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Contests
National and Regional Contests
Indonesia Contests
Indonesia Juniors
2014 Indonesia Juniors
2014 Indonesia Juniors
Part of
Indonesia Juniors
Subcontests
(2)
day 2
1
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Indonesia Juniors 2014 day 2 OSN SMP
p1. Nurbaya's rectangular courtyard will be covered by a number of paving blocks in the form of a regular hexagon or its pieces like the picture below. The length of the side of the hexagon is
12
12
12
cm. https://cdn.artofproblemsolving.com/attachments/6/1/281345c8ee5b1e80167cc21ad39b825c1d8f7b.png Installation of other paving blocks or pieces thereof so that all fully covered page surface. To cover the entire surface The courtyard of the house required
603
603
603
paving blocks. How many paving blocks must be cut into models
A
,
B
,
C
A, B, C
A
,
B
,
C
, and
D
D
D
for the purposes of closing. If
17
17
17
pieces of model
A
A
A
paving blocks are needed, how many the length and width of Nurbaya's yard? Count how much how many pieces of each model
B
,
C
B, C
B
,
C
, and
D
D
D
paving blocks are used. p2. Given the square
P
Q
R
S
PQRS
PQRS
. If one side lies on the line
y
=
2
x
−
17
y = 2x - 17
y
=
2
x
−
17
and its two vertices lie on the parabola
y
=
x
2
y = x^2
y
=
x
2
, find the maximum area of possible squares
P
Q
R
S
PQRS
PQRS
. p3. In the triangular pyramid
T
.
A
B
C
T.ABC
T
.
A
BC
, the points
E
,
F
,
G
E, F, G
E
,
F
,
G
, and
H
H
H
lie at , respectively
A
B
AB
A
B
,
A
C
AC
A
C
,
T
C
TC
TC
, and
T
B
TB
TB
so that
E
A
:
E
B
=
F
A
:
F
C
=
H
B
:
H
T
=
G
C
:
G
T
=
2
:
1
EA : EB = FA : FC = HB : HT = GC : GT = 2:1
E
A
:
EB
=
F
A
:
FC
=
H
B
:
H
T
=
GC
:
GT
=
2
:
1
. Determine the ratio of the volumes of the two halves of the divided triangular pyramid by the plane
E
F
G
H
EFGH
EFG
H
. p4. We know that
x
x
x
is a non-negative integer and
y
y
y
is an integer. Define all pair
(
x
,
y
)
(x, y)
(
x
,
y
)
that satisfy
1
+
2
x
+
2
2
x
+
1
=
y
2
1 + 2^x + 2^{2x + 1} = y^2
1
+
2
x
+
2
2
x
+
1
=
y
2
. p5. The coach of the Indonesian basketball national team will select the players for become a member of the core team. The coach will judge five players
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
and
E
E
E
in one simulation (or trial) match with total time
80
80
80
minute match. At any time there is only one in five players that is playing. There is no limit to the number of substitutions during the match. Total playing time for each player
A
,
B
A, B
A
,
B
, and
C
C
C
are multiples of
5
5
5
minutes, while the total playing time of each players
D
D
D
and
E
E
E
are multiples of
7
7
7
minutes. How many ways each player on the field based on total playing time?
day 1
1
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Indonesia Juniors 2014 day 1 OSN SMP
p1. Bahri lives quite close to the clock gadang in the city of Bukit Tinggi West Sumatra. Bahri has an antique clock. On Monday
4
4
4
th March
2013
2013
2013
at
10.00
10.00
10.00
am, Bahri antique clock is two minutes late in comparison with Clock Tower. A day later, the antique clock was four minutes late compared to the Clock Tower. March
6
6
6
,
2013
2013
2013
the clock is late six minutes compared to Jam Gadang. The following days Bahri observed that his antique clock exhibited the same pattern of delay. On what day and what date in
2014
2014
2014
the antique Bahri clock (hand short and long hands) point to the same number as the Clock Tower? p2. In one season, the Indonesian Football League is participated by
20
20
20
teams football. Each team competes with every other team twice. The result of each match is
3
3
3
if you win,
1
1
1
if you draw, and
0
0
0
if you lose. Every week there are
10
10
10
matches involving all teams. The winner of the competition is the team that gets the highest total score. At the end what week is the fastest possible, the winner of the competition on is the season certain? p3. Look at the following picture. The quadrilateral
A
B
C
D
ABCD
A
BC
D
is a cyclic. Given that
C
F
CF
CF
is perpendicular to
A
F
AF
A
F
,
C
E
CE
CE
is perpendicular to
B
D
BD
B
D
, and
C
G
CG
CG
is perpendicular to
A
B
AB
A
B
. Is the following statements true? Write down your reasons.
B
D
C
E
=
A
B
C
G
+
A
D
C
F
\frac{BD}{CE}=\frac{AB}{CG}+ \frac{AD}{CF}
CE
B
D
=
CG
A
B
+
CF
A
D
https://cdn.artofproblemsolving.com/attachments/b/0/dbd97b4c72bc4ebd45ed6fa213610d62f29459.png p4. Suppose
M
=
201
4
2014
M=2014^{2014}
M
=
201
4
2014
. If the sum of all the numbers (digits) that make up the number
M
M
M
equals
A
A
A
and the sum of all the digits that make up the number
A
A
A
equals
B
B
B
, then find the sum of all the numbers that make up
B
B
B
. p5. Find all positive integers
n
<
200
n < 200
n
<
200
so that
n
2
+
(
n
+
1
)
2
n^2 + (n + 1)^2
n
2
+
(
n
+
1
)
2
is square of an integer.