MathDB
Problems
Contests
National and Regional Contests
Indonesia Contests
Indonesia Juniors
2008 Indonesia Juniors
2008 Indonesia Juniors
Part of
Indonesia Juniors
Subcontests
(2)
day 2
1
Hide problems
Indonesia Juniors 2008 day 2 OSN SMP
p1. Let
A
=
{
(
x
,
y
)
∣
3
x
+
5
y
≥
15
,
x
+
y
2
≤
25
,
x
≥
0
,
x
,
y
A = \{(x, y)|3x + 5y\ge 15, x + y^2\le 25, x\ge 0, x, y
A
=
{(
x
,
y
)
∣3
x
+
5
y
≥
15
,
x
+
y
2
≤
25
,
x
≥
0
,
x
,
y
integer numbers
}
\}
}
. Find all pairs of
(
x
,
z
x
)
∈
A
(x, zx)\in A
(
x
,
z
x
)
∈
A
provided that
z
z
z
is non-zero integer. p2. A shop owner wants to be able to weigh various kinds of weight objects (in natural numbers) with only
4
4
4
different weights. (For example, if he has weights
1
1
1
,
2
2
2
,
5
5
5
and
10
10
10
. He can weighing
1
1
1
kg,
2
2
2
kg,
3
3
3
kg
(
1
+
2
)
(1 + 2)
(
1
+
2
)
,
44
44
44
kg
(
5
−
1
)
(5 - 1)
(
5
−
1
)
,
5
5
5
kg,
6
6
6
kg,
7
7
7
kg,
8
8
8
kg,
9
9
9
kg
(
10
−
1
)
(10 - 1)
(
10
−
1
)
,
10
10
10
kg,
11
11
11
kg,
12
12
12
kg,
13
13
13
kg
(
10
+
1
+
2
)
(10 + 1 + 2)
(
10
+
1
+
2
)
,
14
14
14
kg
(
10
+
5
−
1
)
(10 + 5 -1)
(
10
+
5
−
1
)
,
15
15
15
kg,
16
16
16
kg,
17
17
17
kg and
18
18
18
kg). If he wants to be able to weigh all the weight from
1
1
1
kg to
40
40
40
kg, determine the four weights that he must have. Explain that your answer is correct. p3. Given the following table. https://cdn.artofproblemsolving.com/attachments/d/8/4622407a72656efe77ccaf02cf353ef1bcfa28.png Table
4
×
4
4\times 4
4
×
4
is a combination of four smaller table sections of size
2
×
2
2\times 2
2
×
2
. This table will be filled with four consecutive integers such that:
∙
\bullet
∙
The horizontal sum of the numbers in each row is
10
10
10
.
∙
\bullet
∙
The vertical sum of the numbers in each column is
10
10
10
∙
\bullet
∙
The sum of the four numbers in each part of
2
×
2
2\times 2
2
×
2
which is delimited by the line thickness is also equal to
10
10
10
. Determine how many arrangements are possible. p4. A sequence of real numbers is defined as following:
U
n
=
a
r
n
−
1
U_n=ar^{n-1}
U
n
=
a
r
n
−
1
, if
n
=
4
m
−
3
n = 4m -3
n
=
4
m
−
3
or
n
=
4
m
−
2
n = 4m - 2
n
=
4
m
−
2
U
n
=
−
a
r
n
−
1
U_n=- ar^{n-1}
U
n
=
−
a
r
n
−
1
, if
n
=
4
m
−
1
n = 4m - 1
n
=
4
m
−
1
or
n
=
4
m
n = 4m
n
=
4
m
, where
a
>
0
a > 0
a
>
0
,
r
>
0
r > 0
r
>
0
, and
m
m
m
is a positive integer. Prove that the sum of all the
1
1
1
st to
2009
2009
2009
th terms is
a
(
1
+
r
−
r
2009
+
r
2010
)
1
+
r
2
\frac{a(1+r-r^{2009}+r^{2010})}{1+r^2}
1
+
r
2
a
(
1
+
r
−
r
2009
+
r
2010
)
5. Cube
A
B
C
D
.
