VTRMC 2013

Let $I=3\sqrt2\int^x_0\frac{\sqrt{1+\cos t}}{17-8\cos t}dt$ . If $0<x<\pi$ and $\tan I=\frac2{\sqrt3}$ , what is $x$ ?

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 $\angle ACD$ ?

Define a sequence $(a_n)$ for $n\ge1$ by $a_1=2$ and $a_{n+1}=a_n^{1+n^{-3/2}}$ . Is $(a_n)$ convergent (i.e. $\lim_{n\to\infty}a_n<\infty$ )?

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

(a) $25$ is special; (b) $2014$ is not special; (c) $2015$ is not special.

Prove that $$\frac{x}{\sqrt{1+x^2}}+\frac{y}{\sqrt{1+y^2}}+\frac{z}{\sqrt{1+z^2}} \leq\frac{3\sqrt{3}}{2}$$ for any positive real numbers $x, y,z$ such that $x+y+z = xyz.$ 2008 VTRMC #1 here

Let \begin{align*}X&=\begin{pmatrix}7&8&98&-9&-7-7&-7&9\end{pmatrix}Y&=\begin{pmatrix}9&8&-98&-7&77&9&8\end{pmatrix}.\end{align*}Let $A=Y^{-1}X$ and let $B$ be the inverse of $X^{-1}+A^{-1}$ . Find a matrix $M$ such that $M^2=XY-BY$ (you may assume that $A$ and $X^{-1}+A^{-1}$ are invertible).