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National and Regional Contests
Chile Contests
Chile Classification NMO Juniors
1999 Chile Classification NMO Juniors
1999 Chile Classification NMO Juniors
Part of
Chile Classification NMO Juniors
Subcontests
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1999 Chile Classification / Qualifying NMO Juniors XI
p1. Prove that in every group of
n
n
n
people (
n
>
2
n> 2
n
>
2
), there are always at least
2
2
2
people who they have the same number of friends within the group. [url=https://artofproblemsolving.com/community/c4h1846780p12438053]p2. Given a parallelogram
A
B
C
D
ABCD
A
BC
D
, let
E
,
F
E, F
E
,
F
the midpoints of sides
B
C
BC
BC
and
C
D
CD
C
D
, respectively. Prove that
A
E
AE
A
E
and
A
F
AF
A
F
trisect
B
D
BD
B
D
. p3. Calculate the value of:
(
1
+
2
)
(
1
+
2
2
)
(
1
+
2
4
)
(
1
+
2
8
)
.
.
.
(
1
+
2
2
1999
)
(1 + 2) (1 + 2^2) (1 + 2^4) (1 + 2^8)... (1 + 2^{2^{1999}})
(
1
+
2
)
(
1
+
2
2
)
(
1
+
2
4
)
(
1
+
2
8
)
...
(
1
+
2
2
1999
)
p4. Consider the following ordering of the first
n
n
n
positive integers:
1
(
)
2
(
)
3
(
)
.
.
.
(
)
n
1(\,\,\,) 2(\,\,\,) 3 (\,\,\,)\, ... \,(\,\,\,)n
1
(
)
2
(
)
3
(
)
...
(
)
n
Is it possible to place a sign
+
+
+
or a sign
−
-
−
in the place of each
(
)
(\,\,\,)
(
)
, so that when performing the operation, the resulting result is zero as the result? p5. For the rectangles of the figure it is known that
a
b
=
1
3
\frac{a}{b}=\frac13
b
a
=
3
1
. Determine the value of the ratio
A
r
e
a
△
B
E
C
A
r
e
a
△
D
C
F
\frac{Area_{\vartriangle BEC}}{Area_{\vartriangle DCF}}
A
re
a
△
D
CF
A
re
a
△
BEC
https://cdn.artofproblemsolving.com/attachments/a/2/4ba7e41affb58009a55393c59a00b330f7a9aa.png p6. On an even-sided square board,
2
×
1
2\times 1
2
×
1
rectangular tokens are placed. Show that if we fill the board with the tokens without leaving and without overlapping tokens, then the tokens arranged horizontally they must be an even quantity.