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2009 Indonesia Regional
2009 Indonesia Regional
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Indonesia Regional MO 2009 Part A 20 problems 90' , answer only
Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2009 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2684692p23289182]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. Three black, red, and white dice are rolled together. Write the results of the throw so that the sum of the three dice is
8
8
8
.p2. The number of real numbers x that satisfy the equation
x
4
−
2
x
3
+
5
x
2
−
176
x
+
−
2009
=
0
x^4- 2x^3 + 5x^2-176x +-2009 = 0
x
4
−
2
x
3
+
5
x
2
−
176
x
+
−
2009
=
0
is ...p3. The rational numbers
a
<
b
<
c
a < b < c
a
<
b
<
c
form a arithmetic sequence and
a
b
+
b
c
+
c
a
=
3
\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3
b
a
+
c
b
+
a
c
=
3
. The number of positive numbers a that satisfies is ...p4. Let
N
N
N
represent the set of all positive integers and
S
=
{
n
∈
N
∣
n
2009
+
2
n
+
1
∈
N
}
S= \{n\in N| \frac{n^{2009}+2}{n+1} \in N\}
S
=
{
n
∈
N
∣
n
+
1
n
2009
+
2
∈
N
}
. The number of subsets of
S
S
S
is ...p5. Given a triangle
A
B
C
ABC
A
BC
with
tan
∠
C
A
B
=
22
7
\tan \angle CAB = \frac{22}{7}
tan
∠
C
A
B
=
7
22
. Through the vertex
A
A
A
, a line is drawn such that it divides the side
B
C
BC
BC
into segments of lengths
3
3
3
and
17
17
17
. The area of triangle
A
B
C
ABC
A
BC
is ....p6. The minimum value of
f
(
x
)
=
9
x
2
sin
2
x
+
4
x
sin
x
f(x)=\frac{9x^2\sin^2x+4}{x\sin x}
f
(
x
)
=
x
s
i
n
x
9
x
2
s
i
n
2
x
+
4
for
0
<
x
<
π
0 < x <\pi
0
<
x
<
π
is ...p7. Given a triangle whose lengths from the three altitudes of the triangle are integers. If the lengths of the two lines of altitudes are
10
10
10
and
6
6
6
, then the maximum length of the third line is ...p8. A function
f
:
Z
→
Q
f : Z \to Q
f
:
Z
→
Q
has the property
f
(
x
+
1
)
=
1
+
f
(
x
)
1
−
f
(
x
)
f(x+1)=\frac{1+f(x)}{1-f(x)}
f
(
x
+
1
)
=
1
−
f
(
x
)
1
+
f
(
x
)
for every
x
∈
Z
x\in Z
x
∈
Z
. If
f
(
2
)
=
2
f(2) = 2
f
(
2
)
=
2
, then the value of the function
f
(
2009
)
f(2009)
f
(
2009
)
is ...p9. It is known that a right triangle
A
B
C
ABC
A
BC
has side lengths
a
,
b
a, b
a
,
b
, and
c
c
c
and
a
<
b
<
c
a < b < c
a
<
b
<
c
. Let
r
r
r
and
R
R
R
represent the lengths of the inradii and the circumradii, respectively. If
r
(
a
+
b
+
b
)
R
2
=
3
\frac{r(a+b+b)}{R^2}=\sqrt3
R
2
r
(
a
+
b
+
b
)
=
3
then the value of
r
a
+
b
+
b
\frac{r}{a+b+b}
a
+
b
+
b
r
is ...p10. If
tan
x
+
tan
y
=
25
\tan x + \tan y = 25
tan
x
+
tan
y
=
25
and
cot
x
+
cot
y
=
30
\cot x + \cot y = 30
cot
x
+
cot
y
=
30
, then the value of
tan
(
x
+
y
)
\tan (x + y)
tan
(
x
+
y
)
isp11. On the right of
100
!
100!
100
!
there are digits equal to
0
0
0
in a row as many as ...p12. There are four pairs of shoes, four shoes will be drawn at random. The probability that a pair is drawn is ...p13. It is known that
k
,
m
k, m
k
,
m
, and
n
n
n
are three positive integers that satisfy
k
m
+
m
4
n
=
1
6
\frac{k}{m}+\frac{m}{4n}=\frac{1}{6}
m
k
+
4
n
m
=
6
1
. The smallest number
m
m
m
that satisfies is ...p14. The number of prime numbers
p
p
p
that satisfy
(
2
p
−
1
)
3
+
(
3
p
)
2
=
6
p
(2p-1)^3 + (3p)^2 = 6^p
(
2
p
−
1
)
3
+
(
3
p
)
2
=
6
p
is ...p15. If
x
1
,
x
2
,
.
.
.
