Virginia Tech Regional Math Contest (1991)

Problem 1

An isosceles triangle with an inscribed circle is labeled as shown in the figure. Find an expression, in terms of the angle $\alpha$ and the length $a$, for the area of the curvilinear triangle bounded by sides $AB$ and $AC$ and the arc $BC$.

Problem 2

Find all differentiable functions $f$ which satisfy $f(x)^3 = \displaystyle{\int_{0}^{x} f(t)^2 dt}$ for all real $x$.

Problem 3

Prove that if $\alpha$ is a real root of $(1 - x^2)(1 + x + x^2 + \dots + x^n) - x = 0$ which lies in $(0, 1)$, with $n = 1, 2, \dots$, then $\alpha$ is also a root of $(1 - x^2)(1 + x + x^2 + \dots + x^{n+1}) - 1 = 0$.

Problem 4

Prove that if $x > 0$ and $n > 0$, where $x$ is real and $n$ is an integer, then $$\frac{x^n}{(x+1)^{n+1}} \le \frac{n^n}{(n+1)^{n+1}}.$$

Problem 5

Let $f(x) = x^5 - 5x^3 + 4x$. In each part (i)–(iv), prove or disprove that there exists a real number $c$ for which $f(x) - c = 0$ has a root of multiplicity (i) one, (ii) two, (iii) three, (iv) four.

Problem 6

Let $a_0 = 1$ and for $n > 0$, let $a_n$ be defined by $$a_n = - \sum_{k=1}^{n} \frac{a_{n-k}}{k!}.$$ Prove that $a_n = (-1)^n / n!$, for $n = 0, 1, 2, \dots$.

Problem 7

$A$ and $B$ play the following money game, where $a_n$ and $b_n$ denote the amount of holdings of $A$ and $B$, respectively, after the $n$th round. At each round a player pays one-half his holdings to the bank, then receives one dollar from the bank if the other player had less than $c$ dollars at the end of the previous round. If $a_0 = .5$ and $b_0 = 0$, describe the behavior of $a_n$ and $b_n$ when $n$ is large, for

Problem 8

Mathematical National Park has a collection of trails. There are designated campsites along the trails, including a campsite at each intersection of trails. The rangers call each stretch of trail between adjacent campsites a “segment”. The trails have been laid out so that it is possible to take a hike that starts at any campsite, covers each segment exactly once, and ends at the beginning campsite. Prove that it is possible to plan a collection $\mathscr{C}$ of hikes with all of the following properties: