2014 VTRMC

Part of VTRMC

Subcontests

(7)

1

func. ineq. over M2(Z):det=1

S

denote the set of

2

2

matrices with integer entries and determinant

1

T

denote those matrices of

S

which are congruent to the identity matrix

I\pmod3

\begin{pmatrix}a&b\\c&d\end{pmatrix}\in T

a,b,c,d\in\mathbb Z,ad-bc=1,

3

b,c,a-1,d-1

f:T\to\mathbb R

be a function such that for every

X,Y\in T

Y\ne I

f(XY)>f(X)

f(XY^{-1})>f(X)

. Show that given two finite nonempty subsets

A,B

T

, there are matrices

a\in A

b\in B

a'\in A

b'\in B

a'b'=ab

a'=a

b'=b

. (b) Show that there is no

f:S\to\mathbb R

such that for every

X,Y\in S

Y\ne\pm I

f(XY)>f(X)

f(XY^{-1})>f(X)

1

walking on lattice grid

A,B

be two points in the plane with integer coordinates

A=(x_1,y_1)

B=(x_2,y_2)

x_i,y_i\in\mathbb Z

i=1,2

\pi:A\to B

is a sequence of down and right steps, where each step has an integer length, and the initial step starts from

A

, the last step ending at

B

. In the figure below, we indicated a path from

A_1=(4,9)

B1=(10,3)

d(A,B)

A

B

is the number of such paths. For example, the distance between

A=(0,2)

B=(2,0)

6

. Consider now two pairs of points in the plane

A_i=(x_i,y_i)

B_i=(u_i,z_i)

i=1,2

, with integer coordinates, and in the configuration shown in the picture (but with arbitrary coordinates):

x_2<x_1

y_1>y_2

, which means that

A_1

is North-East of

A_2

u_2<u_1

z_1>z_2

, which means that

B_1

is North-East of

B_2

. Each of the points

A_i

is North-West of the points

B_j

1\le i

j\le2

. In terms of inequalities, this means that

x_i<\min\{u_1,u_2\}

y_i>\max\{z_1,z_2\}

i=1,2

.
https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi9hL2I4ODlmNDAyYmU5OWUyMzVmZmEzMTY1MGY3YjI3YjFlMmMxMTI2LnBuZw==&rn=VlRSTUMgMjAxNC5wbmc=
(a) Find the distance between two points

A

B

as before, as a function of the coordinates of

A

B

A

is North-West of

B

. (b) Consider the

2\times2

M=\begin{pmatrix}d(A_1,B_1)&d(A_1,B_2)\\d(A_2,B_1)&d(A_2,B_2)\end{pmatrix}

. Prove that for any configuration of points

A_1,A_2,B_1,B_2

as described before,

\det M>0

1

n(n+1)(n+2)=m^r, no solutions

n\ge1

r\ge2

be positive integers. Prove that there is no integer

m

n(n+1)(n+2)=m^r

1

tiling 19x19 board with 4x1 and 1x4 rectangles

Suppose we are given a

19\times19

chessboard (a table with

19^2

squares) and remove the central square. Is it possible to tile the remaining

19^2-1=360

4\times1

1\times4

rectangles? (So that each of the

360

squares is covered by exactly one rectangle.) Justify your answer.

1

definite integral

\int^2_0\frac{x(16-x^2)}{16-x^2+\sqrt{(4-x)(4+x)(12+x^2)}}dx

1

sum(n^2-2n-4)/(n^4+4n^2+16) from n to ∞

\sum_{n=2}^\infty\frac{n^2-2n-4}{n^4+4n^2+16}

1

2^2014|19^n-1

Find the least positive integer

n

2^{2014}

19^n-1