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2003 Indonesia Regional
2003 Indonesia Regional
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Indonesia Regional MO 2003 Part A 20 problems 90' , answer only
Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2003 [hide=Part A]Part B consists of 5 essay / proof problems, that one is posted [url=https://artofproblemsolving.com/community/c2476068_2003_indonesia_regional]here Time: 90 minutes
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For each problem you have to submit the answer only.
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Each correct answer is given a value of 1 and the question that is left blank without an answer or an incorrect answer is given a value of 0.p1. If
a
a
a
and
b
b
b
are odd integers with
a
>
b
a > b
a
>
b
, how many even integers are there between
a
a
a
and
b
b
b
?p2. Agung found that the average score of the three math tests he followed was
81
81
81
. The first test score was
85
85
85
. The third test score was
4
4
4
lower than the second test score. What is the value of Agung's second test?p3. What is the set of solutions to the equation
∣
x
+
2
∣
+
∣
3
x
∣
=
14
|x + 2| + |3x| = 14
∣
x
+
2∣
+
∣3
x
∣
=
14
?p4. The four numbers
3
,
5
,
7
3, 5, 7
3
,
5
,
7
and
8
8
8
will be entered into the boxes on the side. What is the greatest yield that can be obtained? https://cdn.artofproblemsolving.com/attachments/a/b/2152b5a3dcc4634087fcc1d535131cae0f30a4.pngp5. Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be three different natural numbers. The third greatest common divisor is
12
12
12
, while the third least common multiple is
840
840
840
. What is the greatest value for
x
+
y
+
z
x + y + z
x
+
y
+
z
?p6. What is the smallest positive integer
k
k
k
such that
20032003...2003
⏟
k
\underbrace{20032003...2003}_{k}
k
20032003...2003
is divisible by
9
9
9
? p7. The quadratic equation
2
x
2
−
2
(
2
a
+
1
)
x
+
a
(
a
−
1
)
=
0
2x^2-2(2a + 1)x + a(a-1) = 0
2
x
2
−
2
(
2
a
+
1
)
x
+
a
(
a
−
1
)
=
0
has two real roots
x
1
x_1
x
1
and
x
2
x_2
x
2
. What value of a satisfies the quadratic equation so that
x
1
<
a
<
x
2
x_1 < a < x_2
x
1
<
a
<
x
2
?p8. In an isosceles triangle
A
B
C
ABC
A
BC
, a square
P
Q
R
S
PQRS
PQRS
is made as follows: Point
P
P
P
is on side
A
B
AB
A
B
, point
Q
Q
Q
is on side
A
C
AC
A
C
, while points
R
R
R
and
S
S
S
are on hypotenuse
B
C
BC
BC
. If the area of triangle
A
B
C
ABC
A
BC
is
x
x
x
, what is the area of the square
P
Q
R
S
PQRS
PQRS
?p9. Upik throws
n
n
n
dice. He calculates the probability that the sum of the dice is
6
6
6
. For
n
n
n
what is the greatest probability?p10. A vertical line divides a triangle with vertices
(
0
,
0
)
(0,0)
(
0
,
0
)
,
(
1
,
1
)
(1,1)
(
1
,
1
)
and
(
9
,
1
)
(9,1)
(
9
,
1
)
into two areas of equal area. What is the equation of the line?p11. Let
m
m
m
and
n
n
n
be two natural numbers that satisfy
m
2
−
2003
=
n
2
m^2-2003 = n^2
m
2
−
2003
=
n
2
. Calculate
m
n
mn
mn
.p12. What is the value of x that satisfies
4
log
(
2
log
x
)
+
2
log
(
4
log
x
)
=
2
^4 \log (^2 \log x) + ^2 \log (^4 \log x) = 2
4
lo
g
(
2
lo
g
x
)
+
2
lo
g
(
4
lo
g
x
)
=
2
?p13. Point
P
P
P
lies inside the square
A
B
C
D
ABCD
A
BC
D
such that
A
P
:
B
P
:
C
P
=
1
:
2
:
3
AP : BP : CP = 1: 2: 3
A
P
:
BP
:
CP
=
1
:
2
:
3
. What is the measure of angle
A
P
B
APB
A
PB
