Virginia Tech Regional Math Contest (1996)

Problem 1

Evaluate $\displaystyle{\int_{0}^{1} \int_{\sqrt{y-y^2}}^{\sqrt{1-y^2}} x e^{(x^2 + y^2)^2} dx dy}$.

Problem 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 $r_1$ and $r_2$ such that $0 < r_1 < r_2 < 1$, $$P\left( \frac{r_1 f(r_1) + r_2 f(r_2)}{f(r_1) + f(r_2)} \right)$$ will be the point of intersection of the line joining $(r_1, 0)$ and $P(r_2)$ with the line joining $P(r_1)$ and $(r_2, 0)$.

Problem 3

Solve the differential equation $y^2 = e^{dy/dx}$ with the initial condition $y = e$ when $x = 1$.

Problem 4

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

Problem 5

Let $a_i$, $i = 1, 2, 3, 4$, be real numbers such that $a_1 + a_2 + a_3 + a_4 = 0$. Show that for arbitrary real numbers $b_i$, $i = 1, 2, 3$, the equation $$a_1 + b_1 x + 3a_2 x^2 + b_2 x^3 + 5a_3 x^4 + b_3 x^5 + 7a_4 x^6 = 0$$ has at least one real root which is on the interval $-1 \le x \le 1$.

Problem 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.

Problem 7

Let us define

$$\begin{align*} f_{n,0}(x) & = x + \dfrac{\sqrt{x}}{n} & \hspace{1cm} & \text{for } x > 0, n \ge 1, \newline f_{n, j+1}(x) & = f_{n,0}(f_{n,j}(x)), & \hspace{1cm} & j = 0, 1, \dots, n-1. \end{align*}$$

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