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Chile Classification NMO Juniors
2010 Chile Classification NMO Juniors
2010 Chile Classification NMO Juniors
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Chile Classification NMO Juniors
Subcontests
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2010 Chile Classification / Qualifying NMO Juniors XXII
p1. Determine what is the smallest positive integer by which the number
2010
2010
2010
must be multiplied so that it is a perfect square. p2. Consider a chessboard of
8
×
8
8\times 8
8
×
8
. An Alba trajectory is defined as any movement that contains
8
8
8
white squares, one by one, that touch in a vertex. For example, the white square diagonal is an Alba trajectory. Determine all possible Alba trajectories that can be built on the board. p3. Find all prime numbers
p
p
p
such that
p
+
2
p + 2
p
+
2
and
2
p
+
5
2p + 5
2
p
+
5
are also prime. p4. Find all natural numbers
n
n
n
such that
1
/
n
1 / n
1/
n
has a decimal representation finite. p5. Prove that all numbers of the form
5
n
5^n
5
n
, with a positive integer
n
n
n
, can be written as the sum of two squares of integer numbers. p6. Let
A
B
C
D
ABCD
A
BC
D
be a square with side
a
a
a
,
N
N
N
be the midpoint of side
B
C
BC
BC
and
M
M
M
be a point on
C
D
CD
C
D
such that
M
C
=
2
M
D
MC = 2 MD
MC
=
2
M
D
. Let
P
P
P
be the intersection point of lines
A
M
AM
A
M
and
D
B
DB
D
B
,
Q
Q
Q
be the intersection point of lines
A
N
AN
A
N
and
B
D
BD
B
D
. Calculate the area of the pentagon
M
P
Q
N
C
MPQNC
MPQNC
in terms of
a
a
a
.PS. Juniors P2 was also proposed as [url=https://artofproblemsolving.com/community/c4h2692797p23378693]Seniors P2.