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2004 Indonesia Regional
2004 Indonesia Regional
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Indonesia Regional MO 2004 Part A 20 problems 90' , answer only
Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2004 [hide=Part A]Part B consists of 5 essay / proof problems, that one is posted [url=https://artofproblemsolving.com/community/c4h2671387p23150599]here Time: 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. p1. Let
x
x
x
and
y
y
y
be non-zero real numbers. If
1
x
+
1
y
=
10
\frac{1}{x}+\frac{1}{y}=10
x
1
+
y
1
=
10
and
x
+
y
=
40
x + y = 40
x
+
y
=
40
, what is
x
y
xy
x
y
?p2. A bottle of syrup can be used to make
60
60
60
glasses of drink if dissolved in water with a ratio of
1
1
1
part syrup to 4 parts water. How many glasses of drink are obtained from a bottle of syrup if the ratio of the solution is
1
1
1
part syrup to
5
5
5
parts water?p3. The population of Central Java is
25
%
25\%
25%
of the population of the island of Java and
15
%
15\%
15%
of the population of Indonesia. What percentage of Indonesia's population lives outside Java?p4. When calculating the volume of a cylinder, Dina made a mistake. He entered the diameter of the base into the formula for the volume of the cylinder, when he should have entered the radius of the base. What is the ratio of the results of the Office's calculations to the expected results?p5. Three circles through the center of the coordinates
(
0
,
0
)
(0, 0)
(
0
,
0
)
. The center of the first circle is in quadrant I, the center of the second circle is in quadrant II and the center of the third circle is in quadrant III. If P is a point inside the three circles, which quadrant is this point in?p6. Given successively (from left to right) the first, second and third images of a row of images. How many black circles are in the nth picture? https://cdn.artofproblemsolving.com/attachments/6/f/2ef5412d46111bca64b9faeea5eab54550c4f1.pngp7. Given a triangle ABC with side length ratio
A
C
:
C
B
=
3
:
4
AC : CB = 3: 4
A
C
:
CB
=
3
:
4
. The bisector of the exterior angle
C
C
C
intersects the extension of
B
A
BA
B
A
at
P
P
P
(point A lies between the points
P
P
P
and
B
B
B
). Determine the ratio of the length of
P
A
:
A
B
PA: AB
P
A
:
A
B
.p8. How many sequences of non-negative integers
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
satisfy the equation
x
+
y
+
z
=
99
x + y + z = 99
x
+
y
+
z
=
99
?p9. Find the set of all natural numbers n such that
n
(
n
−
1
)
(
2
n
−
1
)
n(n-1)(2n-1)
n
(
n
−
1
)
(
2
n
−
1
)
is divisible by
6
6
6
.p10. Find all real numbers x that satisfy
x
2
<
∣
2
x
−
8
∣
x^2 < |2x-8|
x
2
<
∣2
x
−
8∣
.p11. From between
6
6
6
cards numbered
1
1
1
to
6
6
6
two cards are drawn at random. What is the probability of drawing two cards whose sum is
6
6
6
?p12. In a trapezoid of height
4
4
4
, the two diagonals are perpendicular to each other. If one of the diagonals is
5
5
5
, what is the area of the trapezoid?p13. Determine the value of
(
1
−
2
3
)
(
1
−
2
5
)
(
1
−
2
7
)
.
.
.
(
1
−
2
2005
)
\left(1-\frac23\right)\left(1-\frac25\right)\left(1-\frac27\right)...\left(1-\frac{2}{2005}\right)
(
1
−
3
2
)
(
1
−
5
2
)
(
1
−
7
2
)
...
