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Contests
National and Regional Contests
Peru Contests
Peru Iberoamerican Team Selection Test
2009 Peru Iberoamerican Team Selection Test
2009 Peru Iberoamerican Team Selection Test
Part of
Peru Iberoamerican Team Selection Test
Subcontests
(6)
P6
1
Hide problems
Prove that the segment s_0 exists
Let
P
P
P
be a set of
n
≥
2
n \ge 2
n
≥
2
distinct points in the plane, which does not contain any triplet of aligned points. Let
S
S
S
be the set of all segments whose endpoints are points of
P
P
P
. Given two segments
s
1
,
s
2
∈
S
s_1, s_2 \in S
s
1
,
s
2
∈
S
, we write
s
1
⊗
s
2
s_1 \otimes s_2
s
1
⊗
s
2
if the intersection of
s
1
s_1
s
1
with
s
2
s_2
s
2
is a point other than the endpoints of
s
1
s_1
s
1
and
s
2
s_2
s
2
. Prove that there exists a segment
s
0
∈
S
s_0 \in S
s
0
∈
S
such that the set
{
s
∈
S
∣
s
0
⊗
s
}
\{s \in S | s_0 \otimes s\}
{
s
∈
S
∣
s
0
⊗
s
}
has at least
1
15
(
n
−
2
2
)
\frac{1}{15}\dbinom{n-2}{2}
15
1
(
2
n
−
2
)
elements
P5
1
Hide problems
2^n does not divide a^k+b^k+c^k
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive integers whose greatest common divisor is
1
1
1
. Determine whether there always exists a positive integer
n
n
n
such that, for every positive integer
k
k
k
, the number
2
n
2^n
2
n
is not a divisor of
a
k
+
b
k
+
c
k
a^k+b^k+c^k
a
k
+
b
k
+
c
k
.
P4
1
Hide problems
Prove that BN<IM
Let
A
B
C
ABC
A
BC
be a triangle such that
A
B
<
B
C
AB < BC
A
B
<
BC
. Plot the height
B
H
BH
B
H
with
H
H
H
in
A
C
AC
A
C
. Let I be the incenter of triangle
A
B
C
ABC
A
BC
and
M
M
M
the midpoint of
A
C
AC
A
C
. If line
M
I
MI
M
I
intersects
B
H
BH
B
H
at point
N
N
N
, prove that
B
N
<
I
M
BN < IM
BN
<
I
M
.
P3
1
Hide problems
How to make the sum of areas a minimum?
Let
M
,
N
,
P
M, N, P
M
,
N
,
P
be the midpoints of the sides
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
of a triangle
A
B
C
ABC
A
BC
. Let
X
X
X
be a fixed point inside the triangle
M
N
P
MNP
MNP
. The lines
L
1
,
L
2
,
L
3
L_1, L_2, L_3
L
1
,
L
2
,
L
3
that pass through point
X
X
X
are such that
L
1
L_1
L
1
intersects segment
A
B
AB
A
B
at point
C
1
C_1
C
1
and segment
A
C
AC
A
C
at point
B
2
B_2
B
2
;
L
2
L_2
L
2
intersects segment
B
C
BC
BC
at point
A
1
A_1
A
1
and segment
B
A
BA
B
A
at point
C
2
C_2
C
2
;
L
3
L_3
L
3
intersects segment
C
A
CA
C
A
at point
B
1
B_1
B
1
and segment
C
B
CB
CB
at point
A
2
A_2
A
2
. Indicates how to construct the lines
L
1
,
L
2
,
L
3
L_1, L_2, L_3
L
1
,
L
2
,
L
3
in such a way that the sum of the areas of the triangles
A
1
A
2
X
,
B
1
B
2
X
A_1A_2X, B_1B_2X
A
1
A
2
X
,
B
1
B
2
X
and
C
1
C
2
X
C_1C_2X
C
1
C
2
X
is a minimum.
P2
1
Hide problems
A magician and his assistant
A magician and his assistant perform in front of an audience of many people. On the stage there is an
8
8
8
×
8
8
8
board, the magician blindfolds himself, and then the assistant goes inviting people from the public to write down the numbers
1
,
2
,
3
,
4
,
.
.
.
,
64
1, 2, 3, 4, . . . , 64
1
,
2
,
3
,
4
,
...
,
64
in the boxes they want (one number per box) until completing the
64
64
64
numbers. After the assistant covers two adyacent boxes, at her choice. Finally, the magician removes his blindfold and has to
“
g
u
e
s
s
”
“guess”
“
gu
ess
”
what number is in each square that the assistant. Explain how they put together this trick.
C
l
a
r
i
f
i
c
a
t
i
o
n
:
Clarification:
Cl
a
r
i
f
i
c
a
t
i
o
n
:
Two squares are adjacent if they have a common side
P1
1
Hide problems
Prove that the set P is infinite
A set
P
P
P
has the following property: “For any positive integer
k
k
k
, if
p
p
p
is a prime factor of
k
3
+
6
k^3+6
k
3
+
6
, then
p
p
p
belongs to
P
P
P
”. Prove that
P
P
P
is infinite.