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Problems
Contests
National and Regional Contests
Indonesia Contests
Indonesia Juniors
2020 Indonesia Juniors
2020 Indonesia Juniors
Part of
Indonesia Juniors
Subcontests
(2)
day 2
1
Hide problems
Indonesia Juniors 2020 day 2 KSN SMP
p1. Let
U
n
U_n
U
n
be a sequence of numbers that satisfy:
U
1
=
1
U_1=1
U
1
=
1
,
U
n
=
1
+
U
1
U
2
U
3
.
.
.
U
n
−
1
U_n=1+U_1U_2U_3...U_{n-1}
U
n
=
1
+
U
1
U
2
U
3
...
U
n
−
1
for
n
=
2
,
3
,
.
.
.
,
2020
n=2,3,...,2020
n
=
2
,
3
,
...
,
2020
Prove that
1
U
1
+
1
U
2
+
.
.
.
+
1
U
2019
<
2
\frac{1}{U_1}+\frac{1}{U_2}+...+\frac{1}{U_{2019}}<2
U
1
1
+
U
2
1
+
...
+
U
2019
1
<
2
p2. If
a
=
⌈
2020
+
2020
+
.
.
.
+
2020
⌉
a= \left \lceil \sqrt{2020+\sqrt{2020+...+\sqrt{2020}}} \right\rceil
a
=
⌈
2020
+
2020
+
...
+
2020
⌉
,
b
=
⌊
1442
+
1442
+
.
.
.
+
1442
⌋
b= \left \lfloor \sqrt{1442+\sqrt{1442+...+\sqrt{1442}}} \right \rfloor
b
=
⌊
1442
+
1442
+
...
+
1442
⌋
, and
c
=
a
−
b
c=a-b
c
=
a
−
b
, then determine the value of
c
c
c
.p3. Fajar will buy a pair of koi fish in the aquarium. If he randomly picks
2
2
2
fish, then the probability that the
2
2
2
fish are of the same sex is
1
/
2
1/2
1/2
. Prove that the number of koi fish in the aquarium is a perfect square. p4. A pharmacist wants to put
155
155
155
ml of liquid into
3
3
3
bottles. There are 3 bottle choices, namelya. Bottle A
∙
\bullet
∙
Capacity:
5
5
5
ml
∙
\bullet
∙
The price of one bottle is
10
,
000
10,000
10
,
000
Rp
∙
\bullet
∙
If you buy the next bottle, you will get a
20
%
20\%
20%
discount, up to the
4
4
4
th purchase or if you buy
4
4
4
bottles, get
1
1
1
free bottle Ab. Bottle B
∙
\bullet
∙
Capacity:
8
8
8
ml
∙
\bullet
∙
The price of one bottle is
15.000
15.000
15.000
Rp
∙
\bullet
∙
If you buy
2
2
2
:
20
%
20\%
20%
discount
∙
\bullet
∙
If you buy
3
3
3
: Free
1
1
1
bottle of Bc. Bottle C
∙
\bullet
∙
Capacity :
14
14
14
ml
∙
\bullet
∙
Buy
1
1
1
:
25.000
25.000
25.000
Rp
∙
\bullet
∙
Buy
2
2
2
: Free
1
1
1
bottle of A
∙
\bullet
∙
Buy
3
3
3
: Free
1
1
1
bottle of BIf in one purchase, you can only buy a maximum of
4
4
4
bottles, then look for the possibility of pharmacists putting them in bottles so that the cost is minimal (bottles do not have to be filled to capacity). p5. Two circles, let's say
L
1
L_1
L
1
and
L
2
L_2
L
2
have the same center, namely at point
O
O
O
. Radius of
L
1
L_1
L
1
is
10
10
10
cm and radius of
L
2
L_2
L
2
is
5
5
5
cm. The points
A
,
B
,
C
,
D
,
E
,
F
A, B, C, D, E, F
A
,
B
,
C
,
D
,
E
,
F
lie on
L
1
L_1
L
1
so the arcs
A
B
,
B
C
,
C
D
,
D
E
,
E
F
,
F
A
AB,BC,CD,DE,EF,FA
A
B
,
BC
,
C
D
,
D
E
,
EF
,
F
A
are equal. The points
P
,
Q
,
R
P, Q, R
P
,
Q
,
R
lie on
L
