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2016 Indonesia Regional
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Indonesia Regional MO 2016 Part A
Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2016 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2685861p23301381]hereTime: 90 minutes
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Write only the answers to the questions given.
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Some questions can have more than one correct answer. You are asked to provide the most correct or exact answer to a question like this. Scores will only be given to the giver of the most correct or most exact answer.
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Each question is worth 1 (one) point.
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to be more exact:
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in years 2002-08 time was 90' for part A and 120' for part B
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since years 2009 time is 210' for part A and B totally
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each problem in part A is 1 point, in part B is 7 points p1. Suppose
a
,
b
,
c
a, b, c
a
,
b
,
c
are three natural numbers that satisfy
2
a
+
2
b
+
2
c
=
100
2^a + 2^b + 2^c = 100
2
a
+
2
b
+
2
c
=
100
. The value of
a
+
b
+
c
a + b + c
a
+
b
+
c
is ... . p2. A function f has the property
f
(
65
x
+
1
)
=
x
2
−
x
+
1
f(65x+1) = x^2-x+1
f
(
65
x
+
1
)
=
x
2
−
x
+
1
for all real numbers
x
x
x
. The value of
f
(
2016
)
f (2016)
f
(
2016
)
is ... . p3. Three different numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
will be chosen one by one at random from
1
,
2
,
3
,
4
,
.
.
.
,
10
1, 2, 3, 4,..., 10
1
,
2
,
3
,
4
,
...
,
10
with respect to the order. The probability that
a
b
+
c
ab+c
ab
+
c
is even is ... p4. Point
P
P
P
is a point on the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
with
P
A
=
2
PA = 2
P
A
=
2
,
P
B
=
3
PB = 3
PB
=
3
,
P
C
=
5
PC = 5
PC
=
5
, and
P
D
=
6
PD = 6
P
D
=
6
. The maximum area of the quadrilateral
A
B
C
D
ABCD
A
BC
D
is... p5. If
0
<
x
<
π
2
0 < x <\frac{\pi}{2}
0
<
x
<
2
π
and
4
tan
x
+
9
cot
x
≤
12
4 \tan x + 9 \cot x \le 12
4
tan
x
+
9
cot
x
≤
12
, then the possible values of
sin
x
\sin x
sin
x
are ... . p6. For every natural number
n
n
n
, let
s
(
n
)
s(n)
s
(
n
)
represent the result of the sum of the digits of n in decimal writing. For example,
s
(
2016
)
=
2
+
0
+
1
+
6
=
9
s(2016) = 2+0+1+6 = 9
s
(
2016
)
=
2
+
0
+
1
+
6
=
9
. The result of the sum of all natural numbers n such that
n
+
s
(
n
)
=
2016
n + s(n) = 2016
n
+
s
(
n
)
=
2016
is ... p7. Among
30
30
30
students,
15
15
15
students liked athletics,
17
17
17
students liked basketball, and
17
17
17
students love chess. Students who enjoy athletics and basketball as much as students who love basketball and chess. A total of
8
8
8
students enjoy athletics and chess. Students who like basketball and chess are twice as much as students who love happy all three. Meanwhile,
4
4
4
students were not happy with any of the three. From the
30
30
30
