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Problems
Contests
National and Regional Contests
Costa Rica Contests
Costa Rica - Final Round
2018 Costa Rica - Final Round
2018 Costa Rica - Final Round
Part of
Costa Rica - Final Round
Subcontests
(21)
G5
1
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circle tangent to 3 semicircles, 2 ext. tangent and inside the third one
In the accompanying figure, semicircles with centers
A
A
A
and
B
B
B
have radii
4
4
4
and
2
2
2
, respectively. Furthermore, they are internally tangent to the circle of diameter
P
Q
PQ
PQ
. Also the semicircles with centers
A
A
A
and
B
B
B
are externally tangent to each other. The circle with center
C
C
C
is internally tangent to the semicircle with diameter
P
Q
PQ
PQ
and externally tangent to the others two semicircles. Determine the value of the radius of the circle with center
C
C
C
. https://cdn.artofproblemsolving.com/attachments/c/b/281b335f6a2d6230a5b79060e6d85d6ca6f06c.png
G2
1
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BF = CG wanted, BD = EC given, line // angle bisector
Consider
△
A
B
C
\vartriangle ABC
△
A
BC
, with
A
D
AD
A
D
bisecting
∠
B
A
C
\angle BAC
∠
B
A
C
,
D
D
D
on segment
B
C
BC
BC
. Let
E
E
E
be a point on
B
C
BC
BC
, such that
B
D
=
E
C
BD = EC
B
D
=
EC
. Through
E
E
E
we draw the line
ℓ
\ell
ℓ
parallel to
A
D
AD
A
D
and consider a point
P
P
P
on it and inside the
△
A
B
C
\vartriangle ABC
△
A
BC
. Let
G
G
G
be the point where line
B
P
BP
BP
cuts side
A
C
AC
A
C
and let F be the point where line
C
P
CP
CP
to side
A
B
AB
A
B
. Show that
B
F
=
C
G
BF = CG
BF
=
CG
.
G1
1
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3 equal incircles wanted
Let
O
O
O
be the center of the circle circumscribed to
△
A
B
C
\vartriangle ABC
△
A
BC
, and let
P
P
P
be any point on
B
C
BC
BC
(
P
≠
B
P \ne B
P
=
B
and
P
≠
C
P \ne C
P
=
C
). Suppose that the circle circumscribed to
△
B
P
O
\vartriangle BPO
△
BPO
intersects
A
B
AB
A
B
at
R
R
R
(
R
≠
A
R \ne A
R
=
A
and
R
≠
B
R \ne B
R
=
B
) and that the circle circumscribed to
△
C
O
P
\vartriangle COP
△
COP
intersects
C
A
CA
C
A
at point
Q
Q
Q
(
Q
≠
C
Q \ne C
Q
=
C
and
Q
≠
A
Q \ne A
Q
=
A
). 1) Show that
△
P
Q
R
∼
△
A
B
C
\vartriangle PQR \sim \vartriangle ABC
△
PQR
∼
△
A
BC
and that
O
O
O
is orthocenter of
△
P
Q
R
\vartriangle PQR
△
PQR
. 2) Show that the circles circumscribed to the triangles
△
B
P
O
\vartriangle BPO
△
BPO
,
△
C
O
P
\vartriangle COP
△
COP
, and
△
P
Q
R
\vartriangle PQR
△
PQR
all have the same radius.
LRP5
1
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probability of having a repeated flag is 50%, each postcard 2/12 flags
The Matini company released a special album with the flags of the
12
12
12
countries that compete in the CONCACAM Mathematics Cup. Each postcard envelope has two flags chosen randomly. Determine the minimum number of envelopes that need to be opened to that the probability of having a repeated flag is
50
%
50\%
50%
.
LRP4
1
Hide problems
sum of numbers in blue boxes cannot be prime in a 30x30 board
On a
30
×
30
30\times 30
30
×
30
board both rows
1
1
1
to
30
30
30
and columns are numbered, in addition, to each box is assigned the number
i
j
ij
ij
, where the box is in row
i
i
i
and column
j
j
j
.
N
N
N
columns and
m
m
m
rows are chosen, where
1
<
n
1 <n
1
<
n
and
m
<
30
m <30
m
<
30
, and the cells that are simultaneously in any of the rows and in any of the selected columns are painted blue. They paint the others red . (a) Prove that the sum of the numbers in the blue boxes cannot be prime. (b) Can the sum of the numbers in the red cells be prime?
