MathDB
Problems
Contests
National and Regional Contests
Costa Rica Contests
Costa Rica - Final Round
2022 Costa Rica - Final Round
2022 Costa Rica - Final Round
Part of
Costa Rica - Final Round
Subcontests
(6)
3
1
Hide problems
2-player game, staircase with 2022 steps
Shikaku and his son Shikamaru must climb a staircase that has
2022
2022
2022
steps; the steps are listed
1
1
1
,
2
2
2
,
.
.
.
...
...
,
2022
2022
2022
and the floor is considered step
0
0
0
. This bores them both a lot, so so they decide to organize a game. They begin by tying a rope between them, so that At most they can be separated from each other by a distance of
7
7
7
steps, that is, if they are in the steps
m
m
m
and
n
n
n
, then it must always be true that
∣
m
−
n
∣
≤
7
|m-n| \le 7
∣
m
−
n
∣
≤
7
. For the game they establish the following rules: a) They move alternately in turns. b) In his corresponding turn, the player must move to a higher step than in the one that (the same) was previously. c) If a player has just moved to the
n
n
n
-th step, then on the next turn the other player cannot be moved to any of the steps
n
−
1
n-1
n
−
1
,
n
n
n
or
n
+
1
n + 1
n
+
1
, except when it is for reach the last step. d) Whoever reaches the last step (listed with
2022
2022
2022
) wins. Shikamaru is bored to start, so his father starts. Determine which of the two players has a winning strategy and describe it.
4
1
Hide problems
f(n+1) = f(n)-2n+43$
Maria was a brilliant mathematician who found the following property about her year of birth: if
f
f
f
is a function defined in the set of natural numbers
N
=
{
0
,
1
,
2
,
3
,
4
,
5
,
.
.
.
}
N = \{0, 1, 2, 3, 4, 5,...\}
N
=
{
0
,
1
,
2
,
3
,
4
,
5
,
...
}
such that
f
(
1
)
=
1335
f(1) = 1335
f
(
1
)
=
1335
and
f
(
n
+
1
)
=
f
(
n
)
−
2
n
+
43
f(n+1) = f(n)-2n+43
f
(
n
+
1
)
=
f
(
n
)
−
2
n
+
43
for all
n
∈
N
n \in N
n
∈
N
, then his year of birth is the maximum value that
f
(
n
)
f(n)
f
(
n
)
can reach when
n
n
n
takes values in
N
N
N
. Determine the year of birth of Mary.
5
1
Hide problems
good years. when N-edition divides N(N+1)
The
1
1
1
st edition of OLCOMA was organized in
1989
1989
1989
, so in
2022
2022
2022
the
34
34
34
th edition will be celebrated. Suppose that the Olympics will continue to be held annually without interruption. We say that a year
N
N
N
is good if the OLCOMA edition number of that year divides the product
N
(
N
+
1
)
N(N +1)
N
(
N
+
1
)
. For example, the year
2022
2022
2022
is good because
34
34
34
divides
2022
⋅
2023
2022 \cdot 2023
2022
⋅
2023
. Determine the last year
N
N
N
in the
21
21
21
st century,
2000
≤
N
≤
2099
2000\le N \le 2099
2000
≤
N
≤
2099
, which is good.
1
1
Hide problems
BD _|_ CE , DEFO rhombus, ABC equilateral
Let
Γ
\Gamma
Γ
be a circle with center
O
O
O
. Consider the points
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
,
E
E
E
and
F
F
F
in
Γ
\Gamma
Γ
, with
D
D
D
and
E
E
E
in the (minor) arc
B
C
BC
BC
and
C
C
C
in the (minor) arc
E
F
EF
EF
, such that
D
E
F
O
DEFO
D
EFO
is a rhombus and
△
A
B
C
\vartriangle ABC
△
A
BC
It is equilateral. Show that
B
D
↔
\overleftrightarrow{BD}
B
D
and
C
E
↔
\overleftrightarrow{CE}
CE
are perpendicular.
6
1
Hide problems
AI tangent to (BMI) if AI _|_ IN
Consider
A
B
C
ABC
A
BC
with
A
C
>
A
B
AC > AB
A
C
>
A
B
and incenter
I
I
I
. The midpoints of
B
C
‾
\overline{BC}
BC
and
A
C
‾
\overline{AC}
A
C
are
M
M
M
and
N
N
N
, respectively. If
A
I
‾
\overline{AI}
A
I
is perpendicular to
I
N
‾
\overline{IN}
I
N
, then prove that
A
I
‾
\overline{AI}
A
I
is tangent to the circumscribed circle of
△
B
M
I
\vartriangle BMI
△
BM
I
.
2
1
Hide problems
f(x) = x^3 +px^2 +qx+r , f(\sqrt3) = 0, f(s) = 506
Find all functions
f
f
f
, of the form
f
(
x
)
=
x
3
+
p
x
2
+
q
x
+
r
f(x) = x^3 +px^2 +qx+r
f
(
x
)
=
x
3
+
p
x
2
+
q
x
+
r
with
p
p
p
,
q
q
q
and
r
r
r
integers, such that
f
(
s
)
=
506
f(s) = 506
f
(
s
)
=
506
for some integer
s
s
s
and
f
(
3
)
=
0
f(\sqrt3) = 0
f
(
3
)
=
0
.