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Chile Classification NMO Juniors
2014 Chile Classification NMO Juniors
2014 Chile Classification NMO Juniors
Part of
Chile Classification NMO Juniors
Subcontests
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2014 Chile Classification / Qualifying NMO Juniors XXVI
p1. Find
5
5
5
different odd numbers such that the product of any two of them are multiples each of the others. p2. The following game is played on the
3
×
3
3\times 3
3
×
3
board of the figure. A move allowed is to choose one of the squares and change color (black to white, white to black) all those that are still attached to it, either diagonally or sharing one side (the chosen square does not change color). Determine if possible, through movements allowed, make all the squares in the given figure the same color. https://cdn.artofproblemsolving.com/attachments/f/d/151203e973a60bf73ff2e24c8e5ad18b44ae6a.png p3. In an equilateral triangle
A
B
C
ABC
A
BC
with side
2
2
2
, side
A
B
AB
A
B
is extended to a point
D
D
D
so that
B
B
B
is the midpoint of
A
D
AD
A
D
. Let
E
E
E
be the point on
A
C
AC
A
C
such that
∠
A
D
E
=
1
5
o
\angle ADE = 15^o
∠
A
D
E
=
1
5
o
and take a point
F
F
F
on
A
B
AB
A
B
so that
∣
E
F
∣
=
∣
E
C
∣
|EF| = |EC|
∣
EF
∣
=
∣
EC
∣
. Determine the area of the triangle
A
F
E
AFE
A
FE
. p4. For each positive integer
n
n
n
we consider
S
(
n
)
S (n)
S
(
n
)
as the sum of its digits. For example
S
(
1234
)
=
1
+
2
+
3
+
4
=
10
S (1234) = 1 + 2 + 3 + 4 = 10
S
(
1234
)
=
1
+
2
+
3
+
4
=
10
. Calculate
S
(
1
)
−
S
(
2
)
+
S
(
3
)
−
S
(
4
)
+
.
.
.
−
S
(
202
)
+
S
(
203
)
S (1)- S (2) + S (3)- S (4) +...- S (202) + S (203)
S
(
1
)
−
S
(
2
)
+
S
(
3
)
−
S
(
4
)
+
...
−
S
(
202
)
+
S
(
203
)
p5. The four code words
□
∗
⊗
⊕
×
∙
∗
□
∙
⊕
⋄
⊕
\Box * \otimes \,\,\, \oplus \times \bullet \,\,\, * \Box \bullet\,\,\, \oplus \diamond \oplus
□
∗
⊗
⊕
×
∙
∗
□
∙
⊕
⋄
⊕
They are, in some order
A
M
O
S
U
R
R
E
O
M
A
S
AMO \,\,\, SUR \,\,\, REO \,\,\, MAS
A
MO
S
U
R
REO
M
A
S
Decipher
⊗
⋄
□
∗
⊕
×
□
∙
⊕
\otimes \diamond \Box * \oplus \times \Box \bullet \oplus
⊗
⋄
□
∗
⊕
×
□
∙
⊕
p6. Consider a parallelogram
A
B
C
D
ABCD
A
BC
D
such that the angle
∠
D
A
B
\angle DAB
∠
D
A
B
is acute. Let
G
G
G
be a point on the line
A
B
AB
A
B
different from
B
B
B
and such that
∣
B
C
∣
=
∣
G
C
∣
|BC| = |GC|
∣
BC
∣
=
∣
GC
∣
, and let
H
H
H
be a point on the line
B
C
BC
BC
different from
B
B
B
and such that
∣
A
B
∣
=
∣
A
H
∣
|AB| = |AH|
∣
A
B
∣
=
∣
A
H
∣
. Prove that the triangle
G
D
H
GDH
G
DH
is isosceles.PS. Juniors P3, P4 were also proposed as [url=https://artofproblemsolving.com/community/c4h2693874p23392747]Seniors P3, harder P4.