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Problems
Contests
National and Regional Contests
Costa Rica Contests
Costa Rica - Final Round
2019 Costa Rica - Final Round
2019 Costa Rica - Final Round
Part of
Costa Rica - Final Round
Subcontests
(10)
LR3
1
Hide problems
shaded area of n-th square A_n < 2/3 wanted, sequence of squares
Consider the following sequence of squares (side
1
1
1
), in each step the central square is divided into equal parts and colored as shown in the figure: https://cdn.artofproblemsolving.com/attachments/9/0/6874ab5aecadf2112fbe4a196ab3091ab8b31a.png Square 1 Square 2 Square 3Let
A
n
A_n
A
n
with
n
∈
N
n \in N
n
∈
N
,
n
>
1
n> 1
n
>
1
be the shaded area of square
n
n
n
, show that
A
n
<
2
3
A_n <\frac23
A
n
<
3
2
LR2
1
Hide problems
3 dice game
A website offers for
1000
1000
1000
colones, the possibility of playing
4
4
4
shifts a certain game of randomly, in each turn the user will have the same probability
p
p
p
of winning the game and obtaining
1000
1000
1000
colones (per shift). But to calculate
p
p
p
he asks you to roll
3
3
3
dice and add the results, with what p will be the probability of obtaining this sum. Olcoman visits the website, and upon rolling the dice, he realizes that the probability of losing his money is from
(
103
108
)
4
\left( \frac{103}{108}\right)^4
(
108
103
)
4
. a) Determine the probability
p
p
p
that Olcoman wins a game and the possible outcomes with the dice, to get to this one. b) Which sums (with the dice) give the maximum probability of having a profit of exactly
1000
1000
1000
colones? Calculate this probability and the value of
p
p
p
for this case.
A2
1
Hide problems
2x^2 - 3xy + 2y^2 = 1, y^2 - 3yz + 4z^2 = 2, z^2 + 3zx - x^2 = 3$
Let
x
,
y
,
z
∈
R
x, y, z \in R
x
,
y
,
z
∈
R
, find all triples
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
that satisfy the following system of equations:
2
x
2
−
3
x
y
+
2
y
2
=
1
2x^2 - 3xy + 2y^2 = 1
2
x
2
−
3
x
y
+
2
y
2
=
1
y
2
−
3
y
z
+
4
z
2
=
2
y^2 - 3yz + 4z^2 = 2
y
2
−
3
yz
+
4
z
2
=
2
z
2
+
3
z
x
−
x
2
=
3
z^2 + 3zx - x^2 = 3
z
2
+
3
z
x
−
x
2
=
3
G2
1
Hide problems
angle wanted, orthocenter, circumcenter, circle (H,HA)
Let
H
H
H
be the orthocenter and
O
O
O
the circumcenter of the acute triangle
△
A
B
C
\vartriangle ABC
△
A
BC
. The circle with center
H
H
H
and radius
H
A
HA
H
A
intersects the lines
A
C
AC
A
C
and
A
B
AB
A
B
at points
P
P
P
and
Q
Q
Q
, respectively. Let point
O
O
O
be the orthocenter of triangle
△
A
P
Q
\vartriangle APQ
△
A
PQ
, determine the measure of
∠
B
A
C
\angle BAC
∠
B
A
C
.
