Virginia Tech Regional Math Contest (1993)

Problem 1

Prove that $\displaystyle{\int_{0}^{1} \int_{x^2}^{1} e^{y^{3/2}} dy dx = \frac{2e - 2}{3}}$.

Problem 2

Prove that if $f : \mathbb{R} \to \mathbb{R}$ is continuous and $f(x) = \displaystyle{\int_{0}^{x} f(t) dt}$, then $f(x)$ is identically zero.

Problem 3

Let $f_1(x) = x$ and $f_{n+1}(x) = x^{f_n(x)}$, for $n = 1, 2 \dots$. Prove that $f_n'(1) = 1$ and $f_n''(1) = 2$, for all $n \ge 2$.

Problem 4

Prove that a triangle in the plane whose vertices have integer coordinates cannot be equilateral.

Problem 5

Find $\displaystyle{\sum_{n=1}^{\infty} \frac{3^{-n}}{n}}$.

Problem 6

Let $f : \mathbb{R}^2 \to \mathbb{R}^2$ be a surjective map with the property that if the points $A$, $B$ and $C$ are collinear, then so are $f(A)$, $f(B)$ and $f(C)$. Prove that $f$ is bijective.

Problem 7

On a small square billiard table with sides of length $2$ ft., a ball is played from the center and after rebounding off the sides several times, goes into a cup at one of the corners. Prove that the total distance travelled by the ball is not an integer number of feet.

Problem 8

A popular Virginia Tech logo looks something like

Suppose that wire-frame copies of this logo are constructed of 5 equal pieces of wire welded at three places as shown:

If bending is allowed, but no re-welding, show clearly how to cut the maximum possible number of ready-made copies of such a logo from the piece of welded wire mesh shown. Also, prove that no larger number is possible.