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Problems
Contests
National and Regional Contests
Mexico Contests
Mexico National Olympiad
2008 Mexico National Olympiad
2008 Mexico National Olympiad
Part of
Mexico National Olympiad
Subcontests
(3)
3
2
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Chess Board
Consider a chess board, with the numbers
1
1
1
through
64
64
64
placed in the squares as in the diagram below.\begin{tabular}{| c | c | c | c | c | c | c | c |} \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ \hline 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 \\ \hline 25 & 26 & 27 & 28 & 29 & 30 & 31 & 32 \\ \hline 33 & 34 & 35 & 36 & 37 & 38 & 39 & 40 \\ \hline 41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 \\ \hline 49 & 50 & 51 & 52 & 53 & 54 & 55 & 56 \\ \hline 57 & 58 & 59 & 60 & 61 & 62 & 63 & 64 \\ \hline \end{tabular}Assume we have an infinite supply of knights. We place knights in the chess board squares such that no two knights attack one another and compute the sum of the numbers of the cells on which the knights are placed. What is the maximum sum that we can attain?Note. For any
2
×
3
2\times3
2
×
3
or
3
×
2
3\times2
3
×
2
rectangle that has the knight in its corner square, the knight can attack the square in the opposite corner.
Circle Intersections form a Circle!
The internal angle bisectors of
A
A
A
,
B
B
B
, and
C
C
C
in
△
A
B
C
\triangle ABC
△
A
BC
concur at
I
I
I
and intersect the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at
L
L
L
,
M
M
M
, and
N
N
N
, respectively. The circle with diameter
I
L
IL
I
L
intersects
B
C
BC
BC
at
D
D
D
and
E
E
E
; the circle with diameter
I
M
IM
I
M
intersects
C
A
CA
C
A
at
F
F
F
and
G
G
G
; the circle with diameter
I
N
IN
I
N
intersects
A
B
AB
A
B
at
H
H
H
and
J
J
J
. Show that
D
D
D
,
E
E
E
,
F
F
F
,
G
G
G
,
H
H
H
, and
J
J
J
are concyclic.
2
2
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A Bunch of Tangencies
Consider a circle
Γ
\Gamma
Γ
, a point
A
A
A
on its exterior, and the points of tangency
B
B
B
and
C
C
C
from
A
A
A
to
Γ
\Gamma
Γ
. Let
P
P
P
be a point on the segment
A
B
AB
A
B
, distinct from
A
A
A
and
B
B
B
, and let
Q
Q
Q
be the point on
A
C
AC
A
C
such that
P
Q
PQ
PQ
is tangent to
Γ
\Gamma
Γ
. Points
R
R
R
and
S
S
S
are on lines
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, such that
P
Q
∥
R
S
PQ\parallel RS
PQ
∥
RS
and
R
S
RS
RS
is tangent to
Γ
\Gamma
Γ
as well. Prove that
[
A
P
Q
]
⋅
[
A
R
S
]
[APQ]\cdot[ARS]
[
A
PQ
]
⋅
[
A
RS
]
does not depend on the placement of point
P
P
P
.
Numbers in the Vertices of a Cube
We place
8
8
8
distinct integers in the vertices of a cube and then write the greatest common divisor of each pair of adjacent vertices on the edge connecting them. Let
E
E
E
be the sum of the numbers on the edges and
V
V
V
the sum of the numbers on the vertices.a) Prove that
2
3
E
≤
V
\frac23E\le V
3
2
E
≤
V
. b) Can
E
=
V
E=V
E
=
V
?
1
2
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n = d2^2 + d3^3
Let
1
=
d
1
<
d
2
<
d
3
<
⋯
<
d
k
=
n
1=d_1<d_2<d_3<\dots<d_k=n
1
=
d
1
<
d
2
<
d
3
<
⋯
<
d
k
=
n
be the divisors of
n
n
n
. Find all values of
n
n
n
such that
n
=
d
2
2
+
d
3
3
n=d_2^2+d_3^3
n
=
d
2
2
+
d
3
3
.
Knights on a Round Table
A king decides to reward one of his knights by making a game. He sits the knights at a round table and has them call out
1
,
2
,
3
,
1
,
2
,
3
,
…
1,2,3,1,2,3,\dots
1
,
2
,
3
,
1
,
2
,
3
,
…
around the circle (that is, clockwise, and each person says a number). The people who say
2
2
2
or
3
3
3
immediately lose, and this continues until the last knight is left, the winner.Numbering the knights initially as
1
,
2
,
…
,
n
1,2,\dots,n
1
,
2
,
…
,
n
, find all values of
n
n
n
such that knight
2008
2008
2008
is the winner.