Virginia Tech Regional Math Contest (2013)

Problem 1

Let $\displaystyle I = 3 \sqrt{2} \int_0^x \frac{\sqrt{1+\cos t}}{17 - 8\cos t} dt$. If $0 < x < \pi$ and $\tan I = \frac{2}{\sqrt{3}}$, what is $x$?

Problem 2

Let $ABC$ be a right-angled triangle with $\angle ABC = 90^\circ$, and let $D$ be on $AB$ such that $AD = 2DB$. What is the maximum possible value of $\angle ACD$?

Problem 3

Define a sequence $(a_n)$ for $n \ge 1$ by $a_1 = 2$ and $a_{n+1} = a_n^{1 + n^{-3/2}}$. Is $(a_n)$ convergent (i.e., $\displaystyle \lim_{n \to \infty} a_n < \infty$)?

Problem 4

A positive integer $n$ is called special if it can be represented in the form $n = \dfrac{x^2+y^2}{u^2+v^2}$, for some positive integers $x, y, u, v$. Prove that

Problem 5

Prove that $\displaystyle \frac{x}{\sqrt{1+x^2}} + \frac{y}{\sqrt{1+y^2}} + \frac{z}{\sqrt{1+z^2}} \le \frac{3\sqrt3}{2}$ for any positive real numbers $x, y, z$ such that $x + y + z = xyz$.

Problem 6

Let $X = \begin{pmatrix} 7 & 8 & 9 \newline 8 & -9 & -7 \newline -7 & -7 & 9 \end{pmatrix}$, $Y = \begin{pmatrix} 9 & 8 & -9 \newline 8 & -7 & 7 \newline 7 & 9 & 8 \end{pmatrix}$, let $A = Y^{-1} - X$ and let $B$ be the inverse of $X + A^{-1}$. Find a matrix $M$ such that $M^2 = XY - BY$ (you may assume that $A$ and $X^{-1} + A^{-1}$ are invertible).

Problem 7

Find $\displaystyle \sum_{n=1}^\infty \dfrac{n}{(2^n + 2^{-n})^2} + \dfrac{(-1)^n n}{(2^n - 2^{-n})^2}$.