MathDB
Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
1995 Mexico National Olympiad
1995 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
6
1
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arrangements of 0 and 1 on each square of a 4 x 4 board
A
1
1
1
or
0
0
0
is placed on each square of a
4
×
4
4 \times 4
4
×
4
board. One is allowed to change each symbol in a row, or change each symbol in a column, or change each symbol in a diagonal (there are
14
14
14
diagonals of lengths
1
1
1
to
4
4
4
). For which arrangements can one make changes which end up with all
0
0
0
s?
5
1
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5 triangles with equal areas in a convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
is a convex pentagon such that the triangles
A
B
C
,
B
C
D
,
C
D
E
,
D
E
A
ABC, BCD, CDE, DEA
A
BC
,
BC
D
,
C
D
E
,
D
E
A
and
E
A
B
EAB
E
A
B
have equal areas. Show that
(
1
/
4
)
(1/4)
(
1/4
)
area
(
A
B
C
D
E
)
<
(ABCDE) <
(
A
BC
D
E
)
<
area
(
A
B
C
)
<
(
1
/
3
)
(ABC) < (1/3)
(
A
BC
)
<
(
1/3
)
area
(
A
B
C
D
E
)
(ABCDE)
(
A
BC
D
E
)
.
4
1
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26 elements of {1,2,3,.. ,40} such that product of 2of them is never square
Find
26
26
26
elements of
{
1
,
2
,
3
,
.
.
.
,
40
}
\{1, 2, 3, ... , 40\}
{
1
,
2
,
3
,
...
,
40
}
such that the product of two of them is never a square. Show that one cannot find
27
27
27
such elements.
3
1
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collinear points in the regular 7-gon
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
are consecutive vertices of a regular
7
7
7
-gon.
A
L
AL
A
L
and
A
M
AM
A
M
are tangents to the circle center
C
C
C
radius
C
B
CB
CB
.
N
N
N
is the intersection point of
A
C
AC
A
C
and
B
D
BD
B
D
. Show that
L
,
M
,
N
L, M, N
L
,
M
,
N
are collinear.
1
1
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1020 handshakes between N students seated at desks in mxn array
N
N
N
students are seated at desks in an
m
×
n
m \times n
m
×
n
array, where
m
,
n
≥
3
m, n \ge 3
m
,
n
≥
3
. Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are
1020
1020
1020
handshakes, what is
N
N
N
?
2
1
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Nice, maybe easy, geocombi.
Consider 6 points on a plane such that 8 of the distances between them are equal to 1. Prove that there are at least 3 points that form an equilateral triangle.