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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
1996 Mexico National Olympiad
1996 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(6)
6
1
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collinearity as a result of perpendicularity and equality
In a triangle
A
B
C
ABC
A
BC
with
A
B
<
B
C
<
A
C
AB < BC < AC
A
B
<
BC
<
A
C
, points
A
′
,
B
′
,
C
′
A' ,B' ,C'
A
′
,
B
′
,
C
′
are such that
A
A
′
⊥
B
C
AA' \perp BC
A
A
′
⊥
BC
and
A
A
′
=
B
C
,
B
B
′
⊥
C
A
AA' = BC, BB' \perp CA
A
A
′
=
BC
,
B
B
′
⊥
C
A
and
B
B
′
=
C
A
BB'=CA
B
B
′
=
C
A
, and
C
C
′
⊥
A
B
CC' \perp AB
C
C
′
⊥
A
B
and
C
C
′
=
A
B
CC'= AB
C
C
′
=
A
B
, as shown on the picture. Suppose that
∠
A
C
′
B
\angle AC'B
∠
A
C
′
B
is a right angle. Prove that the points
A
′
,
B
′
,
C
′
A',B' ,C'
A
′
,
B
′
,
C
′
are collinear.
5
1
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no 1 to n^2 are written in an nxn squared paper, sum of paths
The numbers
1
1
1
to
n
2
n^2
n
2
are written in an n×n squared paper in the usual ordering. Any sequence of right and downwards steps from a square to an adjacent one (by side) starting at square
1
1
1
and ending at square
n
2
n^2
n
2
is called a path. Denote by
L
(
C
)
L(C)
L
(
C
)
the sum of the numbers through which path
C
C
C
goes. (a) For a fixed
n
n
n
, let
M
M
M
and
m
m
m
be the largest and smallest
L
(
C
)
L(C)
L
(
C
)
possible. Prove that
M
−
m
M-m
M
−
m
is a perfect cube. (b) Prove that for no
n
n
n
can one find a path
C
C
C
with
L
(
C
)
=
1996
L(C ) = 1996
L
(
C
)
=
1996
.
4
1
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no 1 to 16 written in a squared 4x4 paper with divisible sums by n
For which integers
n
≥
2
n\ge 2
n
≥
2
can the numbers
1
1
1
to
16
16
16
be written each in one square of a squared
4
×
4
4\times 4
4
×
4
paper such that the
8
8
8
sums of the numbers in rows and columns are all different and divisible by
n
n
n
?
3
1
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covering a 6×6 square board with eighteen 2×1 rectangles
Prove that it is not possible to cover a
6
×
6
6\times 6
6
×
6
square board with eighteen
2
×
1
2\times 1
2
×
1
rectangles, in such a way that each of the lines going along the interior gridlines cuts at least one of the rectangles. Show also that it is possible to cover a
6
×
5
6\times 5
6
×
5
rectangle with fifteen
2
×
1
2\times 1
2
×
1
rectangles so that the above condition is fulfilled.
2
1
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64 booths around a circular table and on each one there is a chip
There are
64
64
64
booths around a circular table and on each one there is a chip. The chips and the corresponding booths are numbered
1
1
1
to
64
64
64
in this order. At the center of the table there are
1996
1996
1996
light bulbs which are all turned off. Every minute the chips move simultaneously in a circular way (following the numbering sense) as follows: chip
1
1
1
moves one booth, chip
2
2
2
moves two booths, etc., so that more than one chip can be in the same booth. At any minute, for each chip sharing a booth with chip
1
1
1
a bulb is lit. Where is chip
1
1
1
on the first minute in which all bulbs are lit?
1
1
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parallelogram criterion with trisection of a diagonal
Let
P
P
P
and
Q
Q
Q
be the points on the diagonal
B
D
BD
B
D
of a quadrilateral
A
B
C
D
ABCD
A
BC
D
such that
B
P
=
P
Q
=
Q
D
BP = PQ = QD
BP
=
PQ
=
Q
D
. Let
A
P
AP
A
P
and
B
C
BC
BC
meet at
E
E
E
, and let
A
Q
AQ
A
Q
meet
D
C
DC
D
C
at
F
F
F
. (a) Prove that if
A
B
C
D
ABCD
A
BC
D
is a parallelogram, then
E
E
E
and
F
F
F
are the midpoints of the corresponding sides. (b) Prove the converse of (a).