LRP3
1
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Jordan in center of circle, wants to get out, a giant changes his direction
Jordan is in the center of a circle whose radius is
100
100
100
meters and can move one meter at a time, however, there is a giant who at every step can force you to move in the opposite direction to the one he chose (it does not mean returning to the place of departure, but advance but in the opposite direction to the chosen one). Determine the minimum number of steps that Jordan must give to get out of the circle.
LRP1
1
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sums from 1 to a, first reaching b wins, first player without winning strategy
Arnulfo and Berenice play the following game: One of the two starts by writing a number from
1
1
1
to
30
30
30
, the other chooses a number from
1
1
1
to
30
30
30
and adds it to the initial number, the first player chooses a number from
1
1
1
to
30
30
30
and adds it to the previous result, they continue doing the same until someone manages to add
2018
2018
2018
. When Arnulfo was about to start, Berenice told him that it was unfair, because whoever started had a winning strategy, so the numbers had better change. So they asked the following question: Adding chosen numbers from
1
1
1
to
a
a
a
, until reaching the number
b
b
b
, what conditions must meet
a
a
a
and
b
b
b
so that the first player does not have a winning strategy? Indicate if Arnulfo and Berenice are right and answer the question asked by them.
N4
1
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d is prime if prime p = 10^{d -1} + 10^{d-2} + ...+ 10 + 1
Let
p
p
p
be a prime number such that
p
=
1
0
d
−
1
+
1
0
d
−
2
+
.
.
.
+
10
+
1
p = 10^{d -1} + 10^{d-2} + ...+ 10 + 1
p
=
1
0
d
−
1
+
1
0
d
−
2
+
...
+
10
+
1
. Show that
d
d
d
is a prime.
N3
1
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a-b square if 2a^2 + a = 3b^2 + b OLCOMA Costa Rica 2018 SL N3
Let
a
a
a
and
b
b
b
be positive integers such that
2
a
2
+
a
=
3
b
2
+
b
2a^2 + a = 3b^2 + b
2
a
2
+
a
=
3
b
2
+
b
. Prove that
a
−
b
a-b
a
−
b
is a perfect square.
N2
1
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diophantine (c-1) (ab- b -a) = a + b-2
Determine all triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of nonnegative integers that satisfy:
(
c
−
1
)
(
a
b
−
b
−
a
)
=
a
+
b
−
2
(c-1) (ab- b -a) = a + b-2
(
c
−
1
)
(
ab
−
b
−
a
)
=
a
+
b
−
2
N1
1
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2 sets of consecutive exist with sum 100
Prove that there are only two sets of consecutive positive integers that satisfy that the sum of its elements is equal to
100
100
100
.
F3
1
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f(2018) =? f has period 5, f(x)=x^2 when x \in [-2,3]
Consider a function
f
:
R
→
R
f: R \to R
f
:
R
→
R
that fulfills the following two properties:
f
f
f
is periodic of period
5
5
5
(that is, for all
x
∈
R
x\in R
x
∈
R
,
f
(
x
+
5
)
=
f
(
x
)
f (x + 5) = f (x)
f
(
x
+
5
)
=
f
(
x
)
), and by restricting
f
f
f
to the interval
[
−
2
,
3
]
[-2,3]
[
−
2
,
3
]
,
f
f
f
coincides to
x
2
x^2
x
2
. Determine the value of
f
(
2018
)
.
f(2018).
f
(
2018
)
.
