2002 CentroAmerican

Part of CentroAmerican

Subcontests

(6)

1

Lattice points -- Paths

(0,0)

(n,n)

on the lattice is made up of unit moves upward or rightward. It is balanced if the sum of the x-coordinates of its 2n\plus{}1 vertices equals the sum of their y-coordinates. Show that a balanced path divides the square with vertices

(0,0)

(n,0)

(n,n)

(0,n)

into two parts with equal area.

1

Number Theory - Sets and squares

Find a set of infinite positive integers

S

such that for every

n\ge 1

n

distinct elements

x_1,x_2,\cdots, x_n

of S, the number x_1\plus{}x_2\plus{}\cdots \plus{}x_n is not a perfect square.

1

Triangles

ABC

D

be the midpoint of

BC

E

be a point on segment

AC

such that BE\equal{}2AD and

F

is the intersection point of

AD

BE

. If \angle DAC\equal{}60^{\circ}, find the measure of the angle

FEA

1

Sequence

For every integer

a>1

an infinite list of integers is constructed

L(a)

a

is the first number in the list

L(a)

. Given a number

b

L(a)

, the next number in the list is b\plus{}c, where

c

is the largest integer that divides

b

and is smaller than

b

. Find all the integers

a>1

2002

L(a)

1

Triangles

ABC

be an acute triangle, and let

D

E

be the feet of the altitudes drawn from vertexes

A

B

, respectively. Show that if,

Area[BDE]\le Area[DEA]\le Area[EAB]\le Area[ABD]

then, the triangle is isosceles.

1

Number Theory

For what integers

n\ge 3

is it possible to accommodate, in some order, the numbers

1,2,\cdots, n

in a circular form such that every number divides the sum of the next two numbers, in a clockwise direction?