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National and Regional Contests
Costa Rica Contests
Costa Rica - Final Round
2009 Costa Rica - Final Round
2009 Costa Rica - Final Round
Part of
Costa Rica - Final Round
Subcontests
(6)
6
1
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Costa Rican Math Olympiad Problem 6 2009
Let
Δ
A
B
C
\Delta ABC
Δ
A
BC
with incircle
Γ
\Gamma
Γ
, let
D
,
E
D, E
D
,
E
and
F
F
F
the tangency points of
Γ
\Gamma
Γ
with sides
B
C
,
A
C
BC, AC
BC
,
A
C
and
A
B
AB
A
B
, respectively and let
P
P
P
the intersection point of
A
D
AD
A
D
with
Γ
\Gamma
Γ
.
a
)
a)
a
)
Prove that
B
C
,
E
F
BC, EF
BC
,
EF
and the straight line tangent to
Γ
\Gamma
Γ
for
P
P
P
concur at a point
A
′
A'
A
′
.
b
)
b)
b
)
Define
B
′
B'
B
′
and
C
′
C'
C
′
in an anologous way than
A
′
A'
A
′
. Prove that A'\minus{}B'\minus{}C'
3
1
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Costa Rican Math Olympiad Problem 3 2009
Let triangle
A
B
C
ABC
A
BC
acutangle, with
m
∠
A
C
B
≤
m
∠
A
B
C
m \angle ACB\leq\ m \angle ABC
m
∠
A
CB
≤
m
∠
A
BC
.
M
M
M
the midpoint of side
B
C
BC
BC
and
P
P
P
a point over the side
M
C
MC
MC
. Let
C
1
C_{1}
C
1
the circunference with center
C
C
C
. Let
C
2
C_{2}
C
2
the circunference with center
B
B
B
.
P
P
P
is a point of
C
1
C_{1}
C
1
and
C
2
C_{2}
C
2
. Let
X
X
X
a point on the opposite semiplane than
B
B
B
respecting with the straight line
A
P
AP
A
P
; Let
Y
Y
Y
the intersection of side
X
B
XB
XB
with
C
2
C_{2}
C
2
and
Z
Z
Z
the intersection of side
X
C
XC
XC
with
C
1
C_{1}
C
1
. Let m\angle PAX \equal{} \alpha and m\angle ABC \equal{} \beta. Find the geometric place of
X
X
X
if it satisfies the following conditions: (a) \frac {XY}{XZ} \equal{} \frac {XC \plus{} CP}{XB \plus{} BP} (b) \cos(\alpha) \equal{} AB\cdot \frac {\sin(\beta )}{AP}
4
1
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Costa Rican Math Olympiad 2009 Problem 4
Show that the number 3^{{4}^{5}} \plus{} 4^{{5}^{6}} can be expresed as the product of two integers greater than
1
0
2009
10^{2009}
1
0
2009
2
1
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Costa Rican Math Olympiad 2009 Problem 2
Prove that for that for every positive integer
n
n
n
, the smallest integer that is greater than (\sqrt {3} \plus{} 1)^{2n} is divisible by 2^{n \plus{} 1}.
5
1
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Costa Rican Math Olympiad 2009 Problem 5
Suppose the polynomial x^{n} \plus{} a_{n \minus{} 1}x^{n \minus{} 1} \plus{} ... \plus{} a_{1} \plus{} a_{0} can be factorized as (x \plus{} r_{1})(x \plus{} r_{2})...(x \plus{} r_{n}), with
r
1
,
r
2
,
.
.
.
,
r
n
r_{1}, r_{2}, ..., r_{n}
r
1
,
r
2
,
...
,
r
n
real numbers. Show that (n \minus{} 1)a_{n \minus{} 1}^{2}\geq\ 2na_{n \minus{} 2}
1
1
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Costa Rican Math Olympiad 2009 Problem 1
Let
x
x
x
and
y
y
y
positive real numbers such that (1\plus{}x)(1\plus{}y)\equal{}2. Show that xy\plus{}\frac{1}{xy}\geq\ 6