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Indonesia Regional
2012 Indonesia Regional
2012 Indonesia Regional
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Indonesia Regional MO 2012 Part A
Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2012 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2685241p23295291]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. Let
O
O
O
and
I
I
I
represent the center of the circumscribed and inscribed circle, respectively inside on a triangle with side length
3
3
3
,
4
4
4
, and
5
5
5
. The length of
O
I
OI
O
I
is ... p2. Let
x
,
y
x, y
x
,
y
and
z
z
z
be prime numbers that satisfy the equation
34
x
−
51
y
=
2012
z
34x- 51y = 2012z
34
x
−
51
y
=
2012
z
. The value of
x
+
y
+
z
x + y + z
x
+
y
+
z
is ... p3. It is known that four dice are balanced and different, each of which is in the form of a regular octagon with with numbers
1
,
2
,
3
,
.
.
.
,
8
1, 2, 3, ..., 8
1
,
2
,
3
,
...
,
8
at faces. The four dice are tossed (tossed) together once. The probability that there are two dice with the same number of points is ... p4. The real value functions
f
f
f
and
g
g
g
each have the equations
f
(
x
)
=
⌊
x
−
⌋
−
a
,
g
(
x
)
=
x
2
−
x
2
a
f(x)=\sqrt{\lfloor x- \rfloor - a} \,\,\, , \,\,\, g(x)=\sqrt{x^2 - \frac{x\sqrt2}{\sqrt{a}}}
f
(
x
)
=
⌊
x
−
⌋
−
a
,
g
(
x
)
=
x
2
−
a
x
2
where
a
a
a
is positive integer. If the domain
g
o
f
gof
g
o
f
is
{
x
∣
3
1
2
≤
x
<
4
}
\{x|3 \frac1 2\le x <4\}
{
x
∣3
2
1
≤
x
<
4
}
, then the numbers of
a
a
a
are ... p5. Given a prime number
p
>
2
p> 2
p
>
2
. If
S
S
S
is the sum of all natural numbers n that if
n
2
+
p
n
n^2 + pn
n
2
+
p
n
is the square of an integer number then
S
=
S =
S
=
... p6. For any real number
x
x
x
defined
{
x
}
\{x\}
{
x
}
as the integer nearest to
x
x
x
. for example
{
1
,
9
}
=
2
\{1,9\} = 2
{
1
,
9
}
=
2
,
{
0
,
501
}
=
1
\{ 0, 501\} = 1
{
0
,
501
}
=
1
, etc. If
n
n
n
is a positive integer number multiple of 2012, the number of many positive integers
k
k
k
that satisfy
{
k
3
}
\{\sqrt[3]{k}\}
{
3
k
}
is ... p7. A large number of natural numbers
n
<
100
n <100
n
<
100
that have multiples of the form
123456789123456789...123456789
123456789123456789... 123456789
123456789123456789...123456789
is... p8. Given a parallelogram
A
B
C
D
ABCD
A
BC
D
. The point
M
M
M
on
A
B
AB
A
B
is such that
A
M
/
A
B
=
0
,
017
AM/AB = 0, 017
A
M
/
A
B
=
0
,
017
, and point
N
N
N
on
A
D
AD
A
D
so that
A
N
/
A
D
=
17
/
2009
AN/AD = 17/2009
A
N
/
A
D
=
17/2009
. Let
A
C
∩
M
N
=
P
AC \cap MN = P
A
C
∩
MN
=
P
, then
A
C
/
A
P
=
AC/AP =
A
C
/
A
P
=
... p9. In a meeting,
5
5
5
married couples will be seated at a round table. How many ways are there to arrange the sitting positions of the
5
5
5
married couples in such a way, such that exactly
3
3
3
husbands sit next to their wives? p10. If
p
,
q
p, q
p
,
q
and
r
r
r
are the roots of
x
3
−
x
2
+
x
−
2
=
0
x^3 - x^2 + x - 2 = 0
x
3
−
x
2
+
x
−
2
=
0
, then
p
3
+
q
3
+
r
3
=
p^3 + q^3 + r^3 =
p
3
+
q
3
+
r
3
=
.... p11. If
m
m
m
and
n
n
n
are positive integers that satisfy
m
2
+
n
5
=
252
m^2 + n^5 = 252
m
2
+
n
5
=
252
, then
m
+
n
=
m + n =
m
+
n
=
... p12. At
△
A
B
C
\vartriangle ABC
△
A
BC
, point
D
D
D
lies on the line
B
C
BC
BC
.
B
C
=
3
BC = 3
BC
=
3
,
∠
A
B
C
=
3
0
o
\angle ABC = 30^o
∠
A
BC
=
3
0
o
, and
∠
A
D
C
=
4
5
o
\angle ADC = 45^o
∠
A
D
C
=
4
5
o
. Lentgh of
A
C
AC
A
C
is ... p13. Five students,
A
,
B
,
C
,
D
,
E
A,B,C,D,E
A
,
B
,
C
,
D
,
E
are in one group in the relay race. If
A
A
A
is not can run first and
D
D
D
cannot run last, then the number of possible arrangements is ... p14. It is known that
H
H
H
is the set of all natural numbers less than
2012
2012
2012
whose prime factors are not more than
3
3
3
. Next defined set
S
=
{
1
n
∣
n
∈
H
}
.
