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Contests
National and Regional Contests
Indonesia Contests
Indonesia Juniors
2018 Indonesia Juniors
2018 Indonesia Juniors
Part of
Indonesia Juniors
Subcontests
(2)
day 2
1
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Indonesian Junior MO (Nationals) 2018, Day 2
P6. It is given the integer
Y
Y
Y
with
Y
=
2018
+
20118
+
201018
+
2010018
+
⋯
+
201
00
…
0
⏟
100 digits
18.
Y = 2018 + 20118 + 201018 + 2010018 + \cdots + 201 \underbrace{00 \ldots 0}_{\textrm{100 digits}} 18.
Y
=
2018
+
20118
+
201018
+
2010018
+
⋯
+
201
100 digits
00
…
0
18.
Determine the sum of all the digits of such
Y
Y
Y
. (It is implied that
Y
Y
Y
is written with a decimal representation.)P7. Three groups of lines divides a plane into
D
D
D
regions. Every pair of lines in the same group are parallel. Let
x
,
y
x, y
x
,
y
and
z
z
z
respectively be the number of lines in groups 1, 2, and 3. If no lines in group 3 go through the intersection of any two lines (in groups 1 and 2, of course), then the least number of lines required in order to have more than 2018 regions is ....P8. It is known a frustum
A
B
C
D
.
E
F
G
H
ABCD.EFGH
A
BC
D
.
EFG
H
where
A
B
C
D
ABCD
A
BC
D
and
E
F
G
H
EFGH
EFG
H
are squares with both planes being parallel. The length of the sides of
A
B
C
D
ABCD
A
BC
D
and
E
F
G
H
EFGH
EFG
H
respectively are
6
a
6a
6
a
and
3
a
3a
3
a
, and the height of the frustum is
3
t
3t
3
t
. Points
M
M
M
and
N
N
N
respectively are intersections of the diagonals of
A
B
C
D
ABCD
A
BC
D
and
E
F
G
H
EFGH
EFG
H
and the line
M
N
MN
MN
is perpendicular to the plane
E
F
G
H
EFGH
EFG
H
. Construct the pyramids
M
.
E
F
G
H
M.EFGH
M
.
EFG
H
and
N
.
A
B
C
D
N.ABCD
N
.
A
BC
D
and calculate the volume of the 3D figure which is the intersection of pyramids
N
.
A
B
C
D
N.ABCD
N
.
A
BC
D
and
M
.
E
F
G
H
M.EFGH
M
.
EFG
H
.P9. Look at the arrangement of natural numbers in the following table. The position of the numbers is determined by their row and column numbers, and its diagonal (which, the sequence of numbers is read from the bottom left to the top right). As an example, the number
19
19
19
is on the 3rd row, 4th column, and on the 6th diagonal. Meanwhile the position of the number
26
26
26
is on the 3rd row, 5th column, and 7th diagonal.(Image should be placed here, look at attachment.)a) Determine the position of the number
2018
2018
2018
based on its row, column, and diagonal. b) Determine the average of the sequence of numbers whose position is on the "main diagonal" (quotation marks not there in the first place), which is the sequence of numbers read from the top left to the bottom right: 1, 5, 13, 25, ..., which the last term is the largest number that is less than or equal to
2018
2018
2018
.P10. It is known that
A
A
A
is the set of 3-digit integers not containing the digit
0
0
0
. Define a gadang number to be the element of
A
A
A
whose digits are all distinct and the digits contained in such number are not prime, and (a gadang number leaves a remainder of 5 when divided by 7. If we pick an element of
A
A
A
at random, what is the probability that the number we picked is a gadang number?