students, three students were chosen randomly. Probability of each each student who is selected only likes chess or basketball is ... p8. Given a cube
A
B
C
D
:
E
F
G
H
ABCD:EFGH
A
BC
D
:
EFG
H
with side length
5
5
5
. Points
I
I
I
and
J
J
J
any point on
B
F
BF
BF
with
I
J
=
1
IJ = 1
I
J
=
1
. Any point
K
K
K
and
L
L
L
on
C
G
CG
CG
with
K
L
=
2
KL = 2
K
L
=
2
. Ants move from
A
A
A
to
H
H
H
on the path
A
I
J
K
L
H
AIJKLH
A
I
J
K
L
H
. The shortest path is... p9. The number of triple prime numbers
(
p
,
q
,
r
)
(p, q, r)
(
p
,
q
,
r
)
that satisfies
15
p
+
7
p
q
+
q
r
=
p
q
r
15p+7pq+qr = pqr
15
p
+
7
pq
+
q
r
=
pq
r
is ... p10. If
x
2
+
x
y
+
8
x
=
−
9
x^2 + xy + 8x =-9
x
2
+
x
y
+
8
x
=
−
9
and
4
y
2
+
3
x
y
+
16
y
=
−
7
4y^2 + 3xy + 16y =-7
4
y
2
+
3
x
y
+
16
y
=
−
7
, then the value of
x
+
2
y
x + 2y
x
+
2
y
maybe is ... p11. The lengths of the sides of a triangular pyramid are all integers. Its five ribs are
14
14
14
long each;
20
20
20
,
40
40
40
,
52
52
52
, and
70
70
70
. The number of possible lengths of the sixth side is... p12. A chess player plays a minimum of once every day for seven day with a total of
m
m
m
matches. Maximum value of
m
m
m
such that there are two or more consecutive days with a total of four matches is ... p13. Mr. Adi's house has a broken water meter, that cannot show numbers
3
3
3
and
9
9
9
. For example, the number shown on the meter after
22
22
22
is
24
24
24
and also the number shown after
28
28
28
is
40
40
40
. For example, in one month, Pak Adi's water meter shows
478
478
478
m
3
^3
3
. The actual loss of Mr. Adi because the broken meter is ... m
3
^3
3
. p14. The result of the sum of all the real numbers
x
x
x
that fulfil
⌊
8
x
−
1008
⌋
+
⌊
x
⌋
=
2016
\lfloor 8x-1008 \rfloor + \lfloor x \rfloor = 2016
⌊
8
x
−
1008
⌋
+
⌊
x
⌋
=
2016
is ... . p15. Suppose
a
1
,
a
2
,
.
.
.
,
a
120
a_1, a_2, ..., a_{120}
a
1
,
a
2
,
...
,
a
120
are
120
120
120
permutations of the word MEDAN which is sort alphabetically like in a dictionary, for example
a
1
=
A
D
E
M
N
a_1 = ADEMN
a
1
=
A
D
EMN
,
a
2
=
A
D
E
N
M
a_2 = ADENM
a
2
=
A
D
ENM
,
a
3
=
A
D
M
E
N
a_3 = ADMEN
a
3
=
A
D
MEN
, and so on. The result of the sum of all
k
k
k
indexes until the letter
A
A
A
is the third letter in the permutation
a
k
a_k
a
k
is ... p16. Suppose
A
B
C
D
E
ABCDE
A
BC
D
E
is a regular pentagon with area
2
2
2
. The points
P
,
Q
,
R
,
S
,
T
P, Q, R, S, T
P
,
Q
,
R
,
S
,
T
are the intersection of the diagonals of the pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
such that
P
Q
R
S
T
PQRST
PQRST
is a regular pentagon. If the area of
P
Q
R
S
T
PQRST
PQRST
is written in the form
a
−
b
a-\sqrt{b}
a
−
b
with
a
a
a
and
b
b
b
natural numbers, then the value of
a
+
b
a + b
a
+
b
is ... p17. Triangle
A
B
C
ABC
A
BC
has a circumcircle of radius
1
1
1
. If the two medians of triangle ABC each have a length of
1
1
1
, then the perimeter of the triangle
A
B
C
ABC
A
BC
is .... p18. Sequence
x
0
,
x
1
,
x
2
,
.
.
.
,
x
n
x_0, x_1, x_2, ... , x_n
x
0
,
x
1
,
x
2
,
...
,
x
n
is defined as
x
0
=
10
x_0 = 10
x
0
=
10
,
x
1
=
5
_x1 = 5
x
1
=
5
, and
x
k
+
1
=
x
k
−
1
−
1
x
k
x_{k+1} = x_{k-1}-\frac{1}{x_k}
x
k
+
1
=
x
k
−
1
−
x
k
1
for
k
=
1
,
2
,
3
,
.
.
.
,
n
−
1
k = 1, 2, 3, ..., n-1
k
=
1
,
2
,
3
,
...