S =\left\{ \frac{1}{n}| n \in H \right\}.
S
=
{
n
1
∣
n
∈
H
}
.
If
x
x
x
is the sum of all the elements of
S
S
S
, then
⌊
x
⌋
=
\lfloor x \rfloor =
⌊
x
⌋
=
... p15. Given two circles
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
which intersect at two points, namely
A
A
A
and
B
B
B
with
A
B
=
10
AB = 10
A
B
=
10
. The line segment joining the centers of the two circles intersects the circle
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
at P and Q, respectively. If
P
Q
=
3
PQ = 3
PQ
=
3
and the radius of the circle
Γ
1
\Gamma_1
Γ
1
is
13
13
13
, then the radius of the circle
Γ
2
\Gamma_2
Γ
2
is ... p16. The number of pairs of integers
(
x
,
y
)
(x, y)
(
x
,
y
)
that satisfy
1
x
+
1
y
−
1
x
y
3
=
3
4
\frac{1}{x}+ \frac{1}{y}- \frac{1}{xy^3}= \frac{3}{4}
x
1
+
y
1
−
x
y
3
1
=
4
3
is ... p17. For positive real numbers
x
x
x
and
y
y
y
with
x
y
=
1
3
xy = \frac13
x
y
=
3
1
, the minimum value of
1
9
x
6
+
1
4
y
6
\frac{1}{9x^6} + \frac{1}{4y^6}
9
x
6
1
+
4
y
6
1
is ... p18. The number of pairs of positive integers
(
a
,
b
)
(a, b)
(
a
,
b
)
that satisfy
4
a
+
4
a
2
+
4
=
b
2
4^a + 4a^2 + 4 = b^2
4
a
+
4
a
2
+
4
=
b
2
is .. p19. Given a triangle
A
B
C
ABC
A
BC
, the length of
A
B
AB
A
B
is equal to twice the length of
A
C
AC
A
C
. Suppose
D
D
D
and
E
E
E
in segments
A
B
AB
A
B
and
B
C
BC
BC
, respectively, so
∠
B
A
E
=
∠
A
C
D
\angle BAE = \angle ACD
∠
B
A
E
=
∠
A
C
D
. If
F
=
A
E
∩
C
D
F = AE\cap CD
F
=
A
E
∩
C
D
and
C
E
F
CEF
CEF
is an equilateral triangle, then the measure of the angles of triangle
A
B
C
ABC
A
BC
is ...... p20. The number of positive integers
n
n
n
that satisfy
n
≤
2012
n\le 2012
n
≤
2012
and is a square number perfect or cubic or to the power of
4
4
4
or to the power of
5
5
5
or ... or to the power of
10
10
10
, is...
Indonesia Regional MO 2012 Part B
p1. Determine all pairs of non -negative integers
(
a
,
b
,
x
,
y
)
(a, b, x, y)
(
a
,
b
,
x
,
y
)
that satisfy the system of equations:
a
+
b
=
x
y
a + b = xy
a
+
b
=
x
y
x
+
y
=
a
b
x + y = ab
x
+
y
=
ab
p2. Find all pairs of real numbers
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
that satisfy the system of equations
x
=
1
+
y
−
z
2
x=1+\sqrt{y-z^2}
x
=
1
+
y
−
z
2
y
=
1
+
z
−
x
2
y=1+\sqrt{z-x^2}
y
=
1
+
z
−
x
2
z
=
1
+
x
−
y
2
z=1+\sqrt{x-y^2}
z
=
1
+
x
−
y
2
p3. A man has
6
6
6
friends. One night at a restaurant, he met with each of them
11
11
11
times, every
2
2
2
of them
6
6
6
times, every
3
3
3
of them
4
4
4
times, every
4
4
4
of them
3
3
3
times, every
5
5
5
of them
3
3
3
times, and all of them
10
10
10
times. He eats out
9
9
9
times without meeting them. How many times did he eat at the restaurant in total?[url=https://artofproblemsolving.com/community/c6h2371607p19389327]p4. Given an acute triangle
A
B
C
ABC
A
BC
. Point
H
H
H
denotes the foot of the altitude drawn from
A
A
A
. Prove that
A
B
+
A
C
≥
B
C
c
o
s
∠
B
A
C
+
2
A
H
s
i
n
∠
B
A
C
AB + AC \ge BC cos \angle BAC + 2AH sin \angle BAC
A
B
+
A
C
≥
BC
cos
∠
B
A
C
+
2
A
Hs
in
∠
B
A
C
p5. It is known that
p
0
=
1
p_0 = 1
p
0
=
1
and
p
i
p_i
p
i
is the i-th prime number, for
i
=
1
,
2
,
.
.
.
i = 1, 2, ...
i
=
1
,
2
,
...
namely
p
1
=
2
p_1 = 2
p
1
=
2
,
p
2
=
3
p_2 = 3
p
2
=
3
,
.
.
.
...
...
. The prime number of
p
i
p_i
p
i
is said to be moderate if
p
i
(
n
2
)
>
p
i
−
1
(
n
!
)
4
p_i^{(n^2)} >p_{i-1} (n!)^4
p
i
(
n
2
)
>
p
i
−
1
(
n
!
)
4
for all positive integers
n
n
n
. Determine all moderate prime numbers.