2
L_2
L
2
so that the arcs
P
Q
,
Q
R
,
R
S
PQ,QR,RS
PQ
,
QR
,
RS
are equal and
P
A
=
P
F
=
Q
B
=
Q
C
=
R
D
=
R
D
PA=PF=QB=QC=RD=RD
P
A
=
PF
=
QB
=
QC
=
R
D
=
R
D
. Determine the area of the shaded region. https://cdn.artofproblemsolving.com/attachments/b/5/0729eca97488ddfc82ab10eda02c708fecd7ae.png
day 1
1
Hide problems
Indonesia Juniors 2020 day 1 KSN SMP
p1. Let
A
B
AB
A
B
be the diameter of the circle and
P
P
P
is a point outside the circle. The lines
P
Q
PQ
PQ
and
P
R
PR
PR
are tangent to the circles at points
Q
Q
Q
and
R
R
R
. The lines
P
H
PH
P
H
is perpendicular on line
A
B
AB
A
B
at
H
H
H
. Line
P
H
PH
P
H
intersects
A
R
AR
A
R
at
S
S
S
. If
∠
Q
P
H
=
4
0
o
\angle QPH =40^o
∠
QP
H
=
4
0
o
and
∠
Q
S
A
=
3
0
o
\angle QSA =30^o
∠
QS
A
=
3
0
o
, find
∠
R
P
S
\angle RPS
∠
RPS
. p2. There is a meeting consisting of
40
40
40
seats attended by
16
16
16
invited guests. If each invited guest must be limited to at least
1
1
1
chair, then determine the number of arrangements. p3. In the crossword puzzle, in the following crossword puzzle, each box can only be filled with numbers from
1
1
1
to
9
9
9
. https://cdn.artofproblemsolving.com/attachments/2/e/224b79c25305e8ed9a8a4da51059f961b9fbf8.png Across: 1. Composite factor of
1001
1001
1001
3. Non-polyndromic numbers 5.
p
×
q
3
p\times q^3
p
×
q
3
, with
p
≠
q
p\ne q
p
=
q
and
p
,
q
p,q
p
,
q
primesDown: 1.
a
−
1
a-1
a
−
1
and
b
+
1
b+1
b
+
1
,
a
≠
b
a\ne b
a
=
b
and
p
,
q
p,q
p
,
q
primes 2. multiple of
9
9
9
4.
p
3
×
q
p^3 \times q
p
3
×
q
, with
p
≠
q
p\ne q
p
=
q
and
p
,
q
p,q
p
,
q
primes p4. Given a function
f
:
R
→
R
f:R \to R
f
:
R
→
R
and a function
g
:
R
→
R
g:R \to R
g
:
R
→
R
, so that it fulfills the following figure: https://cdn.artofproblemsolving.com/attachments/b/9/fb8e4e08861a3572412ae958828dce1c1e137a.png Find the number of values of
x
x
x
, such that
(
f
(
x
)
)
2
−
2
g
(
x
)
−
x
∈
{
−
10
,
−
9
,
−
8
,
…
,
9
,
10
}
(f(x))^2-2g(x)-x \in\{-10,-9,-8,…,9,10\}
(
f
(
x
)
)
2
−
2
g
(
x
)
−
x
∈
{
−
10
,
−
9
,
−
8
,
…
,
9
,
10
}
. p5. In a garden that is rectangular in shape, there is a watchtower in each corner and in the garden there is a monitoring tower. Small areas will be made in the shape of a triangle so that the corner points are towers (free of monitoring and/or supervisory towers). Let
k
(
m
,
n
)
k(m,n)
k
(
m
,
n
)
be the number of small areas created if there are
m
m
m
control towers and
n
n
n
monitoring towers. a. Find the values of
k
(
4
,
1
)
k(4,1)
k
(
4
,
1
)
,
k
(
4
,
2
)
k(4,2)
k
(
4
,
2
)
,
k
(
4
,
3
)
k(4,3)
k
(
4
,
3
)
, and
k
(
4
,
4
)
k(4,4)
k
(
4
,
4
)
b. Find the general formula
k
(
m
,
n
)
k(m,n)
k
(
m
,
n
)
with
m
m
m
and
n
n
n
natural numbers .