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Problems
Contests
National and Regional Contests
Peru Contests
Peru Cono Sur TST
2003 Peru Cono Sur TST
2003 Peru Cono Sur TST
Part of
Peru Cono Sur TST
Subcontests
(4)
P4
1
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Distance between two tiles in 8x8
Eight tiles are located on an
8
×
8
8\times 8
8
×
8
board in such a way that no pair of them are in the same row or in the same column. Prove that, among the distances between each pair of tiles, we can find two of them that are equal (the distance between two tiles is the distance between the centers of the squares in which they are located).
P3
1
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Spiral center?
Let
M
M
M
and
N
N
N
be points on the side
B
C
BC
BC
of a triangle
A
B
C
ABC
A
BC
such that
B
M
=
C
N
BM = CN
BM
=
CN
(
M
M
M
lies between
B
B
B
and
N
N
N
). Points
P
P
P
and
Q
Q
Q
lie on
A
N
AN
A
N
and
A
M
AM
A
M
respectively, so that
∠
P
M
C
=
∠
M
A
B
\angle PMC =\angle MAB
∠
PMC
=
∠
M
A
B
and
∠
Q
N
B
=
∠
N
A
C
\angle QNB = \angle NAC
∠
QNB
=
∠
N
A
C
. Prove that
∠
Q
B
C
=
∠
P
C
B
\angle QBC = \angle PCB
∠
QBC
=
∠
PCB
.
P2
1
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Sum of p perfect squares
Let
p
p
p
and
n
n
n
be positive integers such that
p
p
p
is prime and
1
+
n
p
1 + np
1
+
n
p
is a perfect square. Prove that the number
n
+
1
n + 1
n
+
1
can be expressed as the sum of
p
p
p
perfect squares, where some of them can be equal.
P1
1
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Four real roots
Determine all real numbers
a
a
a
such that the equation:
x
8
+
a
x
4
+
1
=
0
x^8+ax^4+1=0
x
8
+
a
x
4
+
1
=
0
have four real roots that form an arithmetic progression.