,
n
−
1
and we get
x
n
=
0
x_n = 0
x
n
=
0
. The value of
n
n
n
is ... p19. In a soccer tournament involving
n
n
n
teams, each team plays against the other team exactly once. In one game,
3
3
3
points will be awarded to the winning team and
0
0
0
points to the losing team, while
1
1
1
point is given to each team when the match ends in a draw. After the match ends, only one team gets the most points and only that team gets the number of wins at least. The smallest value of
n
n
n
so that this is possible ... p20. The sequence of non-negative numbers
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3, ...
a
1
,
a
2
,
a
3
,
...
is defined with
a
1
=
1001
a_1 = 1001
a
1
=
1001
and
a
n
+
2
=
∣
a
n
+
1
−
a
n
∣
an+2 = |a_{n+1}-a_n|
an
+
2
=
∣
a
n
+
1
−
a
n
∣
for
n
≥
1
n\ge 1
n
≥
1
. If it is known that a_2 < 1001 and
a
2016
=
1
a_{2016} = 1
a
2016
=
1
, so the number of possible values of
a
2
a_2
a
2
is...
Indonesia Regional MO 2016 Part B
p1. Let
a
a
a
and
b
b
b
be different positive real numbers so that
a
+
a
b
a +\sqrt{ab}
a
+
ab
and
b
+
a
b
b +\sqrt{ab}
b
+
ab
are rational numbers. Prove that
a
a
a
and
b
b
b
are rational numbers.p2. Find the number of ordered pairs of natural numbers
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
that satisfy
a
b
+
b
c
+
c
d
+
d
a
=
2016.
ab + bc + cd + da = 2016.
ab
+
b
c
+
c
d
+
d
a
=
2016.
p3. For natural numbers
k
k
k
, we say a rectangle of size
1
×
k
1 \times k
1
×
k
or
k
×
1
k\times 1
k
×
1
as strips. A rectangle of size
2016
×
n
2016 \times n
2016
×
n
is cut into strips of all different sizes . Find the largest natural number
n
≤
2016
n\le 2016
n
≤
2016
so we can do that.Note:
1
×
k
1\times k
1
×
k
and
k
×
1
k \times 1
k
×
1
strips are considered the same size.[url=https://artofproblemsolving.com/community/c6h1549013p9410660]p4. Let
P
A
PA
P
A
and
P
B
PB
PB
be the tangent of a circle
ω
\omega
ω
from a point
P
P
P
outside the circle. Let
M
M
M
be any point on
A
P
AP
A
P
and
N
N
N
is the midpoint of segment
A
B
AB
A
B
.
M
N
MN
MN
cuts
ω
\omega
ω
at
C
C
C
such that
N
N
N
is between
M
M
M
and
C
C
C
. Suppose
P
C
PC
PC
cuts
ω
\omega
ω
at
D
D
D
and
N
D
ND
N
D
cuts
P
B
PB
PB
at
Q
Q
Q
. Prove
M
Q
MQ
MQ
is parallel to
A
B
AB
A
B
. p5. Given a triple of different natural numbers
(
x
0
,
y
0
,
z
0
)
(x_0, y_0, z_0)
(
x
0
,
y
0
,
z
0
)
that satisfy
x
0
+
y
0
+
z
0
=
2016
x_0 + y_0 + z_0 = 2016
x
0
+
y
0
+
z
0
=
2016
. Every
i
i
i
-hour, with
i
≥
1
i\ge 1
i
≥
1
, a new triple is formed
(
x
i
,
y
i
,
z
i
)
=
(
y
i
−
1
+
z
i
−
1
x
i
−
1
,
z
i
−
1
+
x
i
−
1
−
y
i
−
1
,
x
i
−
1
+
y
i
−
1
−
z
i
−
1
)
(x_i,y_i, z_i) = (y_{i-1} + z_{i-1} x_{i-1}, z_{i-1} + x_{i-1}-y_{i-1}, x_{i-1} + y_{i-1}- z_{i-1})
(
x
i
,
y
i
,
z
i
)
=
(
y
i
−
1
+
z
i
−
1
x
i
−
1
,
z
i
−
1
+
x
i
−
1
−
y
i
−
1
,
x
i
−
1
+
y
i
−
1
−
z
i
−
1
)
Find the smallest natural number
n
n
n
so that at the
n
n
n
-th hour at least one of the found
x
n
x_n
x
n
,
y
n
y_n
y
n
, or
z
n
z_n
z
n
are negative numbers.