day 1
1
Hide problems
Indonesian Junior MO 2018 (Nationals), Day 1
The problems are really difficult to find online, so here are the problems.P1. It is known that two positive integers
m
m
m
and
n
n
n
satisfy
10
n
−
9
m
=
7
10n - 9m = 7
10
n
−
9
m
=
7
dan
m
≤
2018
m \leq 2018
m
≤
2018
. The number
k
=
20
−
18
m
n
k = 20 - \frac{18m}{n}
k
=
20
−
n
18
m
is a fraction in its simplest form. a) Determine the smallest possible value of
k
k
k
. b) If the denominator of the smallest value of
k
k
k
is (equal to some number)
N
N
N
, determine all positive factors of
N
N
N
. c) On taking one factor out of all the mentioned positive factors of
N
N
N
above (specifically in problem b), determine the probability of taking a factor who is a multiple of 4.I added this because my translation is a bit weird. [hide=Indonesian Version] Diketahui dua bilangan bulat positif
m
m
m
dan
n
n
n
dengan
10
n
−
9
m
=
7
10n - 9m = 7
10
n
−
9
m
=
7
dan
m
≤
2018
m \leq 2018
m
≤
2018
. Bilangan
k
=
20
−
18
m
n
k = 20 - \frac{18m}{n}
k
=
20
−
n
18
m
merupakan suatu pecahan sederhana. a) Tentukan bilangan
k
k
k
terkecil yang mungkin. b) Jika penyebut bilangan
k
k
k
terkecil tersebut adalah
N
N
N
, tentukan semua faktor positif dari
N
N
N
. c) Pada pengambilan satu faktor dari faktor-faktor positif
N
N
N
di atas, tentukan peluang terambilnya satu faktor kelipatan 4.P2. Let the functions
f
,
g
:
R
→
R
f, g : \mathbb{R} \to \mathbb{R}
f
,
g
:
R
→
R
be given in the following graphs. [hide=Graph Construction Notes]I do not know asymptote, can you please help me draw the graphs? Here are its complete description: For both graphs, draw only the X and Y-axes, do not draw grids. Denote each axis with
X
X
X
or
Y
Y
Y
depending on which line you are referring to, and on their intercepts, draw a small node (a circle) then mark their
X
X
X
or
Y
Y
Y
coordinates only (since their other coordinates are definitely 0). Graph (1) is the function
f
f
f
, who is a quadratic function with -2 and 4 as its
X
X
X
-intercepts and 4 as its
Y
Y
Y
-intercept. You also put
f
f
f
right besides the curve you have, preferably just on the right-up direction of said curve. Graph (2) is the function
g
g
g
, which is piecewise. For
x
≥
0
x \geq 0
x
≥
0
,
g
(
x
)
=
1
2
x
−
2
g(x) = \frac{1}{2}x - 2
g
(
x
)
=
2
1
x
−
2
, whereas for
x
<
0
x < 0
x
<
0
,
g
(
x
)
=
−
x
−
2
g(x) = - x - 2
g
(
x
)
=
−
x
−
2
. You also put
g
g
g
right besides the curve you have, on the lower right of the line, on approximately
x
=
2
x = 2
x
=
2
. Define the function
g
∘
f
g \circ f
g
∘
f
with
(
g
∘
f
)
(
x
)
=
g
(
f
(
x
)
)
(g \circ f)(x) = g(f(x))
(
g
∘
f
)
(
x
)
=
g
(
f
(
x
))
for all
x
∈
D
f
x \in D_f
x
∈
D
f
where
D
f
D_f
D
f
is the domain of
f
f
f
. a) Draw the graph of the function
g
∘
f
g \circ f
g
∘
f
. b) Determine all values of
x
x
x
so that
−
1
2
≤
(
g
∘
f
)
(
x
)
≤
6
-\frac{1}{2} \leq (g \circ f)(x) \leq 6
−
2
1
≤
(
g
∘
f
)
(
x
)
≤
6
.P3. The quadrilateral
A
B
C
D
ABCD
A
BC
D
has side lengths
A
B
=
B
C
=
4
3
AB = BC = 4\sqrt{3}
A
B
=
BC
=
4
3
cm and
C
D
=
D
A
=
4
CD = DA = 4
C
D
=
D
A
=
4
cm. All four of its vertices lie on a circle. Calculate the area of quadrilateral
A
B
C
D
ABCD
A
BC
D
.P4. There exists positive integers
x
x
x
and
y
y
y
, with
x
<
100
x < 100
x
<
100
and
y
>
9
y > 9
y
>
9
. It is known that
y
=
p
777
x
y = \frac{p}{777} x
y
=
777
p
x
, where
p
p
p
is a 3-digit number whose number in its tens place is 5. Determine the number/quantity of all possible values of
y
y
y
.P5. The 8-digit number
a
b
c
d
e
f
g
h
‾
\overline{abcdefgh}
ab
c
d
e
f
g
h
(the original problem does not have an overline, which I fixed) is arranged from the set
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
}
\{1, 2, 3, 4, 5, 6, 7, 8\}
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
}
. Such number satisfies
a
+
c
+
e
+
g
≥
b
+
d
+
f
+
h
a + c + e + g \geq b + d + f + h
a
+
c
+
e
+
g
≥
b
+
d
+
f
+
h
. Determine the quantity of different possible (such) numbers.