MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2003 Mexico National Olympiad
2003 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
4
1
Hide problems
simple geometry
The quadrilateral
A
B
C
D
ABCD
A
BC
D
has
A
B
AB
A
B
parallel to
C
D
CD
C
D
.
P
P
P
is on the side
A
B
AB
A
B
and
Q
Q
Q
on the side
C
D
CD
C
D
such that
A
P
P
B
=
D
Q
C
Q
\frac{AP}{PB}= \frac{DQ}{CQ}
PB
A
P
=
CQ
D
Q
. M is the intersection of
A
Q
AQ
A
Q
and
D
P
DP
D
P
, and
N
N
N
is the intersection of
P
C
PC
PC
and
Q
B
QB
QB
. Find
M
N
MN
MN
in terms of
A
B
AB
A
B
and
C
D
CD
C
D
.
6
1
Hide problems
compatible numbers
Given a positive integer
n
n
n
, an allowed move is to form
2
n
+
1
2n+1
2
n
+
1
or
3
n
+
2
3n+2
3
n
+
2
. The set
S
n
S_{n}
S
n
is the set of all numbers that can be obtained by a sequence of allowed moves starting with
n
n
n
. For example, we can form
5
→
11
→
35
5 \rightarrow 11 \rightarrow 35
5
→
11
→
35
so
5
,
11
5, 11
5
,
11
and
35
35
35
belong to
S
5
S_{5}
S
5
. We call
m
m
m
and
n
n
n
compatible if
S
m
S_{m}
S
m
and
S
n
S_{n}
S
n
has a common element. Which members of
{
1
,
2
,
3
,
.
.
.
,
2002
}
\{1, 2, 3, ... , 2002\}
{
1
,
2
,
3
,
...
,
2002
}
are compatible with
2003
2003
2003
?
3
1
Hide problems
men and women liking
At a party there are
n
n
n
women and
n
n
n
men. Each woman likes
r
r
r
of the men, and each man likes
s
s
s
of then women. For which
r
r
r
and
s
s
s
must there be a man and a woman who like each other?
5
1
Hide problems
card strategy
Some cards each have a pair of numbers written on them. There is just one card for each pair
(
a
,
b
)
(a,b)
(
a
,
b
)
with
1
≤
a
<
b
≤
2003
1 \leq a < b \leq 2003
1
≤
a
<
b
≤
2003
. Two players play the following game. Each removes a card in turn and writes the product
a
b
ab
ab
of its numbers on the blackboard. The first player who causes the greatest common divisor of the numbers on the blackboard to fall to
1
1
1
loses. Which player has a winning strategy?
1
1
Hide problems
insert a zero
Find all positive integers with two or more digits such that if we insert a
0
0
0
between the units and tens digits we get a multiple of the original number.
2
1
Hide problems
three circles
A
,
B
,
C
A, B, C
A
,
B
,
C
are collinear with
B
B
B
betweeen
A
A
A
and
C
C
C
.
K
1
K_{1}
K
1
is the circle with diameter
A
B
AB
A
B
, and
K
2
K_{2}
K
2
is the circle with diameter
B
C
BC
BC
. Another circle touches
A
C
AC
A
C
at
B
B
B
and meets
K
1
K_{1}
K
1
again at
P
P
P
and
K
2
K_{2}
K
2
again at
Q
Q
Q
. The line
P
Q
PQ
PQ
meets
K
1
K_{1}
K
1
again at
R
R
R
and
K
2
K_{2}
K
2
again at
S
S
S
. Show that the lines
A
R
AR
A
R
and
C
S
CS
CS
meet on the perpendicular to
A
C
AC
A
C
at
B
B
B
.