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Problems
Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
1990 Chile National Olympiad
1990 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(7)
3
1
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numbering vertices of n-gon from 0 to n-1 (Chile NMO 1990 P3)
Given a polygon with
n
n
n
sides, we assign the numbers
0
,
1
,
.
.
.
,
n
−
1
0,1,...,n-1
0
,
1
,
...
,
n
−
1
to the vertices, and to each side is assigned the sum of the numbers assigned to its ends. The figure shows an example for
n
=
5
n = 5
n
=
5
. Notice that the numbers assigned to the sides are still in arithmetic progression. https://cdn.artofproblemsolving.com/attachments/c/0/975969e29a7953dcb3e440884461169557f9a7.png
∙
\bullet
∙
Make the respective assignment for a
9
9
9
-sided polygon, and generalize for odd
n
n
n
.
∙
\bullet
∙
Prove that this is not possible if
n
n
n
is even.
2
1
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odd naturals with indicator same as 1990 (Chile NMO 1990 P2)
Find all the odd naturals whose indicator is the same as
1990
1990
1990
.We clarify that, if a natural decomposes into prime factors in the form
Π
j
=
1
r
p
j
a
j
\Pi_{j=1}^r p_j^{a_j}
Π
j
=
1
r
p
j
a
j
, define the indicator as :
ϕ
(
n
)
=
r
Π
j
=
1
r
p
j
a
j
−
1
(
p
j
+
1
)
\phi (n) = r\Pi_{j=1}^r p_j^{a_j-1} (p_j + 1)
ϕ
(
n
)
=
r
Π
j
=
1
r
p
j
a
j
−
1
(
p
j
+
1
)
.[hide=official wording for first sentence]Encuentre todos los naturales impares cuyo indicador es el mismo que el de 1990.
7
1
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mastermind game in olympiad problem (Chile NMO 1990 P7)
It is about deciphering the code
C
1
C
2
C
3
C
4
C_1C_2C_3C_4
C
1
C
2
C
3
C
4
in which each letter is one of the colors: white
(
B
)
(B)
(
B
)
, blue
(
A
)
(A)
(
A
)
, red
(
R
)
(R)
(
R
)
, green
(
V
)
(V)
(
V
)
, black
(
N
)
(N)
(
N
)
and brown
(
C
)
(C)
(
C
)
with allowed repetitions. Four were made attempts to decipher it.
N
A
V
B
NAVB
N
A
V
B
and
A
C
R
C
ACRC
A
CRC
have two color hits, but in wrong places.
R
B
A
C
RBAC
RB
A
C
and
V
R
B
A
VRBA
V
RB
A
have one color match in the correct place, and two other color matches, in places incorrect. Determine all combinations compatible with the information.
5
1
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996 <= \sum_{k = 1}^{n} 1/k (Chile NMO 1990 P5)
Determine a natural
n
n
n
such that
996
≤
∑
k
=
1
n
1
k
996 \le \sum_{k = 1}^{n}\frac{1}{k}
996
≤
k
=
1
∑
n
k
1
4
1
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g (1990) =? if g (x) <=x, g (x + y) <=g (x) + g (y) (Chile NMO 1990 P4)
The function
g
g
g
, with domain and real numbers, fulfills the following:
∙
\bullet
∙
g
(
x
)
≤
x
g (x) \le x
g
(
x
)
≤
x
, for all real
x
x
x
∙
\bullet
∙
g
(
x
+
y
)
≤
g
(
x
)
+
g
(
y
)
g (x + y) \le g (x) + g (y)
g
(
x
+
y
)
≤
g
(
x
)
+
g
(
y
)
for all real
x
,
y
x,y
x
,
y
Find
g
(
1990
)
g (1990)
g
(
1990
)
.
6
1
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regular polygons with equal perimeters and doubles sides (Chile 1990 L2 P6)
Given a regular polygon with apothem
A
A
A
and circumradius
R
R
R
. Find for a regular polygon of equal perimeter and with double number of sides, the apothem
a
a
a
and the circumcircle
r
r
r
in terms of
A
,
R
A,R
A
,
R
1
1
Hide problems
any triangle can be divided into isosceles triangles (Chile 1990 L2 P1)
Show that any triangle can be subdivided into isosceles triangles.