MathDB

1

covering a ladder (part of a grid) with n squares

n

can be covered a ladder like that of the figure (but with

n

steps instead of

4

) with

n

squares of integer sides, not necessarily the same size, without these squares overlapping and without standing out from the edge of the figure ?

2

1

triangles AEM , MCA are similar iff <ABC = 60^o

Let

ABC

be a right triangle with a right angle at

A

, such that

AB < AC

. Let

M

be the midpoint of

BC

and

D

the intersection of

AC

with the perpendicular on

BC

passing through

M

. Let

E

be the intersection of the parallel to

AC

that passes through

M

, with the perpendicular on

BD

passing through

B

. Show that the triangles

AEM

and

MCA

are similar if and only if

\angle ABC = 60^o

.

5

1

Mexican midpoint

Let

ABC

be an acute triangle , with altitudes

AD,BE

and

CF

. Circle of diameter

AD

intersects the sides

AB,AC

in

M,N

respevtively. Let

P,Q

be the intersection points of

AD

with

EF

and

MN

respectively. Show that

Q

is the midpoint of

PD

.

3

1

ways of filling the 2xn grid with numbers 1,2,..., 2n under one condition

Let

n

be an integer greater than

1

. In how many ways can we fill all the numbers

1, 2,..., 2n

in the boxes of a grid of

2\times n

, one in each box, so that any two consecutive numbers are they in squares that share one side of the grid?

1

Relatives of a number

Let

ab

be a two digit number. A positive integer

n

is a relative of

ab

if:
[*] The units digit of

n

is

b

. [*] The remaining digits of

n

are nonzero and add up to

a

. Find all two digit numbers which divide all of their relatives.

6

1

"Surtido" Numbers

Let n be the sum of the digits in a natural number A. The number A it's said to be "surtido" if every number 1,2,3,4....,n can be expressed as a sum of digits in A. a)Prove that, if 1,2,3,4,5,6,7,8 are sums of digits in A, then A is "Surtido" b)If 1,2,3,4,5,6,7 are sums of digits in A, does it follow that A is "Surtido"?

2006 Mexico National Olympiad

Subcontests

covering a ladder (part of a grid) with n squares

triangles AEM , MCA are similar iff <ABC = 60^o

Mexican midpoint

ways of filling the 2xn grid with numbers 1,2,..., 2n under one condition

Relatives of a number

"Surtido" Numbers

2006 Mexico National Olympiad

Subcontests

covering a ladder (part of a grid) with n squares

triangles AEM , MCA are similar iff &lt;ABC = 60^o

Mexican midpoint

ways of filling the 2xn grid with numbers 1,2,..., 2n under one condition

Relatives of a number

&quot;Surtido&quot; Numbers

triangles AEM , MCA are similar iff <ABC = 60^o

"Surtido" Numbers