2005 VTRMC

Part of VTRMC

Subcontests

(7)

1

tranpose, 5x10 matrices

A

5\times10

matrix with real entries, and let

A^{\text T}

denote its transpose. Suppose every

5\times1

matrix with real entries can be written in the form

A\mathbf u

\mathbf u

10\times1

matrix with real entries. Prove that every

5\times1

matrix with real entries can be written in the form

AA^{\text T}\mathbf v

\mathbf v

5\times1

matrix with real entries.

1

another integral

\int^1_0\left((e-1)\sqrt{\ln(1+ex-x)}+e^{x^2}\right)dx

1

limit in two variables

f(x,y)=\frac{xy}{x^2+y^2\ln(x^2)^2}

x\ne0

f(0,y)=0

y\ne0

. Determine whether

\lim_{(x,y)\to(0,0)}f(x,y)

exists, and find its value is if the limit does exist.

1

cube, lightbeam reflecting inside

A cubical box with sides of length

7

has vertices at

(0,0,0)

(7,0,0)

(0,7,0)

(7,7,0)

(0,0,7)

(7,0,7)

(0,7,7)

(7,7,7)

. The inside of the box is lined with mirrors and from the point

(0,1,2)

, a beam of light is directed to the point

(1,3,4)

. The light then reflects repeatedly off the mirrors on the inside of the box. Determine how far the beam of light travels before it first returns to its starting point at

(0,1,2)

1

covering tiled squares with dominoes, recurrence

We wish to tile a strip of

n

1

1

-inch squares. We can use dominos which are made up of two tiles that cover two adjacent squares, or

1

-inch square tiles which cover one square. We may cover each square with one or two tiles and a tile can be above or below a domino on a square, but no part of a domino can be placed on any part of a different domino. We do not distinguish whether a domino is above or below a tile on a given square. Let

t(n)

denote the number of ways the strip can be tiled according to the above rules. Thus for example,

t(1)=2

t(2)=8

. Find a recurrence relation for

t(n)

, and use it to compute

t(6)

1

k+σ(k)=2^n, permutation of [20]

Find, and write out explicitly, a permutation

\{p(1),p(2),\ldots,p(20)\}

\{1,2,\ldots,20\}

k+p(k)

2

k=1,2,\ldots,20

, and prove that only one such permutation exists.

1

n+6(p^3+1) is prime when p is prime

Find the largest positive integer

n

with the property that

n+6(p^3+1)

is prime whenever

p

is a prime number such that

2\le p<n

. Justify your answer.