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Contests
National and Regional Contests
Peru Contests
Peru Iberoamerican Team Selection Test
2014 Peru Iberoamerican Team Selection Test
2014 Peru Iberoamerican Team Selection Test
Part of
Peru Iberoamerican Team Selection Test
Subcontests
(6)
P6
1
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A simple graph G of 2014 vertices that satisfies the following conditions at the
Determine the largest positive integer
k
k
k
for which there exists a simple graph
G
G
G
of
2014
2014
2014
vertices that simultaneously satisfies the following conditions:
a
)
a)
a
)
G
G
G
does not contain triangles
b
)
b)
b
)
For each
i
i
i
between
1
1
1
and
k
k
k
, inclusive, at least one vertex of
G
G
G
has degree
i
i
i
c
)
c)
c
)
No vertex of
G
G
G
has a degree greater than
k
k
k
P4
1
Hide problems
$$x^{2014} + 2x^{2013} + 3x^{2012} + 4x^{2011} +\ldots + 2014x + 2015$$
Determine the minimum value of
x
2014
+
2
x
2013
+
3
x
2012
+
4
x
2011
+
…
+
2014
x
+
2015
x^{2014} + 2x^{2013} + 3x^{2012} + 4x^{2011} +\ldots + 2014x + 2015
x
2014
+
2
x
2013
+
3
x
2012
+
4
x
2011
+
…
+
2014
x
+
2015
where
x
x
x
is a real number.
P3
1
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2012 special numbers
A positive integer
n
n
n
is called
s
p
e
c
i
a
l
special
s
p
ec
ia
l
if there exist integers
a
>
1
a > 1
a
>
1
and
b
>
1
b > 1
b
>
1
such that
n
=
a
b
+
b
n=a^b + b
n
=
a
b
+
b
. Is there a set of
2014
2014
2014
consecutive positive integers that contains exactly
2012
2012
2012
s
p
e
c
i
a
l
special
s
p
ec
ia
l
numbers?
P2
1
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Prove that two different boards can be obtained
Let
n
≥
4
n\ge 4
n
≥
4
be an integer. You have two
n
×
n
n\times n
n
×
n
boards. Each board contains the numbers
1
1
1
to
n
2
n^2
n
2
inclusive, one number per square, arbitrarily arranged on each board. A move consists of exchanging two rows or two columns on the first board (no moves can be made on the second board). Show that it is possible to make a sequence of moves such that for all
1
≤
i
≤
n
1 \le i \le n
1
≤
i
≤
n
and
1
≤
j
≤
n
1 \le j \le n
1
≤
j
≤
n
, the number that is in the
i
−
t
h
i-th
i
−
t
h
row and
j
−
t
h
j-th
j
−
t
h
column of the first board is different from the number that is in the
i
−
t
h
i-th
i
−
t
h
row and
j
−
t
h
j-th
j
−
t
h
column of the second board.
P1
1
Hide problems
starting with intersecting circles, line passes through midpoint wanted
Circles
C
1
C_1
C
1
and
C
2
C_2
C
2
intersect at different points
A
A
A
and
B
B
B
. The straight lines tangents to
C
1
C_1
C
1
that pass through
A
A
A
and
B
B
B
intersect at
T
T
T
. Let
M
M
M
be a point on
C
1
C_1
C
1
that is out of
C
2
C_2
C
2
. The
M
T
MT
MT
line intersects
C
1
C_1
C
1
at
C
C
C
again, the
M
A
MA
M
A
line intersects again to
C
2
C_2
C
2
in
K
K
K
and the line
A
C
AC
A
C
intersects again to the circumference
C
2
C_2
C
2
in
L
L
L
. Prove that the
M
C
MC
MC
line passes through the midpoint of the
K
L
KL
K
L
segment.
P5
1
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The midpoint of the altitude lies on radical axis
The incircle
⊙
(
I
)
\odot (I)
⊙
(
I
)
of
△
A
B
C
\triangle ABC
△
A
BC
touch
A
C
AC
A
C
and
A
B
AB
A
B
at
E
E
E
and
F
F
F
respectively. Let
H
H
H
be the foot of the altitude from
A
A
A
, if
R
≡
I
C
∩
A
H
,
Q
≡
B
I
∩
A
H
R \equiv IC \cap AH, \ \ Q \equiv BI \cap AH
R
≡
I
C
∩
A
H
,
Q
≡
B
I
∩
A
H
prove that the midpoint of
A
H
AH
A
H
lies on the radical axis between
⊙
(
R
E
C
)
\odot (REC)
⊙
(
REC
)
and
⊙
(
Q
F
B
)
\odot (QFB)
⊙
(
QFB
)
I hope that this is not repost :)