2
1
Hide problems
area among 2 circumferences inside a parallelogram
Consider the parallelogram
A
B
C
D
ABCD
A
BC
D
, with
∠
A
B
C
=
60
\angle ABC = 60
∠
A
BC
=
60
and sides
A
B
=
3
AB =\sqrt3
A
B
=
3
,
B
C
=
1
BC = 1
BC
=
1
. Let
ω
\omega
ω
be the circle of center
B
B
B
and radius
B
A
BA
B
A
, and let
τ
\tau
τ
be the circle of center
D
D
D
and radius
D
A
DA
D
A
. Determine the area of the region between the circumferences
ω
\omega
ω
and
τ
\tau
τ
, within the parallelogram
A
B
C
D
ABCD
A
BC
D
(the area of the shaded region). https://cdn.artofproblemsolving.com/attachments/5/a/02b17ec644289d95b6fce78cb5f1ecb3d3ba5b.png
6
1
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collinear wanted, starting with right isosceles triangle
Consider the right isosceles
△
A
B
C
\vartriangle ABC
△
A
BC
at
A
A
A
. Let
L
L
L
be the intersection of the bisector of
∠
A
C
B
\angle ACB
∠
A
CB
with
A
B
AB
A
B
and
K
K
K
the intersection point of
C
L
CL
C
L
with the bisector of
B
C
BC
BC
. Let
X
X
X
be the point on line
A
K
AK
A
K
such that
∠
K
C
X
=
9
0
o
\angle KCX = 90^o
∠
K
CX
=
9
0
o
and let
Y
Y
Y
be the point of intersection of
C
X
CX
CX
with the circumcircle of
△
A
B
C
\vartriangle ABC
△
A
BC
. Let
Y
′
Y'
Y
′
the reflection of point
Y
Y
Y
wrt
B
C
BC
BC
. Prove that
B
−
K
−
Y
′
B - K -Y'
B
−
K
−
Y
′
.Notation:
A
−
B
−
C
A-B-C
A
−
B
−
C
means than points
A
,
B
,
C
A,B,C
A
,
B
,
C
are collinear in that order i.e.
B
B
B
lies between
A
A
A
and
C
C
C
.
5
1
Hide problems
p <=19 divides a_n = 2 x 10^{n + 1} + 19
We have an a sequence such that
a
n
=
2
⋅
1
0
n
+
1
+
19
a_n = 2 \cdot 10^{n + 1} + 19
a
n
=
2
⋅
1
0
n
+
1
+
19
. Determine all the primes
p
p
p
, with
p
≤
19
p \le 19
p
≤
19
, for which there exists some
n
≥
1
n \ge 1
n
≥
1
such that
p
p
p
divides
a
n
a_n
a
n
.
4
1
Hide problems
f (2019)=? g (x^2) = f (x) , f (x + 1) - f (x - 1) = x , g (1) = 0
Let
g
:
R
→
R
g: R \to R
g
:
R
→
R
be a linear function such that
g
(
1
)
=
0
g (1) = 0
g
(
1
)
=
0
. If
f
:
R
→
R
f: R \to R
f
:
R
→
R
is a quadratic function such what
g
(
x
2
)
=
f
(
x
)
g (x^2) = f (x)
g
(
x
2
)
=
f
(
x
)
and
f
(
x
+
1
)
−
f
(
x
−
1
)
=
x
f (x + 1) - f (x - 1) = x
f
(
x
+
1
)
−
f
(
x
−
1
)
=
x
for all
x
∈
R
x \in R
x
∈
R
. Determine the value of
f
(
2019
)
f (2019)
f
(
2019
)
.
3
1
Hide problems
\sqrt{xy} is an integer with the digits of n in reverse order, 2n = x + y
Let
x
,
y
x, y
x
,
y
be two positive integers, with
x
>
y
x> y
x
>
y
, such that
2
n
=
x
+
y
2n = x + y
2
n
=
x
+
y
, where n is a number two-digit integer. If
x
y
\sqrt{xy}
x
y
is an integer with the digits of
n
n
n
but in reverse order, determine the value of
x
−
y
x - y
x
−
y
1
1
Hide problems
villain hides a medal in a vertex of a regular polygon with 2019 sides
In a faraway place in the Universe, a villain has a medal with special powers and wants to hide it so that no one else can use it. For this, the villain hides it in a vertex of a regular polygon with
2019
2019
2019
sides. Olcoman, the savior of the Olcomita people, wants to get the medal to restore peace in the Universe, for which you have to pay
1000
1000
1000
olcolones for each time he makes the following move: on each turn he chooses a vertex of the polygon, which turns green if the medal is on it or in one of the four vertices closest to it, or otherwise red. Find the fewest olcolones Olcoman needs to determine with certainty the position of the medal.