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Problems
Contests
National and Regional Contests
Indonesia Contests
Indonesia Juniors
2006 Indonesia Juniors
2006 Indonesia Juniors
Part of
Indonesia Juniors
Subcontests
(2)
day 2
1
Hide problems
Indonesia Juniors 2006 day 2 OSN SMP
p1. Two integers
m
m
m
and
n
n
n
are said to be coprime if there are integers
a
a
a
and
b
b
b
such that
a
m
+
b
n
=
1
am + bn = 1
am
+
bn
=
1
. Show that for each integer
p
p
p
, the pair of numbers formed by
21
p
+
4
21p + 4
21
p
+
4
and
14
p
+
3
14p + 3
14
p
+
3
are always coprime. p2. Two farmers, Person
A
A
A
and Person
B
B
B
intend to change the boundaries of their land so that it becomes like a straight line, not curvy as in image below. They do not want the area of their origin to be reduced. Try define the boundary line they should agree on, and explain why the new boundary does not reduce the area of their respective origins. https://cdn.artofproblemsolving.com/attachments/4/d/ec771d15716365991487f3705f62e4566d0e41.png p3. The system of equations of four variables is given:
{
23
x
+
47
y
−
3
z
=
434
47
x
−
23
y
−
4
w
=
183
19
z
+
17
w
=
91
\left\{\begin{array}{l} 23x + 47y - 3z = 434 \\ 47x - 23y - 4w = 183 \\ 19z + 17w = 91 \end{array} \right.
⎩
⎨
⎧
23
x
+
47
y
−
3
z
=
434
47
x
−
23
y
−
4
w
=
183
19
z
+
17
w
=
91
where
x
,
y
,
z
x, y, z
x
,
y
,
z
, and
w
w
w
are positive integers. Determine the value of
(
13
x
−
14
y
)
3
−
(
15
z
+
16
w
)
3
(13x - 14y)^3 - (15z + 16w)^3
(
13
x
−
14
y
)
3
−
(
15
z
+
16
w
)
3
p4. A person drives a motorized vehicle so that the material used fuel is obtained at the following graph. https://cdn.artofproblemsolving.com/attachments/6/f/58e9f210fafe18bfb2d9a3f78d90ff50a847b2.png Initially the vehicle contains
3
3
3
liters of fuel. After two hours, in the journey of fuel remains
1
1
1
liter. a. If in
1
1
1
liter he can cover a distance of
32
32
32
km, what is the distance taken as a whole? Explain why you answered like that? b. After two hours of travel, is there any acceleration or deceleration? Explain your answer. c. Determine what the average speed of the vehicle is. p5. Amir will make a painting of the circles, each circle to be filled with numbers. The circle's painting is arrangement follows the pattern below. https://cdn.artofproblemsolving.com/attachments/8/2/533bed783440ea8621ef21d88a56cdcb337f30.png He made a rule that the bottom four circles would be filled with positive numbers less than
10
10
10
that can be taken from the numbers on the date of his birth, i.e.
26
−
12
−
1961
26 \,\, - \,\, 12 \,\, - \,\,1961
26
−
12
−
1961
without recurrence. Meanwhile, the circles above will be filled with numbers which is the product of the two numbers on the circles in underneath.a. In how many ways can he place the numbers from left to right, right on the bottom circles in order to get the largest value on the top circle? Explain.b. On another occasion, he planned to put all the numbers on the date of birth so that the number of the lowest circle now, should be as many as
8
8
8
circles. He no longer cares whether the numbers are repeated or not .i. In order to get the smallest value in the top circle, how should the numbers be arranged? ii. How many arrays are worth considering to produce the smallest value?
day 1
1
Hide problems
Indonesia Juniors 2006 day 1 OSN SMP
p1. Given
N
=
9
+
99
+
999
+
.
.
.
+
9999...9
⏟
121
numbers
N = 9 + 99 + 999 + ... +\underbrace{\hbox{9999...9}}_{\hbox{121\,\,numbers}}
N
=
9
+
99
+
999
+
...
+
121
numbers
9999...9
. Determine the value of N. p2. The triangle
A
B
C
ABC
A
BC
in the following picture is isosceles, with
A
B
=
A
C
=
90
AB = AC =90
A
B
=
A
C
=
90
cm and
B
C
=
108
BC = 108
BC
=
108
cm. The points
P
P
P
and
Q
Q
Q
are located on
B
C
BC
BC
, respectively such that
B
P
:
P
Q
:
Q
C
=
1
:
2
:
1
BP: PQ: QC = 1: 2: 1
BP
:
PQ
:
QC
=
1
:
2
:
1
. Points
S
S
S
and
R
R
R
are the midpoints of
A
B
AB
A
B
and
A
C
AC
A
C
respectively. From these two points draw a line perpendicular to
P
R
PR
PR
so that it intersects at
P
R
PR
PR
at points
M
M
M
and
N
N
N
respectively. Determine the length of
M
N
MN
MN
. https://cdn.artofproblemsolving.com/attachments/7/1/e1d1c4e6f067df7efb69af264f5c8de5061a56.png p3. If eight equilateral triangles with side
12
12
12
cm are arranged as shown in the picture on the side, we get a octahedral net. Define the volume of the octahedron. https://cdn.artofproblemsolving.com/attachments/4/8/18cdb8b15aaf4d92f9732880784facf9348a84.png p4. It is known that
a
2
+
b
2
=
1
a^2 + b^2 = 1
a
2
+
b
2
=
1
and
x
2
+
y
2
=
1
x^2 + y^2 = 1
x
2
+
y
2
=
1
. Continue with the following algebraic process.
(
a
2
+
b
2
)
(
x
2
+
y
2
)
–
(
a
x
+
b
y
)
2
=
.
.
.
(a^2 + b^2)(x^2 + y^2) – (ax + by)^2 = ...
(
a
2
+
b
2
)
(
x
2
+
y
2
)
–
(
a
x
+
b
y
)
2
=
...
a. What relationship can be concluded between
a
x
+
b
y
ax + by
a
x
+
b
y
and
1
1
1
? b. Why? p5. A set of questions consists of
3
3
3
questions with a choice of answers True (
T
T
T
) or False (
F
F
F
), as well as
3
3
3
multiple choice questions with answers
A
,
B
,
C
A, B, C
A
,
B
,
C
, or
D
D
D
. Someone answer all questions randomly. What is the probability that he is correct in only
2
2
2
questions?