?p14. By combining the three basic colors red, yellow, and blue other colors can be formed. Suppose there are
5
5
5
cans of red paint,
5
5
5
cans of yellow, and
5
5
5
cans of blue. Budi can choose any can to mix colors, and all paint in a can must be used all. How many color choices are there?p15. Mr. Oto bought two cars for resale. He made a
30
%
30\%
30%
profit on the first car, but suffered a
20
%
20\%
20%
loss on the second car. The selling price for both cars is the same. What is the percentage profit (or loss) of Pak Oto as a whole? [Note: All percentage to the purchase price. For the answer, use the '-' sign to represent the loss and the '+' sign to represent the gain.]p16. Four married couples watch an orchestra performance. Their seats must be separated between the husband's group and the wife's group. For each group, there are 4 seats next to each other in a row. How many ways are there to give them a seat?p17. A ball with fingers
r
r
r
is kicked from
B
B
B
to
A
A
A
. The ball rolls exactly
10
10
10
laps before hitting an inclined plane and stopping. What is the distance from
B
B
B
to
A
A
A
? https://cdn.artofproblemsolving.com/attachments/e/5/f5e9b117e0815601fdf0b0b046607d798e10fd.pngp18. What is the remainder of the division
1
⋅
1
!
+
2
⋅
2
!
+
3
⋅
!
+
.
.
.
+
99
⋅
99
!
+
100
⋅
100
!
1\cdot 1! + 2\cdot 2! + 3\cdot ! + ... + 99\cdot 99! + 100\cdot 100!
1
⋅
1
!
+
2
⋅
2
!
+
3
⋅
!
+
...
+
99
⋅
99
!
+
100
⋅
100
!
by
101
101
101
?p19. A circle has a diameter
A
B
AB
A
B
whose length is a
2
2
2
-digit integer. The arc string
C
D
CD
C
D
is perpendicular to
A
B
AB
A
B
and intersects
A
B
AB
A
B
at point
H
H
H
. The length of
C
D
CD
C
D
is equal to the number obtained by changing the position of the two digits from the length of
A
B
AB
A
B
. If the distance from
H
H
H
to the center of the circle is a rational number, what is the length of
A
B
AB
A
B
?p20. How many ways to choose three different numbers so that there are no two consecutive numbers, if the numbers are chosen from the set
{
1
,
2
,
3
,
.
.
.
,
9
,
10
}
\{1, 2, 3,..., 9, 10\}
{
1
,
2
,
3
,
...
,
9
,
10
}
?
Indonesia Regional MO 2003
Problem 1. Andi, Beni, Coki, Doni, and Edo are playing truths and lies. Each player becomes a mousedeer or a wolf (This expression is a bit common in Indonesian). A mousedeer always tells the truth, whereas a wolf always lies. Andi says that Beni is a mousedeer. Coki says Doni is a wolf, while Edo says Andi is not a wolf. Beni says Coki is not a mousedeer, Doni says that Edo and Andi are different animals ("mousedeer" and "wolf" are animals). Determine the number of wolves in the game.Problem 2. Determine all integers
a
a
a
and
b
b
b
such thar
2
+
a
3
+
b
\frac{\sqrt{2} + \sqrt{a}}{\sqrt{3} + \sqrt{b}}
3
+
b
2
+
a
is a rational number.Problem 3. Points
P
P
P
and
Q
Q
Q
are the midpoints of edges
A
E
AE
A
E
and
C
G
CG
CG
(respectively) on cube
A
B
C
D
.
E
F
G
H
ABCD.EFGH
A
BC
D
.
EFG
H
. If the length of an edge of the cube is 1 unit, determine the area of quadrilateral
D
P
F
Q
DPFQ
D
PFQ
.Problem 4. Prove that
999
!
<
50
0
999
999! < 500^{999}
999
!
<
50
0
999
.Problem 5. Three points are located on an area bounded by the
Y
Y
Y
-axis and the graph of the equation
7
x
−
3
y
2
+
21
=
0
7x - 3y^2 + 21 = 0
7
x
−
3
y
2
+
21
=
0
. Prove that at least two (a pair) of the three points have a distance of less than 4 units from each other.