2013 VTRMC

Part of VTRMC

Subcontests

(7)

1

image of (x^2+y^2)/(u^2+v^2)

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.

1

maximize angle in triangle, side equality given

ABC

be a right-angled triangle with

\angle ABC=90^\circ

D

AB

AD=2DB

. What is the maximum possible value of

\angle ACD

1

matrix inverse problem

Let \begin{align*}X&=\begin{pmatrix}7&8&9\\8&-9&-7\\-7&-7&9\end{pmatrix}\\Y&=\begin{pmatrix}9&8&-9\\8&-7&7\\7&9&8\end{pmatrix}.\end{align*}Let

A=Y^{-1}X

B

be the inverse of

X^{-1}+A^{-1}

. Find a matrix

M

M^2=XY-BY

(you may assume that

A

X^{-1}+A^{-1}

are invertible).

1

convergence of sequence

Define a sequence

(a_n)

n\ge1

a_1=2

a_{n+1}=a_n^{1+n^{-3/2}}

(a_n)

convergent (i.e.

\lim_{n\to\infty}a_n<\infty

1

integral equation

I=3\sqrt2\int^x_0\frac{\sqrt{1+\cos t}}{17-8\cos t}dt

0<x<\pi

\tan I=\frac2{\sqrt3}

x

1

2013 VTRMC #5

\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

x+y+z = xyz.

[url=https://artofproblemsolving.com/community/c7h236610p10925499]2008 VTRMC #1 [url=http://www.math.vt.edu/people/plinnell/Vtregional/solutions.pdf]here

1

Infinite Sum

\sum_{n=1}^\infty \frac{n}{(2^n-2^{-n})^2}+\frac{(-1)^nn}{(2^n-2^{-n})^2}