MathDB
Problems
Contests
National and Regional Contests
El Salvador Contests
El Salvador Correspondence
2006 El Salvador Correspondence
2006 El Salvador Correspondence
Part of
El Salvador Correspondence
Subcontests
(1)
1
Hide problems
2006 El Salvador Correspondence / Qualifying NMO VI
p1. Given the square
A
B
C
D
ABCD
A
BC
D
, find all points
P
P
P
interior squared such that the area of quadrilateral
A
B
C
P
ABCP
A
BCP
is equal to three times the area of quadrilateral
A
P
C
D
APCD
A
PC
D
. p2. Find all possible pairs of positive integers
(
x
,
y
)
(x, y)
(
x
,
y
)
such that neither
x
x
x
nor
y
y
y
has a prime factor greater than
5
5
5
, and are also a solution of the equation
x
2
−
y
2
=
2
100
x^2-y^2= 2^{100}
x
2
−
y
2
=
2
100
. p3. On the right pan of a scale is a bag that weighs
11
,
111
11,111
11
,
111
grams. A person place weights on one or another plate of the balance; the first weight is one gram, and each weight placed has twice the weight of the previous one, that is, the weights are successively
1
,
2
,
4
,
8
,
.
.
.
1, 2, 4, 8, ...
1
,
2
,
4
,
8
,
...
grams. At some point the plates on the scale are in balance. Determine and argue on which the
16
16
16
gram plate weight is, the right or the left one. p4. How many integer numbers between
1
1
1
and
1000
1000
1000
inclusive can be written as the sum of a positive multiple of
7
7
7
plus a positive multiple of
4
4
4
? p5. We will say that two numbers are concatenated when one is written below from the other, for example by concatenating 200 and
6
6
6
you get
2006
2006
2006
and by concatenating
6
6
6
and
200
200
200
gives
6200
6200
6200
. Find two
6
6
6
-digit numbers such that when you concatenate them the resulting number is divisible by its product.