(
1
−
2005
2
)
.p14. Santi and Tini run along a circular track. Both start running at the same time from point
P
P
P
, but take opposite directions. Santi runs
1
1
2
1\frac12
1
2
1
times faster than Tini. If
P
Q
PQ
PQ
is the center line of the circular path and the two meet for the first time at point
R
R
R
, what is the measure of
∠
R
P
Q
\angle RPQ
∠
RPQ
?p15. On the sides
S
U
SU
S
U
,
T
S
TS
TS
and
U
T
UT
U
T
of triangle
S
T
U
STU
ST
U
, the points
P
,
Q
P, Q
P
,
Q
and
R
R
R
are selected so that
S
P
=
1
4
S
U
SP = \frac14 SU
SP
=
4
1
S
U
,
T
Q
=
1
2
T
S
TQ = \frac12 TS
TQ
=
2
1
TS
and
U
R
=
1
3
U
T
UR = \frac13 UT
U
R
=
3
1
U
T
. If the area of triangle
S
T
U
STU
ST
U
is
1
1
1
, what is the area of triangle
P
Q
R
PQR
PQR
?p16. Two real numbers
x
,
y
x, y
x
,
y
satisfy
(
x
+
x
2
+
1
)
(
y
+
y
2
+
1
)
=
1
\left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=1
(
x
+
x
2
+
1
)
(
y
+
y
2
+
1
)
=
1
. What is the value of
x
+
y
x + y
x
+
y
?p17. What is the minimum number of points that must be taken from a square with a side length of
2
2
2
, so that it is guaranteed to always pick two points whose distance between them is not more than
1
2
2
\frac12 \sqrt2
2
1
2
?p18. Let
f
f
f
be a function that satisfies
f
(
x
)
f
(
y
)
−
f
(
x
y
)
=
x
+
y
f(x)f(y)-f(xy) = x + y
f
(
x
)
f
(
y
)
−
f
(
x
y
)
=
x
+
y
, for every integer
x
x
x
and
y
y
y
. What is the value of
f
(
2004
)
f(2004)
f
(
2004
)
?p19. The notation
g
c
d
(
a
,
b
)
gcd \, (a, b)
g
c
d
(
a
,
b
)
expresses the greatest common divisor of the integers
a
a
a
and
b
b
b
. Three natural numbers
a
1
<
a
2
<
a
3
a_1 < a_2 < a_3
a
1
<
a
2
<
a
3
satisfy
g
c
d
(
a
1
,
a
2
,
a
3
)
=
1
gcd \,(a_1, a_2, a_3) = 1
g
c
d
(
a
1
,
a
2
,
a
3
)
=
1
, but
g
c
d
(
a
i
,
a
j
)
>
1
gcd \,(a_i, a_j) > 1
g
c
d
(
a
i
,
a
j
)
>
1
if
i
≠
j
i\ne j
i
=
j
,
i
,
j
=
1
,
2
,
3
i, j = 1, 2, 3
i
,
j
=
1
,
2
,
3
. Find
(
a
1
,
a
2
,
a
3
)
(a_1, a_2, a_3)
(
a
1
,
a
2
,
a
3
)
so that
a
1
+
a
2
+
a
3
a_1 + a_2 + a_3
a
1
+
a
2
+
a
3
is minimal.p20. Define
a
∗
b
=
a
+
b
+
a
b
a * b = a + b + ab
a
∗
b
=
a
+
b
+
ab
, for all integers
a
,
b
a, b
a
,
b
. We say that the integer
a
a
a
is a factor of the integer
c
c
c
if there is an integer
b
b
b
that satisfies
a
∗
b
=
c
a * b = c
a
∗
b
=
c
. Find all the positive factors of
67
67
67
.
Indonesia Regional MO 2004
Problem 1. Determine all triples
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
where
x
,
y
,
z
x, y, z
x
,
y
,
z
are reals, which satisfy the following simultaneous system of equations: \begin{align*} x^2 + 4 &= y^3 + 4x - z^3 \\ y^2 + 4&= z^3 + 4y - x^3 \\ z^2 + 4 &= x^3 + 4z - y^3. \end{align*}Problem 2. On triangle
A
B
C
ABC
A
BC
, it is given points
D
D
D
,
E
E
E
, and
F
F
F
on sides
B
C
,
C
A
BC, CA
BC
,
C
A
, and
A
B
AB
A
B
such that
A
D
,
B
E
,
AD, BE,
A
D
,
BE
,
and
C
F
CF
CF
concur at a point
O
O
O
. Prove that
A
O
A
D
+
B
O
B
E
+
C
O
C
F
=
2.
\frac{AO}{AD} + \frac{BO}{BE} + \frac{CO}{CF} = 2.
A
D
A
O
+
BE
BO
+
CF
CO
=
2.
Problem 3. Beni, Coki, and Dodi are living in the same house (cohabitating, don't ask me why :v) and are going to the same school. It takes Beni, Coki, and Dodi 2, 4, and 8 minutes to travel from the house to school, respectively. With a bike, it takes each person 1 minute to travel. Show that it is possible for the three of them to travel to school in no more than
2
3
4
2 \frac{3}{4}
2
4
3
minutes.Problem 4. Prove that there is no natural number
m
m
m
so that there exists
k
,
e
k, e
k
,
e
where
e
≥
2
e \geq 2
e
≥
2
, which satisfy
m
(
m
2
+
1
)
=
k
e
.
m(m^2 + 1) = k^e.
m
(
m
2
+
1
)
=
k
e
.
Problem 5. A lattice point is a point whose coordinates are a pair of integers. Let
P
1
,
P
2
,
P
3
,
P
4
,
P
5
P_1, P_2, P_3, P_4, P_5
P
1
,
P
2
,
P
3
,
P
4
,
P
5
are five lattice points on a plane. Prove that there exists (at least) a pair
(
P
i
,
P
j
)
(P_i, P_j)
(
P
i
,
P
j
)
, where
i
≠
j
i \neq j
i
=
j
, such that the segment
P
i
P
j
P_iP_j
P
i
P
j
crosses another lattice point besides
P
i
P_i
P
i
and
P
j
P_j
P
j
.