{2}+*y*^{2}__<__1}.) - Two circles with radii 1 and 2 are placed so that they are tangent to
each other and a straight line. A third circle is nestled between
them so that it is tangent to the first two circles and the line.
Find the radius of the third circle.

- For each positive integer
*n*, let*S*_{n}denote the total number of squares in an*n*X*n*square grid. Thus*S*_{1}= 1 and*S*_{2}= 5, because a 2 X 2 square grid has four 1 X 1 squares and one 2 X 2 square. Find a recurrence relation for*S*_{n}, and use it to calculate the total number of squares on a chess board (i.e. determine*S*_{8}). - Let
*a*_{n}be the*n*th positive integer*k*such that the greatest integer not exceeding divides*k*, so the first few terms of {*a*_{n}} are {1, 2, 3, 4, 6, 8, 9, 12,...}. Find*a*_{10000}and give reasons to substantiate your answer. - Determine the interval of convergence of the power series
S
_{n = 1}^{}*n*^{n}*x*^{n}/*n*!. (That is, determine the real numbers*x*for which the above power series converges; you must determine correctly whether the series is convergent at the end points of the interval.) - Find a function
*f*:**R**^{+}- >**R**^{+}such that*f*(*f*(*x*)) = (3*x*+ 1)/(*x*+ 3) for all positive real numbers*x*(here**R**^{+}denotes the positive (nonzero) real numbers). - Let
*G*denote a set of invertible 2 X 2 matrices (matrices with complex numbers as entries and determinant nonzero) with the property that if*a*,*b*are in*G*, then so are*ab*and*a*^{-1}. Suppose there exists a function*f*:*G*- >**R**with the property that either*f*(*ga*) >*f*(*a*) or*f*(*g*^{-1}*a*) >*f*(*a*) for all*a*,*g*in*G*with*g*=/=*I*(here*I*denotes the identity matrix,**R**denotes the real numbers, and the inequality signs are strict inequality). Prove that given nonempty finite subsets*A*,*B*of*G*, there is a matrix in*G*which can be written in exactly one way in the form*xy*with*x*in*A*and*y*in*B*.