MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
1990 Mexico National Olympiad
1990 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
1
1
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counting paths in a triangular grid, moves only to one direction
How many paths are there from
A
A
A
to the line
B
C
BC
BC
if the path does not go through any vertex twice and always moves to the left? https://cdn.artofproblemsolving.com/attachments/e/6/a4bc3a9decc06eaeed6f7e99cb58f7b2524471.jpg
6
1
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similar triangles around a right triangle
A
B
C
ABC
A
BC
is a triangle with
∠
C
=
9
0
o
\angle C = 90^o
∠
C
=
9
0
o
.
E
E
E
is a point on
A
C
AC
A
C
, and
F
F
F
is the midpoint of
E
C
EC
EC
.
C
H
CH
C
H
is an altitude.
I
I
I
is the circumcenter of
A
H
E
AHE
A
H
E
, and
G
G
G
is the midpoint of
B
C
BC
BC
. Show that
A
B
C
ABC
A
BC
and
I
G
F
IGF
I
GF
are similar.
5
1
Hide problems
19 lattice points with no 3 collinear, always exist 3 with lattice centroid
Given
19
19
19
points in the plane with integer coordinates, no three collinear, show that we can always find three points whose centroid has integer coordinates.
4
1
Hide problems
0/1 + 1/1 + 0/2 + 1/2 + 2/2 + 0/3 + 1/3 + 2/3 + 3/3 + ...+5/6 + 6/6
Find
0
/
1
+
1
/
1
+
0
/
2
+
1
/
2
+
2
/
2
+
0
/
3
+
1
/
3
+
2
/
3
+
3
/
3
+
0
/
4
+
1
/
4
+
2
/
4
+
3
/
4
+
4
/
4
+
0
/
5
+
1
/
5
+
2
/
5
+
3
/
5
+
4
/
5
+
5
/
5
+
0
/
6
+
1
/
6
+
2
/
6
+
3
/
6
+
4
/
6
+
5
/
6
+
6
/
6
0/1 + 1/1 + 0/2 + 1/2 + 2/2 + 0/3 + 1/3 + 2/3 + 3/3 + 0/4 + 1/4 + 2/4 + 3/4 + 4/4 + 0/5 + 1/5 + 2/5 + 3/5 + 4/5 + 5/5 + 0/6 + 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6
0/1
+
1/1
+
0/2
+
1/2
+
2/2
+
0/3
+
1/3
+
2/3
+
3/3
+
0/4
+
1/4
+
2/4
+
3/4
+
4/4
+
0/5
+
1/5
+
2/5
+
3/5
+
4/5
+
5/5
+
0/6
+
1/6
+
2/6
+
3/6
+
4/6
+
5/6
+
6/6
3
1
Hide problems
(n-1)^2 \ n^{n-1}-1 for n > 2
Show that
n
n
−
1
−
1
n^{n-1}-1
n
n
−
1
−
1
is divisible by
(
n
−
1
)
2
(n-1)^2
(
n
−
1
)
2
for
n
>
2
n > 2
n
>
2
.
2
1
Hide problems
right ABC, altitiude BH, relation between 3 inradii wanted
A
B
C
ABC
A
BC
is a triangle with
∠
B
=
9
0
o
\angle B = 90^o
∠
B
=
9
0
o
and altitude
B
H
BH
B
H
. The inradii of
A
B
C
,
A
B
H
,
C
B
H
ABC, ABH, CBH
A
BC
,
A
B
H
,
CB
H
are
r
,
r
1
,
r
2
r, r_1, r_2
r
,
r
1
,
r
2
. Find a relation between them.