,
x
2009
x_1, x_2, ..., x_{2009}
x
1
,
x
2
,
...
,
x
2009
are real numbers, then the smallest value of
cos
x
1
sin
x
2
+
cos
x
2
sin
x
3
+
.
.
.
+
cos
x
2009
sin
x
1
\cos x1 \sin x_2 + \cos x_2 \sin x_3 + ...+ \cos x_{2009} \sin x_1
cos
x
1
sin
x
2
+
cos
x
2
sin
x
3
+
...
+
cos
x
2009
sin
x
1
is ...p16. Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be the roots of the polynomial
x
3
−
8
x
2
+
4
x
−
2
x^3-8x2 + 4x-2
x
3
−
8
x
2
+
4
x
−
2
. If
f
(
x
)
=
x
3
+
p
x
2
+
q
x
+
r
f(x) = x^3 + px^2 + qx + r
f
(
x
)
=
x
3
+
p
x
2
+
q
x
+
r
is a polynomial with roots
a
+
b
−
c
a + b - c
a
+
b
−
c
,
b
+
c
−
a
b + c - a
b
+
c
−
a
,
c
+
a
−
b
c + a - b
c
+
a
−
b
then
f
(
1
)
=
.
.
.
.
f(1) = ....
f
(
1
)
=
....
p17. The number of obtuse triangles with a natural number side that has the longest side
10
10
10
is ... Note: two congruent triangles are considered equal.p18. Let
n
n
n
be the smallest natural number that has exactly
2009
2009
2009
factors and
n
n
n
is a multiple of
2009
2009
2009
. The smallest prime factor of
n
n
n
is ...p19. Let
p
(
x
)
=
x
2
−
6
p(x) = x^2-6
p
(
x
)
=
x
2
−
6
and
A
=
{
x
∈
R
∣
p
(
p
(
x
)
)
=
x
}
A = \{x\in R | p(p(x)) = x\}
A
=
{
x
∈
R
∣
p
(
p
(
x
))
=
x
}
. The maximum value of
{
∣
x
∣
:
x
∈
A
}
\{|x| : x\in A\}
{
∣
x
∣
:
x
∈
A
}
is ...p20. Let
q
=
5
+
1
2
q=\frac{\sqrt5+1}{2}
q
=
2
5
+
1
and
[
x
]
[x]
[
x
]
represent the largest integer that is less than or equal to
x
x
x
. The value of
[
q
[
q
n
]
]
−
[
q
2
n
]
[q[qn]] - [q^2n]
[
q
[
q
n
]]
−
[
q
2
n
]
for any
n
∈
N
n\in N
n
∈
N
is ...
Indonesia Regional MO 2009 Part B
p1. An ant is about to step on food that is
10
10
10
steps in front of it. The ant is being punished, he can only step forward a multiple of three steps and the rest must step backwards. Determine how many steps he takes to reach the food, if he has to take no more than twenty steps. (Note: if the ant takes two steps, each one steps backwards, then it is considered to be the same as two steps back.)p2. Given that
n
n
n
is a natural number. Let's say
x
=
6
+
2009
n
x=6+2009 \sqrt{n}
x
=
6
+
2009
n
. If
x
2009
−
x
x
3
−
x
\frac{x^{2009}-x}{x^3-x}
x
3
−
x
x
2009
−
x
is a rational number, show that
n
n
n
is the square of a natural number.[url=https://artofproblemsolving.com/community/c6h2372256p19397380]p3. Given triangle
A
B
C
ABC
A
BC
and point
D
D
D
on side
A
C
AC
A
C
. Let
r
1
r_1
r
1
,
r
2
r_2
r
2
and
r
r
r
represent the radii of the inscribed circles of triangles
A
B
D
ABD
A
B
D
,
B
C
D
BCD
BC
D
, and
A
B
C
ABC
A
BC
, respectively. Prove that
r
1
+
r
2
>
r
r_1 + r_2 > r
r
1
+
r
2
>
r
.p4. It is known that
p
p
p
is a prime number so that the equations
7
p
=
8
x
2
−
1
7p = 8x^2 -1
7
p
=
8
x
2
−
1
and
p
2
=
2
y
2
−
1
p^2 = 2y^2 - 1
p
2
=
2
y
2
−
1
have solutions
x
x
x
and
y
y
y
are integers. Find all
p
p
p
-values that satisfy.p5. It is known that the set
H
H
H
has five elements from
{
0
,
1
,
2
,
3
,
.
.
.
,
9
}
\{0, 1, 2, 3,..., 9\}
{
0
,
1
,
2
,
3
,
...
,
9
}
. Prove that there are two subsets of
H
H
H
, which are non-empty and mutually exclusive, in which all the elements have the same sum.[url=https://artofproblemsolving.com/community/c2476066_indonesia_regional]more