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Problems
Contests
International Contests
CentroAmerican
2006 CentroAmerican
2006 CentroAmerican
Part of
CentroAmerican
Subcontests
(6)
6
1
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Product of ratios
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral.
I
=
A
C
∩
B
D
I=AC\cap BD
I
=
A
C
∩
B
D
, and
E
E
E
,
H
H
H
,
F
F
F
and
G
G
G
are points on
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
and
D
A
DA
D
A
respectively, such that
E
F
∩
G
H
=
I
EF \cap GH= I
EF
∩
G
H
=
I
. If
M
=
E
G
∩
A
C
M=EG \cap AC
M
=
EG
∩
A
C
,
N
=
H
F
∩
A
C
N=HF \cap AC
N
=
H
F
∩
A
C
, show that
A
M
I
M
⋅
I
N
C
N
=
I
A
I
C
.
\frac{AM}{IM}\cdot \frac{IN}{CN}=\frac{IA}{IC}.
I
M
A
M
⋅
CN
I
N
=
I
C
I
A
.
4
1
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Minimum of a sum
The product of several distinct positive integers is divisible by
2006
2
{2006}^{2}
2006
2
. Determine the minimum value the sum of such numbers can take.
2
1
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Segment of constant length
Let
Γ
\Gamma
Γ
and
Γ
′
\Gamma'
Γ
′
be two congruent circles centered at
O
O
O
and
O
′
O'
O
′
, respectively, and let
A
A
A
be one of their two points of intersection.
B
B
B
is a point on
Γ
\Gamma
Γ
,
C
C
C
is the second point of intersection of
A
B
AB
A
B
and
Γ
′
\Gamma'
Γ
′
, and
D
D
D
is a point on
Γ
′
\Gamma'
Γ
′
such that
O
B
D
O
′
OBDO'
OB
D
O
′
is a parallelogram. Show that the length of
C
D
CD
C
D
does not depend on the position of
B
B
B
.
1
1
Hide problems
Find the last digit of a sum
For
0
≤
d
≤
9
0 \leq d \leq 9
0
≤
d
≤
9
, we define the numbers
S
d
=
1
+
d
+
d
2
+
⋯
+
d
2006
S_{d}=1+d+d^{2}+\cdots+d^{2006}
S
d
=
1
+
d
+
d
2
+
⋯
+
d
2006
Find the last digit of the number
S
0
+
S
1
+
⋯
+
S
9
.
S_{0}+S_{1}+\cdots+S_{9}.
S
0
+
S
1
+
⋯
+
S
9
.
3
1
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Function involving floors
For every natural number
n
n
n
we define
f
(
n
)
=
⌊
n
+
n
+
1
2
⌋
f(n)=\left\lfloor n+\sqrt{n}+\frac{1}{2}\right\rfloor
f
(
n
)
=
⌊
n
+
n
+
2
1
⌋
Show that for every integer
k
≥
1
k \geq 1
k
≥
1
the equation
f
(
f
(
n
)
)
−
f
(
n
)
=
k
f(f(n))-f(n)=k
f
(
f
(
n
))
−
f
(
n
)
=
k
has exactly
2
k
−
1
2k-1
2
k
−
1
solutions.
5
1
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Building bridges between islands
The Olimpia country is formed by
n
n
n
islands. The most populated one is called Panacenter, and every island has a different number of inhabitants. We want to build bridges between these islands, which we'll be able to travel in both directions, under the following conditions: a) No pair of islands is joined by more than one bridge. b) Using the bridges we can reach every island from Panacenter. c) If we want to travel from Panacenter to every other island, in such a way that we use each bridge at most once, the number of inhabitants of the islands we visit is strictly decreasing. Determine the number of ways we can build the bridges.