23rd Annual
Virginia Tech Regional Mathematics Contest
From 8:30a.m. to 11:00a.m., November 3, 2001

Fill out the individual registration form

  1. 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 {y2 + z2 < 1}, {z2 + x2 < 1}, {x2 + y2 < 1}.)

  2. 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.


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  3. For each positive integer n, let Sn denote the total number of squares in an n X n square grid. Thus S1 = 1 and S2 = 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 Sn, and use it to calculate the total number of squares on a chess board (i.e. determine S8).

  4. Let an be the nth positive integer k such that the greatest integer not exceeding $ \sqrt{k}$ divides k, so the first few terms of {an} are {1, 2, 3, 4, 6, 8, 9, 12,...}. Find a10000 and give reasons to substantiate your answer.

  5. Determine the interval of convergence of the power series Sn = 1$\scriptstyle \infty$nnxn/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.)

  6. Find a function f : R+ - > R+ such that f (f (x)) = (3x + 1)/(x + 3) for all positive real numbers x (here R+ denotes the positive (nonzero) real numbers).

  7. 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-1a) > 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.





Peter Linnell
2001-11-04