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 $$\int_{0}^{1}$$$$\int_{\sqrt{y-y^2}}^{\sqrt{1-y^2}}$$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 y^y = e^dy/dx with the initial condition y = e when x = 1.

4. Let f (x) be a twice continuously differentiable in the interval (0,$\infty$). If
   lim_{x - > $\infty$}(x²f''(x) + 4xf'(x) + 2f (x)) = 1,
   find lim_{x - > $\infty$}f (x) and lim_{x - > $\infty$}xf'(x). Do not assume any special form of f (x). Hint: use l'Hôpital's rule.

5. Let a_i, i = 1, 2, 3, 4, be real numbers such that a₁ + a₂ + a₃ + a₄ = 0. Show that for arbitrary real numbers b_i, 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 -1$\le$x$\le$1.

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

   $$\sqrt{x} \ge$$ f_{n, 0}(x)

   f_{n, j + 1}(x) = f_{n, 0}(f_{n, j}(x)), j = 0, 1,..., n - 1.

   
   
   Find lim_{n - > $\infty$}f_{n, n}(x) for x > 0.