E
F
G
H
ABCD.EFGH
A
BC
D
.
EFG
H
is cut into four parts by two planes. The first plane is parallel to side
A
B
C
D
ABCD
A
BC
D
and passes through the midpoint of edge
B
F
BF
BF
. The sceond plane passes through the midpoints
A
B
AB
A
B
,
A
D
AD
A
D
,
G
H
GH
G
H
, and
F
G
FG
FG
. Determine the ratio of the volumes of the smallest part to the largest part.
day 1
1
Hide problems
Indonesia Juniors 2008 day 1 OSN SMP
p1. Circle
M
M
M
is the incircle of ABC, while circle
N
N
N
is the incircle of
A
C
D
ACD
A
C
D
. Circles
M
M
M
and
N
N
N
are tangent at point
E
E
E
. If side length
A
D
=
x
AD = x
A
D
=
x
cm,
A
B
=
y
AB = y
A
B
=
y
cm,
B
C
=
z
BC = z
BC
=
z
cm, find the length of side
D
C
DC
D
C
(in terms of
x
,
y
x, y
x
,
y
, and
z
z
z
). https://cdn.artofproblemsolving.com/attachments/d/5/66ddc8a27e20e5a3b27ab24ff1eba3abee49a6.png p2. The address of the house on Jalan Bahagia will be numbered with the following rules:
∙
\bullet
∙
One side of the road is numbered with consecutive even numbers starting from number
2
2
2
.
∙
\bullet
∙
The opposite side is numbered with an odd number starting from number
3
3
3
.
∙
\bullet
∙
In a row of even numbered houses, there is some land vacant house that has not been built.
∙
\bullet
∙
The first house numbered
2
2
2
has a neighbor next door. When the RT management ordered the numbers of the house, it is known that the cost of making each digit is
12.000
12.000
12.000
Rp. For that, the total cost to be incurred is
1.020.000
1.020.000
1.020.000
Rp. It is also known that the cost of all even-sided house numbers is
132.000
132.000
132.000
Rp. cheaper than the odd side. When the land is empty later a house has been built, the number of houses on the even and odd sides is the same. Determine the number of houses that are now on Jalan Bahagia . p3. Given the following problem: Each element in the set
A
=
{
10
,
11
,
12
,
.
.
.
,
2008
}
A = \{10, 11, 12,...,2008\}
A
=
{
10
,
11
,
12
,
...
,
2008
}
multiplied by each element in the set
B
=
{
21
,
22
,
23
,
.
.
.
,
99
}
B = \{21, 22, 23,...,99\}
B
=
{
21
,
22
,
23
,
...
,
99
}
. The results are then added together to give value of
X
X
X
. Determine the value of
X
X
X
. Someone answers the question by multiplying
2016991
2016991
2016991
with
4740
4740
4740
. How can you explain that how does that person make sense? p4. Let
P
P
P
be the set of all positive integers between
0
0
0
and
2008
2008
2008
which can be expressed as the sum of two or more consecutive positive integers . (For example:
11
=
5
+
6
11 = 5 + 6
11
=
5
+
6
,
90
=
29
+
30
+
31
90 = 29 + 30 + 31
90
=
29
+
30
+
31
,
100
=
18
+
19
+
20
+
21
+
22
100 = 18 + 19 +20 + 21 + 22
100
=
18
+
19
+
20
+
21
+
22
. So
11
,
90
,
100
11, 90, 100
11
,
90
,
100
are some members of
P
P
P
.) Find the sum of of all members of
P
P
P
. p5. A four-digit number will be formed from the numbers at
0
,
1
,
2
,
3
,
4
,
5
0, 1, 2, 3, 4, 5
0
,
1
,
2
,
3
,
4
,
5
provided that the numbers in the number are not repeated, and the number formed is a multiple of
3
3
3
. What is the probability that the number formed has a value less than
3000
3000
3000
?