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2013 Indonesia Regional
2013 Indonesia Regional
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Indonesia Regional MO 2013 Part A
Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2013 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2685416p23297006]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 pointsp1. Given three circles with radius
r
=
2
r = 2
r
=
2
, which are tangent to each other. The total area of the three circles and their bounded area is equal to...p2.
2013
2013
2013
lights are controlled by
2013
2013
2013
switch buttons which are numbered
1
1
1
,
2
2
2
,
3
3
3
,
.
.
.
...
...
,
2013
2013
2013
. Pressing the switch button once will change the light on (on or off). At first all the lights were off. On the first day, all the switch buttons are pressed once. On the second day, all switch buttons numbered
2
2
2
or multiples of
2
2
2
are pressed once. By doing the same thing on day
n
n
n
, all light switch buttons numbered
n
n
n
or multiples of
n
n
n
are pressed once. And so on. How many lights were on after the operation on day
2013
2013
2013
was performed?p3. Given a real function
f
f
f
with
f
(
x
)
=
c
x
2
x
−
3
f(x)=\frac{cx}{2x-3}
f
(
x
)
=
2
x
−
3
c
x
,
x
≠
3
2
x\ne \frac32
x
=
2
3
and
f
(
f
(
x
)
)
=
x
f(f(x))=x
f
(
f
(
x
))
=
x
for all
x
≠
3
2
x\ne \frac32
x
=
2
3
. The value of
f
(
2013
)
f(2013)
f
(
2013
)
is ...p4. Pairs of positive integers
(
x
,
y
)
(x, y)
(
x
,
y
)
that satisfy
x
y
2
x
+
y
\frac{xy^2}{x+y}
x
+
y
x
y
2
prime number are ...p5. If
∣
x
∣
+
x
+
y
=
10
|x|+x+y=10
∣
x
∣
+
x
+
y
=
10
and
x
+
∣
y
∣
−
y
=
12
x+|y|-y = 12
x
+
∣
y
∣
−
y
=
12
, then the value of
x
+
y
x + y
x
+
y
is ...p6. The number of positive integers
n
n
n
, that satisfy
n
2
−
660
n^2 - 660
n
2
−
660
is a perfect square number, is...p7. How many sequences of nine terms are
a
1
a_1
a
1
,
a
2
a_2
a
2
,
.
.
.
...
...
,
a
9
a_9
a
9
, each of which is
0
0
0
,
1
1
1
,
2
2
2
,
3
3
3
,
4
4
4
,
5
5
5
,
6
6
6
,
7
7
7
,
8
8
8
, or
9
9
9
, and contains exactly one sequence
a
i
a_i
a
i
,
a
j
a_j
a
j
where
a
i
a_i
a
i
is even and
a
j
a_j
a
j
is odd?p8. Natural number
n
n
n
is said to be "beautiful" if
n
n
n
consists of
3
3
3
different digits or more and the constituent digits form an arithmetic sequence or geometric sequence. For example, number
123
123
123
is beautiful because
1
,
2
,
3
1, 2, 3
1
,
2
,
3
form an arithmetic sequence. There are ... beautiful numbers .p9. Let
M
M
M
be the midpoint of side
B
C
BC
BC
in triangle
A
B
C
ABC
A
BC
and
∠
C
A
B
=
4
5
o
\angle CAB = 45^o
∠
C
A
B
=
4
5
o
,
∠
A
B
C
=
3
0
o
\angle ABC = 30^o
∠
A
BC
=
3
0
o
, then
tan
∠
A
M
C
\tan\angle AMC
tan
∠
A
MC
is ...p10. Given a prime number
p
>
2013
p > 2013
p
>
2013
. Suppose
a
a
a
and
b
b
b
are natural numbers so that
a
+
b
a + b
a
+
b
is divisible by
p
p
p
but not divisible by
p
2
p^2
p
2
. If it is known that
a
2013
+
b
2013
a^{2013} + b^{2013}
a
2013
+
b
2013
is divisible by
p
2
p^2
p
2
then the number of natural numbers
n
≤
2013
n\le 2013
n
≤
2013
so that
a
2013
+
b
2013
a^{2013} + b^{2013}
a
2013
+
b
2013
is divisible by
p
n
p^n
p
n
is ...p11. There are six kindergarten children each bringing a meal. They will hold a cross gift, where the food is collected and then divided again so that each child receives food that is not the food that was brought back. The number of ways to do this is...p12. The graphs of the parabola
y
=
x
2
−
a
y = x^2 - a
y
=
x
2
−
a
and
x
=
y
2
−
b
x = y^2 - b
x
=
y
2
−
b
with
a
>
0
a > 0
a
>
0
and
b
>
0
b > 0
b
>
0
, intersect at four points
(
x
1
,
y
1
)
(x_1, y_1)
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
(x_2, y_2)
(
x
2
,
y
2
)
,
(
x
3
,
y
3
)
(x_3, y_3)
(
x
3
,
y
3
)
, and
(
x
4
,
y
4
)
(x_4, y_4)
(
x
4
,
y
4
)
. The value of
(
x
1
+
x
2
)
(
x
1
+
x
3
)
(
x
1
+
x
4
)
(x_1 + x_2)(x_1 + x_3)(x_1 + x_4)
(
x
1
+
x
2
)
(
x
1
+
x
3
)
(
x
1
+
x
4
)
is ...p13. A dice is tossed
2
2
2
(two) times. Let
a
a
a
and
b
b
b
be the numbers that appear on the first and second tosses, respectively. The probability that there are real numbers
x
x
x
,
y
y
y
, and
z
z
z
that satisfy the equations
x
+
y
+
z
=
a
x + y + z = a
x
+
y
+
z
=
a
and
x
2
+
y
2
+
z
2
=
b
x^2 + y^2 + z^2 = b
x
2
+
y
2
+
z
2
=
b
is ... p14. Let
Δ
1
\Delta_1
Δ
1
,
Δ
2
\Delta_2
Δ
2
,
Δ
3
\Delta_3
Δ
3
, be a sequence of equilateral triangles whose side length
Δ
1
\Delta_1
Δ
1
is
1
1
1
. For
n
≥
1
n\ge 1
n
≥
1
, triangle
Δ
n
+
1
\Delta_{n+1}
Δ
n
+
1
is defined in the following way: first define
P
n
P_n
P
n
as a square whose vertices the angle lies on the sides of
Δ
n
\Delta_n
Δ
n
, then defined
L
n
L_n
L
n
as the largest circle in
P
n
P_n
P
n
, then defined
Δ
n
+
1
\Delta_{n+1}
Δ
n
+
1
, an equilateral triangle whose vertices lie on the circumference of the circle. The side length of
Δ
2013
\Delta_{2013}
Δ
2013
is ...p15. A sequence
x
1
.
