MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
1989 Mexico National Olympiad
1989 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
6
1
Hide problems
no of paths in a triangular grid
Determine the number of paths from
A
A
A
to
B
B
B
on the picture that go along gridlines only, do not pass through any point twice, and never go upwards? https://cdn.artofproblemsolving.com/attachments/0/2/87868e24a48a2e130fb5039daeb85af42f4b9d.png
5
1
Hide problems
centers of 4 circles are vertices of a rectangle
Let
C
1
C_1
C
1
and
C
2
C_2
C
2
be two tangent unit circles inside a circle
C
C
C
of radius
2
2
2
. Circle
C
3
C_3
C
3
inside
C
C
C
is tangent to the circles
C
,
C
1
,
C
2
C,C_1,C_2
C
,
C
1
,
C
2
, and circle
C
4
C_4
C
4
inside
C
C
C
is tangent to
C
,
C
1
,
C
3
C,C_1,C_3
C
,
C
1
,
C
3
. Prove that the centers of
C
,
C
1
,
C
3
C,C_1,C_3
C
,
C
1
,
C
3
and
C
4
C_4
C
4
are vertices of a rectangle.
4
1
Hide problems
smallest n so that r =\overline{a_1a_0a_m ..._20} = 2n, in decimal
Find the smallest possible natural number
n
=
a
m
.
.
.
a
2
a
1
a
0
‾
n = \overline{a_m ...a_2a_1a_0}
n
=
a
m
...
a
2
a
1
a
0
(in decimal system) such that the number
r
=
a
1
a
0
a
m
.
.
.
2
0
‾
r = \overline{a_1a_0a_m ..._20}
r
=
a
1
a
0
a
m
..
.
2
0
equals
2
n
2n
2
n
.
3
1
Hide problems
1989-digit natural number such that product of its digits = their sum and ...
Prove that there is no
1989
1989
1989
-digit natural number at least three of whose digits are equal to
5
5
5
and such that the product of its digits equals their sum.
2
1
Hide problems
a|b^2, b^2|a^3, a^3|b^4, b^4|a^5$, but a^5 does not divide b^6
Find two positive integers
a
,
b
a,b
a
,
b
such that
a
∣
b
2
,
b
2
∣
a
3
,
a
3
∣
b
4
,
b
4
∣
a
5
a | b^2, b^2 | a^3, a^3 | b^4, b^4 | a^5
a
∣
b
2
,
b
2
∣
a
3
,
a
3
∣
b
4
,
b
4
∣
a
5
, but
a
5
a^5
a
5
does not divide
b
6
b^6
b
6
1
1
Hide problems
perpendicular medians in a triangle with given third side and area
In a triangle
A
B
C
ABC
A
BC
the area is
18
18
18
, the length
A
B
AB
A
B
is
5
5
5
, and the medians from
A
A
A
and
B
B
B
are orthogonal. Find the lengths of the sides
B
C
,
A
C
BC,AC
BC
,
A
C
.