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National and Regional Contests
Costa Rica Contests
Costa Rica - Final Round
2008 Costa Rica - Final Round
2008 Costa Rica - Final Round
Part of
Costa Rica - Final Round
Subcontests
(6)
6
1
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Costa Rican math olympiad 2008 Problem 6
Let
O
O
O
be the circumcircle of a
Δ
A
B
C
\Delta ABC
Δ
A
BC
and let
I
I
I
be its incenter, for a point
P
P
P
of the plane let
f
(
P
)
f(P)
f
(
P
)
be the point obtained by reflecting
P
′
P'
P
′
by the midpoint of
O
I
OI
O
I
, with
P
′
P'
P
′
the homothety of
P
P
P
with center
O
O
O
and ratio
R
r
\frac{R}{r}
r
R
with
r
r
r
the inradii and
R
R
R
the circumradii,(understand it by \frac{OP}{OP'}\equal{}\frac{R}{r}). Let
A
1
A_1
A
1
,
B
1
B_1
B
1
and
C
1
C_1
C
1
the midpoints of
B
C
BC
BC
,
A
C
AC
A
C
and
A
B
AB
A
B
, respectively. Show that the rays
A
1
f
(
A
)
A_1f(A)
A
1
f
(
A
)
,
B
1
f
(
B
)
B_1f(B)
B
1
f
(
B
)
and
C
1
f
(
C
)
C_1f(C)
C
1
f
(
C
)
concur on the incircle.
5
1
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Costa Rican math olympiad 2008 Problem 5
Let
p
p
p
be a prime number such that p\minus{}1 is a perfect square. Prove that the equation a^{2}\plus{}(p\minus{}1)b^{2}\equal{}pc^{2} has infinite many integer solutions
a
a
a
,
b
b
b
and
c
c
c
with (a,b,c)\equal{}1
4
1
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Costa Rican math olympiad 2008 Problem 4
Let
x
x
x
,
y
y
y
and
z
z
z
be non negative reals, such that there are not two simultaneously equal to
0
0
0
. Show that \frac {x \plus{} y}{y \plus{} z} \plus{} \frac {y \plus{} z}{x \plus{} y} \plus{} \frac {y \plus{} z}{z \plus{} x} \plus{} \frac {z \plus{} x}{y \plus{} z} \plus{} \frac {z \plus{} x}{x \plus{} y} \plus{} \frac {x \plus{} y}{z \plus{} x}\geq\ 5 \plus{} \frac {x^{2} \plus{} y^{2} \plus{} z^{2}}{xy \plus{} yz \plus{} zx} and determine the equality cases.
3
1
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Costa Rican math olympiad 2008 Problem 3
Find all polinomials
P
(
x
)
P(x)
P
(
x
)
with real coefficients, such that P(\sqrt {3}(a \minus{} b)) \plus{} P(\sqrt {3}(b \minus{} c)) \plus{} P(\sqrt {3}(c \minus{} a)) \equal{} P(2a \minus{} b \minus{} c) \plus{} P( \minus{} a \plus{} 2b \minus{} c) \plus{} P( \minus{} a \minus{} b \plus{} 2c) for any
a
a
a
,
b
b
b
and
c
c
c
real numbers
2
1
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Costa Rican math olympiad 2008 Problem 2
Let
A
B
C
ABC
A
BC
be a triangle and let
P
P
P
be a point on the angle bisector
A
D
AD
A
D
, with
D
D
D
on
B
C
BC
BC
. Let
E
E
E
,
F
F
F
and
G
G
G
be the intersections of
A
P
AP
A
P
,
B
P
BP
BP
and
C
P
CP
CP
with the circumcircle of the triangle, respectively. Let
H
H
H
be the intersection of
E
F
EF
EF
and
A
C
AC
A
C
, and let
I
I
I
be the intersection of
E
G
EG
EG
and
A
B
AB
A
B
. Determine the geometric place of the intersection of
B
H
BH
B
H
and
C
I
CI
C
I
when
P
P
P
varies.
1
1
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Costa Rican math olympiad 2008 Problem 1
We want to colour all the squares of an
n
x
n
nxn
n
x
n
board of red or black. The colorations should be such that any subsquare of
2
x
2
2x2
2
x
2
of the board have exactly two squares of each color. If
n
≥
2
n\geq 2
n
≥
2
how many such colorations are possible?