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Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
1997 Chile National Olympiad
1997 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(7)
4
1
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triangular domino game (1997 Chile NMO P4)
The triangular domino is a game that uses the tokens shown below, with equilateral triangle shape with side
1
1
1
. The idea of the game is to construct an equilateral triangle with side
n
n
n
, no gaps, following the rules of the domino or classic.
∙
\bullet
∙
Show that the sum
S
S
S
of the values corresponding to the edges that are part of the sides of the greater triangle, it depends only on n, and not on the way in which the tokens are paired.
∙
\bullet
∙
For each value of
n
n
n
, calculate
S
S
S
. https://cdn.artofproblemsolving.com/attachments/e/9/898664fac380725a7398dfe470298a90b8c69b.png
2
1
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a^1 + a^2 +...+ a^n = 1 mod 10 (1997 Chile NMO P2)
Given integers
a
>
0
a> 0
a
>
0
,
n
>
0
n> 0
n
>
0
, suppose that
a
1
+
a
2
+
.
.
.
+
a
n
≡
1
m
o
d
10
a^1 + a^2 +...+ a^n \equiv 1 \mod 10
a
1
+
a
2
+
...
+
a
n
≡
1
mod
10
. Prove that
a
≡
n
≡
1
m
o
d
10
a \equiv n \equiv 1 \mod 10
a
≡
n
≡
1
mod
10
.
6
1
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min no of points, set of 1997 planes with at least 3 points (1997 Chile NMO P6)
For each set
C
C
C
of points in space, we designate by
P
C
P_C
P
C
the set of planes containing at least three points of
C
C
C
.
∙
\bullet
∙
Prove that there exists
C
C
C
such that
ϕ
(
P
C
)
=
1997
\phi (P_C) = 1997
ϕ
(
P
C
)
=
1997
, where
ϕ
\phi
ϕ
corresponds to the cardinality.
∙
\bullet
∙
Determine the least number of points that
C
C
C
must have so that the previous property can be fulfilled.
7
1
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7 courses for students (1997 Chile NMO P7)
In a career in mathematics,
7
7
7
courses are taught, among which students can choose the ones you want. Determine the number of students in the career, knowing that:
∙
\bullet
∙
No two students have chosen the same courses.
∙
\bullet
∙
Any two students have at least one course in common.
∙
\bullet
∙
If the race had one more student, it would not be possible to do both.
1
1
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integer grades for Lautaro, Camilo and Rafael (1997 Chile NMO P1)
Lautaro, Camilo and Rafael give the same exams. Each note is a positive integer. Camilo was the first in physics. Lautaro obtained a total score of
20
20
20
, Camilo, a total of
10
10
10
and Rafael, a total of
9
9
9
. Among all the tests, there were no two scores that were repeated. Determine how many They took exams, and who was second in math.
5
1
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concurrency wanted. 3 circles intersecting in pairs given (Chile 1997 P5)
Let:
C
1
,
C
2
,
C
3
C_1, C_2, C_3
C
1
,
C
2
,
C
3
three circles , intersecting in pairs, such that the secant line common to two of them (any) passes through the center of the third. Prove that the three lines thus defined are concurrent.
3
1
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ABCD is a trapezoid, areas related (Chile 1997 P3)
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral, whose diagonals intersect at
O
O
O
. The triangles
△
A
O
B
\triangle AOB
△
A
OB
,
△
B
O
C
\triangle BOC
△
BOC
,
△
C
O
D
\triangle COD
△
CO
D
have areas
1
,
2
,
4
1, 2, 4
1
,
2
,
4
, respectively. Find the area of
△
A
O
D
\triangle AOD
△
A
O
D
and prove that
A
B
C
D
ABCD
A
BC
D
is a trapezoid.