18th Annual
Virginia Tech Regional Mathematics Contest
From 9:00a.m. to 11:30a.m., October 26, 1996

Fill out the individual registration form

1. Evaluate xe(x4 + 2x2y2 + y4) dxdy.

2. For each rational number r, define f (r) to be the smallest positive integer n such that r = m/n for some integer m, and denote by P(r) the point in the (x, y) plane with coordinates P(r) = (r, 1/f (r)). Find a necessary and sufficient condition that, given two rational numbers r1 and r2 such that 0 < r1 < r2 < 1,

P((r1f (r1) + r2f (r2))/(f (r1) + f (r2)))

will be the point of intersection of the line joining (r1, 0) and P(r2) with the line joining P(r1) and (r2, 0).

3. Solve the differential equation yy = edy/dx with the initial condition y = e when x = 1.

4. Let f (x) be a twice continuously differentiable in the interval (0,). If

limx - > (x2f''(x) + 4xf'(x) + 2f (x)) = 1,

find limx - > f (x) and limx - > xf'(x). Do not assume any special form of f (x). Hint: use l'Hôpital's rule.

5. Let ai, i = 1, 2, 3, 4, be real numbers such that a1 + a2 + a3 + a4 = 0. Show that for arbitrary real numbers bi, i = 1, 2, 3, the equation

a1 + b1x + 3a2x2 + b2x3 + 5a3x4 + b3x5 + 7a4x6 = 0

has at least one real root which is on the interval -1x1.

6. There are 2n balls in the plane such that no three balls are on the same line and such that no two balls touch each other. n balls are red and the other n balls are green. Show that there is at least one way to draw n line segments by connecting each ball to a unique different colored ball so that no two line segments intersect.

7. Let us define

 fn, 0(x) = x + ()/n forx > 0, n1, fn, j + 1(x) = fn, 0(fn, j(x)), j = 0, 1,..., n - 1.

Find limn - > fn, n(x) for x > 0.

Peter Linnell
2000-09-08