2006 IberoAmerican

Part of IberoAmerican

Subcontests

(3)

2

Problem 3

1,\, 2,\, \ldots\, , n^{2}

are written in the squares of an

n \times n

board in some order. Initially there is a token on the square labelled with

n^{2}.

In each step, the token can be moved to any adjacent square (by side). At the beginning, the token is moved to the square labelled with the number

1

along a path with the minimum number of steps. Then it is moved to the square labelled with

2,

3,

etc, always taking the shortest path, until it returns to the initial square. If the total trip takes

N

steps, find the smallest and greatest possible values of

N.

Problem 6

Consider a regular

n

n

odd. Given two adjacent vertices

A_{1}

A_{2},

define the sequence

(A_{k})

of vertices of the

n

-gon as follows: For

k\ge 3,\, A_{k}

is the vertex lying on the perpendicular bisector of

A_{k-2}A_{k-1}.

n

for which each vertex of the

n

-gon occurs in this sequence.

2

Problem 2

For n real numbers

a_{1},\, a_{2},\, \ldots\, , a_{n},

d

denote the difference between the greatest and smallest of them and

S = \sum_{i<j}\left |a_i-a_j \right|.

(n-1)d\le S\le\frac{n^{2}}{4}d

and find when each equality holds.

Problem 5

AD

CD

of a tangent quadrilateral

ABCD

touch the incircle

\varphi

P

Q,

respectively. If

M

is the midpoint of the chord

XY

\varphi

on the diagonal

BD,

\angle AMP = \angle CMQ.

2

Isosceles triangle

In a scalene triangle

ABC

\angle A = 90^\circ,

the tangent line at

A

to its circumcircle meets line

BC

M

and the incircle touches

AC

S

AB

R.

RS

BC

N,

while the lines

AM

SR

U.

Prove that the triangle

UMN

Problem 4

(a,\, b)

of positive integers such that

2a-1

2b+1

are coprime and

a+b

4ab+1.