MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2000 Mexico National Olympiad
2000 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
1
1
Hide problems
4 circles, externally tangent in pairs
Circles
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
are given on the plane such that circles
A
A
A
and
B
B
B
are externally tangent at
P
,
B
P, B
P
,
B
and
C
C
C
at
Q
,
C
Q, C
Q
,
C
and
D
D
D
at
R
R
R
, and
D
D
D
and
A
A
A
at
S
S
S
. Circles
A
A
A
and
C
C
C
do not meet, and so do not
B
B
B
and
D
D
D
. (a) Prove that the points
P
,
Q
,
R
,
S
P,Q,R,S
P
,
Q
,
R
,
S
lie on a circle. (b) Suppose that
A
A
A
and
C
C
C
have radius
2
,
B
2, B
2
,
B
and
D
D
D
have radius
3
3
3
, and the distance between the centers of
A
A
A
and
C
C
C
is
6
6
6
. Compute the area of the quadrilateral
P
Q
R
S
PQRS
PQRS
.
2
1
Hide problems
triangle of numbers, under any two consecutive numbers their sum is written
A triangle of numbers is constructed as follows. The first row consists of the numbers from
1
1
1
to
2000
2000
2000
in increasing order, and under any two consecutive numbers their sum is written. (See the example corresponding to
5
5
5
instead of
2000
2000
2000
below.) What is the number in the lowermost row?1 2 3 4 5 3 5 7 9 8 12 16 20 28 4
3
1
Hide problems
constructing set with +, - of elements of given set
Given a set
A
A
A
of positive integers, the set
A
′
A'
A
′
is composed from the elements of
A
A
A
and all positive integers that can be obtained in the following way: Write down some elements of
A
A
A
one after another without repeating, write a sign
+
+
+
or
−
-
−
before each of them, and evaluate the obtained expression. The result is included in
A
′
A'
A
′
. For example, if
A
=
{
2
,
8
,
13
,
20
}
A = \{2,8,13,20\}
A
=
{
2
,
8
,
13
,
20
}
, numbers
8
8
8
and
14
=
20
−
2
+
8
14 = 20-2+8
14
=
20
−
2
+
8
are elements of
A
′
A'
A
′
. Set
A
′
′
A''
A
′′
is constructed from
A
′
A'
A
′
in the same manner. Find the smallest possible number of elements of
A
A
A
, if
A
′
′
A''
A
′′
contains all the integers from
1
1
1
to
40
40
40
.
6
1
Hide problems
prove equal angles starting with an obtuse triangle
Let
A
B
C
ABC
A
BC
be a triangle with
∠
B
>
9
0
o
\angle B > 90^o
∠
B
>
9
0
o
such that there is a point
H
H
H
on side
A
C
AC
A
C
with
A
H
=
B
H
AH = BH
A
H
=
B
H
and BH perpendicular to
B
C
BC
BC
. Let
D
D
D
and
E
E
E
be the midpoints of
A
B
AB
A
B
and
B
C
BC
BC
respectively. A line through
H
H
H
parallel to
A
B
AB
A
B
cuts
D
E
DE
D
E
at
F
F
F
. Prove that
∠
B
C
F
=
∠
A
C
D
\angle BCF = \angle ACD
∠
BCF
=
∠
A
C
D
.
4
1
Hide problems
max possible no of primes before first composite in a sequence of integers
Let
a
a
a
and
b
b
b
be positive integers not divisible by
5
5
5
. A sequence of integers is constructed as follows: the first term is
5
5
5
, and every consequent term is obtained by multiplying its precedent by
a
a
a
and adding
b
b
b
. (For example, if
a
=
2
a = 2
a
=
2
and
b
=
4
b = 4
b
=
4
, the first three terms are
5
,
14
,
32
5,14,32
5
,
14
,
32
.) What is the maximum possible number of primes that can occur before encoutering the first composite term?
5
1
Hide problems
Combinatorics: Easy or tough?
A board
n
n
n
×
n
n
n
is coloured black and white like a chessboard. The following steps are permitted: Choose a rectangle inside the board (consisting of entire cells)whose side lengths are both odd or both even, but not both equal to
1
1
1
, and invert the colours of all cells inside the rectangle. Determine the values of
n
n
n
for which it is possible to make all the cells have the same colour in a finite number of such steps.