F2
1
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f (n, m) = f (n-1, m-1) + f (n-1, m-2) + f (n-2, m-1) + f (n-2, m-2) wanted
Consider
f
(
n
,
m
)
f (n, m)
f
(
n
,
m
)
the number of finite sequences of
1
1
1
's and
0
0
0
's such that each sequence that starts at
0
0
0
, has exactly n
0
0
0
's and
m
m
m
1
1
1
's, and there are not three consecutive
0
0
0
's or three
1
1
1
's. Show that if
m
,
n
>
1
m, n> 1
m
,
n
>
1
, then
f
(
n
,
m
)
=
f
(
n
−
1
,
m
−
1
)
+
f
(
n
−
1
,
m
−
2
)
+
f
(
n
−
2
,
m
−
1
)
+
f
(
n
−
2
,
m
−
2
)
f (n, m) = f (n-1, m-1) + f (n-1, m-2) + f (n-2, m-1) + f (n-2, m-2)
f
(
n
,
m
)
=
f
(
n
−
1
,
m
−
1
)
+
f
(
n
−
1
,
m
−
2
)
+
f
(
n
−
2
,
m
−
1
)
+
f
(
n
−
2
,
m
−
2
)
A1
1
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min (2x^2 + 98)/(x + 7)^2 OLCOMA Costa Rica 2018 SL A1
If
x
∈
R
−
{
−
7
}
x \in R-\{-7\}
x
∈
R
−
{
−
7
}
, determine the smallest value of the expression
2
x
2
+
98
(
x
+
7
)
2
\frac{2x^2 + 98}{(x + 7)^2}
(
x
+
7
)
2
2
x
2
+
98
A2
1
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x^2-8x+20=2\sqrt{x^2-8x+30 OLCOMA Costa Rica 2018 SL A2
Determine the sum of the real roots of the equation
x
2
−
8
x
+
20
=
2
x
2
−
8
x
+
30
x^2-8x+20=2\sqrt{x^2-8x+30}
x
2
−
8
x
+
20
=
2
x
2
−
8
x
+
30
6
1
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determine the height that the liquid reaches inside the pyramid
The four faces of a right triangular pyramid are equilateral triangles whose edge measures
3
3
3
dm. Suppose the pyramid is hollow, resting on one of its faces at a horizontal surface (see attached figure) and that there is
2
2
2
dm
3
^3
3
of water inside. Determine the height that the liquid reaches inside the pyramid. https://cdn.artofproblemsolving.com/attachments/0/7/6cd6e1077620371e56ed57d19fd3d05a58904e.png
3
1
Hide problems
computational geo, tangent related
In the attached figure, point
C
C
C
is the center of the circle,
A
B
AB
A
B
is tangent to the circle,
P
−
C
−
P
′
P-C-P'
P
−
C
−
P
′
and
A
C
⊥
P
P
′
AC\perp PP'
A
C
⊥
P
P
′
. If
A
T
=
2
AT = 2
A
T
=
2
cm. and
A
B
=
4
AB = 4
A
B
=
4
cm, calculate
B
Q
BQ
BQ
https://cdn.artofproblemsolving.com/attachments/e/e/d47429b82fb87299c40f5224489313909cfd0f.png 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
.
4
1
Hide problems
10^{f (n)} <10n + 1 <10^{f (n) +1}
Determine if there exists a function f:
N
∗
→
N
∗
N^*\to N^*
N
∗
→
N
∗
that satisfies that for all
n
∈
N
∗
n \in N^*
n
∈
N
∗
,
1
0
f
(
n
)
<
10
n
+
1
<
1
0
f
(
n
)
+
1
.
10^{f (n)} <10n + 1 <10^{f (n) +1}.
1
0
f
(
n
)
<
10
n
+
1
<
1
0
f
(
n
)
+
1
.
Justify your answer.Note:
N
∗
N^*
N
∗
denotes the set of positive integers.
5
1
Hide problems
M = (a + b)^2-ab is a multiple of 5 for even a,b
Let
a
a
a
and
b
b
b
be even numbers, such that
M
=
(
a
+
b
)
2
−
a
b
M = (a + b)^2-ab
M
=
(
a
+
b
)
2
−
ab
is a multiple of
5
5
5
. Consider the following statements: I) The unit digits of
a
3
a^3
a
3
and
b
3
b^3
b
3
are different. II)
M
M
M
is divisible by
100
100
100
. Please indicate which of the above statements are true with certainty.
2
1
Hide problems
max of x+y given sums with 6 sums with 4 numbers
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
, and
d
d
d
be real numbers. The six sums of two numbers
x
x
x
and
y
y
y
, different from the previous four, are
117
117
117
,
510
510
510
,
411
411
411
,
252
252
252
, in no particular order. Determine the maximum possible value of
x
+
y
x + y
x
+
y
.
1
1
Hide problems
probability of 2 segments from 10 points on a circle intersect
There are
10
10
10
points on a circle and all possible segments are drawn on the which two of these points are the endpoints. Determine the probability that selecting two segments randomly, they intersect at some point (it could be on the circumference).