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Problems
Contests
International Contests
CentroAmerican
2004 CentroAmerican
2004 CentroAmerican
Part of
CentroAmerican
Subcontests
(3)
3
2
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Cevian triangle of ABC is similar to ABC
A
B
C
ABC
A
BC
is a triangle, and
E
E
E
and
F
F
F
are points on the segments
B
C
BC
BC
and
C
A
CA
C
A
respectively, such that
C
E
C
B
+
C
F
C
A
=
1
\frac{CE}{CB}+\frac{CF}{CA}=1
CB
CE
+
C
A
CF
=
1
and
∠
C
E
F
=
∠
C
A
B
\angle CEF=\angle CAB
∠
CEF
=
∠
C
A
B
. Suppose that
M
M
M
is the midpoint of
E
F
EF
EF
and
G
G
G
is the point of intersection between
C
M
CM
CM
and
A
B
AB
A
B
. Prove that triangle
F
E
G
FEG
FEG
is similar to triangle
A
B
C
ABC
A
BC
.
Lots of pearls and necklaces
With pearls of different colours form necklaces, it is said that a necklace is prime if it cannot be decomposed into strings of pearls of the same length, and equal to each other. Let
n
n
n
and
q
q
q
be positive integers. Prove that the number of prime necklaces with
n
n
n
beads, each of which has one of the
q
n
q^n
q
n
possible colours, is equal to
n
n
n
times the number of prime necklaces with
n
2
n^2
n
2
pearls, each of which has one of
q
q
q
possible colours.Note: two necklaces are considered equal if they have the same number of pearls and you can get the same colour on both necklaces, rotating one of them to match it to the other.
2
2
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Integers expressible as the sum of two terms in the sequence
Define the sequence
(
a
n
)
(a_n)
(
a
n
)
as follows:
a
0
=
a
1
=
1
a_0=a_1=1
a
0
=
a
1
=
1
and for
k
≥
2
k\ge 2
k
≥
2
,
a
k
=
a
k
−
1
+
a
k
−
2
+
1
a_k=a_{k-1}+a_{k-2}+1
a
k
=
a
k
−
1
+
a
k
−
2
+
1
. Determine how many integers between
1
1
1
and
2004
2004
2004
inclusive can be expressed as
a
m
+
a
n
a_m+a_n
a
m
+
a
n
with
m
m
m
and
n
n
n
positive integers and
m
≠
n
m\not= n
m
=
n
.
The point P in the trapezium ABCD
Let
A
B
C
D
ABCD
A
BC
D
be a trapezium such that
A
B
∣
∣
C
D
AB||CD
A
B
∣∣
C
D
and
A
B
+
C
D
=
A
D
AB+CD=AD
A
B
+
C
D
=
A
D
. Let
P
P
P
be the point on
A
D
AD
A
D
such that
A
P
=
A
B
AP=AB
A
P
=
A
B
and
P
D
=
C
D
PD=CD
P
D
=
C
D
.
a
)
a)
a
)
Prove that
∠
B
P
C
=
9
0
∘
\angle BPC=90^{\circ}
∠
BPC
=
9
0
∘
.
b
)
b)
b
)
Q
Q
Q
is the midpoint of
B
C
BC
BC
and
R
R
R
is the point of intersection between the line
A
D
AD
A
D
and the circle passing through the points
B
,
A
B,A
B
,
A
and
Q
Q
Q
. Show that the points
B
,
P
,
R
B,P,R
B
,
P
,
R
and
C
C
C
are concyclic.
1
2
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A border separates a black square from a white square
In a
10
×
10
10\times 10
10
×
10
square board, half of the squares are coloured white and half black. One side common to two squares on the board side is called a border if the two squares have different colours. Determine the minimum and maximum possible number of borders that can be on the board.
Players A and B remove the numbers 1 to 9
On a whiteboard, the numbers
1
1
1
to
9
9
9
are written. Players
A
A
A
and
B
B
B
take turns, and
A
A
A
is first. Each player in turn chooses one of the numbers on the whiteboard and removes it, along with all multiples (if any). The player who removes the last number loses. Determine whether any of the players has a winning strategy, and explain why.