MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
1993 Mexico National Olympiad
1993 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
6
1
Hide problems
odd prime: p \n(n+1)(n+2)(n+3)+1 for n iff p \ (m^2 - 5) for m
p
p
p
is an odd prime. Show that
p
p
p
divides
n
(
n
+
1
)
(
n
+
2
)
(
n
+
3
)
+
1
n(n+1)(n+2)(n+3) + 1
n
(
n
+
1
)
(
n
+
2
)
(
n
+
3
)
+
1
for some integer
n
n
n
iff
p
p
p
divides
m
2
−
5
m^2 - 5
m
2
−
5
for some integer
m
m
m
.
5
1
Hide problems
collinearity of intersections of diameters of 3 chords
O
A
,
O
B
,
O
C
OA, OB, OC
O
A
,
OB
,
OC
are three chords of a circle. The circles with diameters
O
A
,
O
B
OA, OB
O
A
,
OB
meet again at
Z
Z
Z
, the circles with diameters
O
B
,
O
C
OB, OC
OB
,
OC
meet again at
X
X
X
, and the circles with diameters
O
C
,
O
A
OC, OA
OC
,
O
A
meet again at
Y
Y
Y
. Show that
X
,
Y
,
Z
X, Y, Z
X
,
Y
,
Z
are collinear.
4
1
Hide problems
f(n,k): f(n,0)=f(n,n)=1 and f(n,k)=f(n-1,k-1)+f(n-1,k) for 0<k<n , f(3991,1993)
f
(
n
,
k
)
f(n,k)
f
(
n
,
k
)
is defined by (1)
f
(
n
,
0
)
=
f
(
n
,
n
)
=
1
f(n,0) = f(n,n) = 1
f
(
n
,
0
)
=
f
(
n
,
n
)
=
1
and (2)
f
(
n
,
k
)
=
f
(
n
−
1
,
k
−
1
)
+
f
(
n
−
1
,
k
)
f(n,k) = f(n-1,k-1) + f(n-1,k)
f
(
n
,
k
)
=
f
(
n
−
1
,
k
−
1
)
+
f
(
n
−
1
,
k
)
for
0
<
k
<
n
0 < k < n
0
<
k
<
n
. How many times do we need to use (2) to find
f
(
3991
,
1993
)
f(3991,1993)
f
(
3991
,
1993
)
?
3
1
Hide problems
995 points inside the pentagon of area 1993, at least 3 points (ABC)<=1
Given a pentagon of area
1993
1993
1993
and
995
995
995
points inside the pentagon, let
S
S
S
be the set containing the vertices of the pentagon and the
995
995
995
points. Show that we can find three points of
S
S
S
which form a triangle of area
≤
1
\le 1
≤
1
.
2
1
Hide problems
all numbers 100 - 999 which equal the sum of the cubes of their digits
Find all numbers between
100
100
100
and
999
999
999
which equal the sum of the cubes of their digits.
1
1
Hide problems
starting with 3 right triangles, find area of a quadrilateral
A
B
C
ABC
A
BC
is a triangle with
∠
A
=
9
0
o
\angle A = 90^o
∠
A
=
9
0
o
. Take
E
E
E
such that the triangle
A
E
C
AEC
A
EC
is outside
A
B
C
ABC
A
BC
and
A
E
=
C
E
AE = CE
A
E
=
CE
and
∠
A
E
C
=
9
0
o
\angle AEC = 90^o
∠
A
EC
=
9
0
o
. Similarly, take
D
D
D
so that
A
D
B
ADB
A
D
B
is outside
A
B
C
ABC
A
BC
and similar to
A
E
C
AEC
A
EC
.
O
O
O
is the midpoint of
B
C
BC
BC
. Let the lines
O
D
OD
O
D
and
E
C
EC
EC
meet at
D
′
D'
D
′
, and the lines
O
E
OE
OE
and
B
D
BD
B
D
meet at
E
′
E'
E
′
. Find area
D
E
D
′
E
′
DED'E'
D
E
D
′
E
′
in terms of the sides of
A
B
C
ABC
A
BC
.