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Contests
National and Regional Contests
Chile Contests
Chile National Olympiad
1991 Chile National Olympiad
1991 Chile National Olympiad
Part of
Chile National Olympiad
Subcontests
(6)
5
1
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infinite sum a_j/2^j if a_n Fibonacci (Chile NMO 1991 P4)
The sequence
(
a
k
)
(a_k)
(
a
k
)
,
k
>
0
k> 0
k
>
0
is Fibonacci, with
a
0
=
a
1
=
1
a_0 = a_1 = 1
a
0
=
a
1
=
1
. Calculate the value of
∑
j
=
0
∞
a
j
2
j
\sum_{j = 0}^{\infty} \frac{a_j}{2^j}
j
=
0
∑
∞
2
j
a
j
3
1
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18 dominoes in 6x6 board (Chile NMO 1991 P3)
A board of
6
×
6
6\times 6
6
×
6
is totally covered by
18
18
18
dominoes (of
2
×
1
2\times 1
2
×
1
), that is, there are no overlaps, gaps, and the tiles do not come off the board. Prove that, regardless of the arrangement of the tiles, there is always a line that divides the board into two non-empty parts, and without cutting tiles.
4
1
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2x + 3y, 9x + 5y both divisible by 17 (Chile NMO 1991 P4)
Show that the expressions
2
x
+
3
y
2x + 3y
2
x
+
3
y
,
9
x
+
5
y
9x + 5y
9
x
+
5
y
are both divisible by
17
17
17
, for the same values of
x
x
x
and
y
y
y
.
1
1
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2^x-2^y = 1 , where x,y >=0 integers (Chile NMO 1991 P1)
Determine all nonnegative integer solutions of the equation
2
x
−
2
y
=
1
2^x-2^y = 1
2
x
−
2
y
=
1
6
1
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computational in isosceles with vertex angle 36, equal angles (Chile 1991 L2 P6)
Given a triangle with
△
A
B
C
\triangle ABC
△
A
BC
, with:
∠
C
=
3
6
o
\angle C = 36^o
∠
C
=
3
6
o
and
∠
A
=
∠
B
\angle A = \angle B
∠
A
=
∠
B
. Consider the points
D
D
D
on
B
C
BC
BC
,
E
E
E
on
A
D
AD
A
D
,
F
F
F
on
B
E
BE
BE
,
G
G
G
on
D
F
DF
D
F
and
H
H
H
on
E
G
EG
EG
, so that the rays
A
D
,
B
E
,
D
F
,
E
G
,
F
H
AD, BE, DF, EG, FH
A
D
,
BE
,
D
F
,
EG
,
F
H
bisect the angles
A
,
B
,
D
,
E
,
F
A, B, D, E, F
A
,
B
,
D
,
E
,
F
respectively. It is known that
F
H
=
1
FH = 1
F
H
=
1
. Calculate
A
C
AC
A
C
.
2
1
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equiangular inscribed polygon with odd no. of sides is regular (Chile 1991 P2)
If a polygon inscribed in a circle is equiangular and has an odd number of sides, prove that it is regular.