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Problems
Contests
International Contests
IberoAmerican
2022 Iberoamerican
2022 Iberoamerican
Part of
IberoAmerican
Subcontests
(6)
4
1
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Process combo again
Let
n
>
2
n> 2
n
>
2
be a positive integer. Given is a horizontal row of
n
n
n
cells where each cell is painted blue or red. We say that a block is a sequence of consecutive boxes of the same color. Arepito the crab is initially standing at the leftmost cell. On each turn, he counts the number
m
m
m
of cells belonging to the largest block containing the square he is on, and does one of the following:If the square he is on is blue and there are at least
m
m
m
squares to the right of him, Arepito moves
m
m
m
squares to the right;If the square he is in is red and there are at least
m
m
m
squares to the left of him, Arepito moves
m
m
m
cells to the left; In any other case, he stays on the same square and does not move any further. For each
n
n
n
, determine the smallest integer
k
k
k
for which there is an initial coloring of the row with
k
k
k
blue cells, for which Arepito will reach the rightmost cell.
5
1
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Interesting geo with tangents, external bisector and many equal segments
Let
A
B
C
ABC
A
BC
be an acute triangle with circumcircle
Γ
\Gamma
Γ
. Let
P
P
P
and
Q
Q
Q
be points in the half plane defined by
B
C
BC
BC
containing
A
A
A
, such that
B
P
BP
BP
and
C
Q
CQ
CQ
are tangents to
Γ
\Gamma
Γ
and
P
B
=
B
C
=
C
Q
PB = BC = CQ
PB
=
BC
=
CQ
. Let
K
K
K
and
L
L
L
be points on the external bisector of the angle
∠
C
A
B
\angle CAB
∠
C
A
B
, such that
B
K
=
B
A
,
C
L
=
C
A
BK = BA, CL = CA
B
K
=
B
A
,
C
L
=
C
A
. Let
M
M
M
be the intersection point of the lines
P
K
PK
P
K
and
Q
L
QL
Q
L
. Prove that
M
K
=
M
L
MK=ML
M
K
=
M
L
.
6
1
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f(a)f(a+b)-ab is a square
Find all functions
f
:
N
→
N
f:\mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
, such that
f
(
a
)
f
(
a
+
b
)
−
a
b
f(a)f(a+b)-ab
f
(
a
)
f
(
a
+
b
)
−
ab
is a perfect square for all
a
,
b
∈
N
a, b \in \mathbb{N}
a
,
b
∈
N
.
3
1
Hide problems
Bounded function in (0,1)
Find all functions
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
such that
f
(
y
f
(
x
)
)
+
f
(
x
−
1
)
=
f
(
x
)
f
(
y
)
f(yf(x))+f(x-1)=f(x)f(y)
f
(
y
f
(
x
))
+
f
(
x
−
1
)
=
f
(
x
)
f
(
y
)
and
∣
f
(
x
)
∣
<
2022
|f(x)|<2022
∣
f
(
x
)
∣
<
2022
for all
0
<
x
<
1
0<x<1
0
<
x
<
1
.
2
1
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NT game of writing digits
Let
S
=
{
13
,
133
,
⋯
}
S=\{13, 133, \cdots\}
S
=
{
13
,
133
,
⋯
}
be the set of the positive integers of the form
133
⋯
3
133 \cdots 3
133
⋯
3
. Consider a horizontal row of
2022
2022
2022
cells. Ana and Borja play a game: they alternatively write a digit on the leftmost empty cell, starting with Ana. When the row is filled, the digits are read from left to right to obtain a
2022
2022
2022
-digit number
N
N
N
. Borja wins if
N
N
N
is divisible by a number in
S
S
S
, otherwise Ana wins. Find which player has a winning strategy and describe it.
1
1
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Equilateral triangle geo
Given is an equilateral triangle
A
B
C
ABC
A
BC
with circumcenter
O
O
O
. Let
D
D
D
be a point on to minor arc
B
C
BC
BC
of its circumcircle such that
D
B
>
D
C
DB>DC
D
B
>
D
C
. The perpendicular bisector of
O
D
OD
O
D
meets the circumcircle at
E
,
F
E, F
E
,
F
, with
E
E
E
lying on the minor arc
B
C
BC
BC
. The lines
B
E
BE
BE
and
C
F
CF
CF
meet at
P
P
P
. Prove that
P
D
⊥
B
C
PD \perp BC
P
D
⊥
BC
.