x
2
,
.
.
.
,
x
n
,
.
.
.
x_1.x_2, ...,x_n, ...
x
1
.
x
2
,
...
,
x
n
,
...
is defined by:
x
1
=
2
x_1=2
x
1
=
2
and
x
n
+
1
=
(
1
+
1
n
)
x
n
+
2
n
x_{n+1}=\left(1+\frac{1}{n}\right) x_n + \frac{2}{n}
x
n
+
1
=
(
1
+
n
1
)
x
n
+
n
2
for every natural number
n
n
n
. The value of
x
2013
x_{2013}
x
2013
is ...p16. Given a square with side lengths equal to
2
3
2\sqrt3
2
3
. Inside the square there are two equilateral triangles whose bases are opposite sides of the square. The intersection of the two equilateral triangles forms a rhombus. The area of the rhombus is equal to...p17. The number of positive integers
a
a
a
and
b
b
b
that satisfy gcd
(
a
,
b
)
=
1
(a, b) = 1
(
a
,
b
)
=
1
and
a
b
+
25
b
21
a
\frac{a}{b}+\frac{25b}{21a}
b
a
+
21
a
25
b
is an integer , is ...p18. Given triangle
A
B
C
ABC
A
BC
,
A
B
=
20
AB = 20
A
B
=
20
,
A
C
=
21
AC = 21
A
C
=
21
and
B
C
=
29
BC = 29
BC
=
29
. The points
D
D
D
and
E
E
E
lie on the segment
B
C
BC
BC
, so
B
D
=
8
BD = 8
B
D
=
8
and
E
C
=
9
EC = 9
EC
=
9
. The measure of
∠
D
A
E
\angle DAE
∠
D
A
E
is equal to ...p19. A competition is attended by
20
20
20
participants. In each round, two participants compete. Each participant who loses twice is removed from the competition, the last participant in the competition is the winner. If it is known that the winner of the competition has never lost, the number of matches held in that competition is...p20. The sum of all the integers
x
x
x
, such that
2
log
(
x
2
−
4
x
−
1
)
^2 \log (x^2 - 4x - 1)
2
lo
g
(
x
2
−
4
x
−
1
)
is an integer, is ...
Indonesia Regional MO 2013 Part B
p1. There are two glasses, glass
A
A
A
contains
5
5
5
red balls, and glass
B
B
B
contains
4
4
4
red balls and one white ball. One glass is chosen at random and then one ball is drawn at random from the glass. This is done repeatedly until one of the glasses is empty. Determine the probability that the white ball is not drawn.p2. For any real number
x
x
x
, define as
[
x
]
[x]
[
x
]
the largest integer less than or equal to
x
x
x
. Find the number of natural numbers
n
≤
1
,
000
,
000
n \le 1,000,000
n
≤
1
,
000
,
000
such that
n
−
[
n
]
<
1
2013
\sqrt{n}-[\sqrt{n}] <\frac{1}{2013}
n
−
[
n
]
<
2013
1
p3. A natural number
n
n
n
is said to be valid if
1
n
+
2
n
+
3
n
+
.
.
.
+
m
n
1^n + 2^n + 3^n +... + m^n
1
n
+
2
n
+
3
n
+
...
+
m
n
is divisible by
1
+
2
+
3
+
.
.
.
+
m
1 + 2 + 3 +...+ m
1
+
2
+
3
+
...
+
m
for every natural number
m
m
m
. a) Show that
2013
2013
2013
is valid. b) Prove that there are infinitely many invalid numbers.[url=https://artofproblemsolving.com/community/c4h2685414p23296970]p4. Prove that for all positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
where
a
+
b
+
c
≤
6
a + b + c\le 6
a
+
b
+
c
≤
6
holds
a
+
2
a
(
a
+
4
)
+
b
+
2
b
(
b
+
4
)
+
+
c
+
2
c
(
c
+
4
)
≥
1
\frac{a+2}{a(a+4)}+\frac{b+2}{b(b+4)}++\frac{c+2}{c(c+4)} \ge 1
a
(
a
+
4
)
a
+
2
+
b
(
b
+
4
)
b
+
2
+
+
c
(
c
+
4
)
c
+
2
≥
1
[url=https://artofproblemsolving.com/community/c6h2371585p19388892]p5. Given an acute triangle
A
B
C
ABC
A
BC
. The longest line of altitude is the one from vertex
A
A
A
perpendicular to
B
C
BC
BC
, and it's length is equal to the length of the median of vertex
B
B
B
. Prove that
∠
A
B
C
≤
6
0
o
\angle ABC \le 60^o
∠
A
BC
≤
6
0
o