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Contests
National and Regional Contests
Indonesia Contests
Indonesia Juniors
2009 Indonesia Juniors
2009 Indonesia Juniors
Part of
Indonesia Juniors
Subcontests
(2)
day 2
1
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Indonesia Juniors 2009 day 2 OSN SMP
p1. A telephone number with
7
7
7
digits is called a Beautiful Number if the digits are which appears in the first three numbers (the three must be different) repeats on the next three digits or the last three digits. For example some beautiful numbers:
7133719
7133719
7133719
,
7131735
7131735
7131735
,
7130713
7130713
7130713
,
1739317
1739317
1739317
,
5433354
5433354
5433354
. If the numbers are taken from
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
0, 1, 2, 3, 4, 5, 6, 7, 8
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
or
9
9
9
, but the number the first cannot be
0
0
0
, how many Beautiful Numbers can there be obtained? p2. Find the number of natural numbers
n
n
n
such that
n
3
+
100
n^3 + 100
n
3
+
100
is divisible by
n
+
10
n +10
n
+
10
p3. A function
f
f
f
is defined as in the following table. https://cdn.artofproblemsolving.com/attachments/5/5/620d18d312c1709b00be74543b390bfb5a8edc.png Based on the definition of the function
f
f
f
above, then a sequence is defined on the general formula for the terms is as follows:
U
1
=
2
U_1=2
U
1
=
2
and
U
n
+
1
=
f
(
U
n
)
U_{n+1}=f(U_n)
U
n
+
1
=
f
(
U
n
)
, for
n
=
1
,
2
,
3
,
.
.
.
n = 1, 2, 3, ...
n
=
1
,
2
,
3
,
...
p4. In a triangle
A
B
C
ABC
A
BC
, point
D
D
D
lies on side
A
B
AB
A
B
and point
E
E
E
lies on side
A
C
AC
A
C
. Prove for the ratio of areas:
A
D
E
A
B
C
=
A
D
×
A
E
A
B
×
A
C
\frac{ADE }{ABC}=\frac{AD\times AE}{AB\times AC}
A
BC
A
D
E
=
A
B
×
A
C
A
D
×
A
E
p5. In a chess tournament, a player only plays once with another player. A player scores
1
1
1
if he wins,
0
0
0
if he loses, and
1
2
\frac12
2
1
if it's a draw. After the competition ended, it was discovered that
1
2
\frac12
2
1
of the total value that earned by each player is obtained from playing with 10 different players who got the lowest total points. Especially for those in rank bottom ten,
1
2
\frac12
2
1
of the total score one gets is obtained from playing with
9
9
9
other players. How many players are there in the competition?
day 1
1
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Indonesia Juniors 2009 day 1 OSN SMP
p1. A quadratic equation has the natural roots
a
a
a
and
b
b
b
. Another quadratic equation has roots
b
b
b
and
c
c
c
with
a
≠
c
a\ne c
a
=
c
. If
a
a
a
,
b
b
b
, and
c
c
c
are prime numbers less than
15
15
15
, how many triplets
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
that might meet these conditions are there (provided that the coefficient of the quadratic term is equal to
1
1
1
)? p2. In Indonesia, was formerly known the "Archipelago Fraction''. The Archipelago Fraction is a fraction
a
b
\frac{a}{b}
b
a
such that
a
a
a
and
b
b
b
are natural numbers with
a
<
b
a < b
a
<
b
. Find the sum of all Archipelago Fractions starting from a fraction with
b
=
2
b = 2
b
=
2
to
b
=
1000
b = 1000
b
=
1000
. p3. Look at the following picture. The letters
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
, and
e
e
e
in the box will replaced with numbers from
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
1, 2, 3, 4, 5, 6, 7, 8
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
, or
9
9
9
, provided that
a
,
b
,
c
,
d
a,b, c, d
a
,
b
,
c
,
d
, and
e
e
e
must be different. If it is known that
a
e
=
b
d
ae = bd
a
e
=
b
d
, how many arrangements are there? https://cdn.artofproblemsolving.com/attachments/f/2/d676a57553c1097a15a0774c3413b0b7abc45f.png p4. Given a triangle
A
B
C
ABC
A
BC
with
A
A
A
as the vertex and
B
C
BC
BC
as the base. Point
P
P
P
lies on the side
C
A
CA
C
A
. From point
A
A
A
a line parallel to
P
B
PB
PB
is drawn and intersects extension of the base at point
D
D
D
. Point
E
E
E
lies on the base so that
C
E
:
E
D
=
2
:
3
CE : ED = 2 :3
CE
:
E
D
=
2
:
3
. If
F
F
F
is the midpoint between
E
E
E
and
C
C
C
, and the area of triangle ABC is equal with
35
35
35
cm
2
^2
2
, what is the area of triangle
P
E
F
PEF
PEF
? p5. Each side of a cube is written as a natural number. At the vertex of each angle is given a value that is the product of three numbers on three sides that intersect at the vertex. If the sum of all the numbers at the points of the angle is equal to
1001
1001
1001
, find the sum of all the numbers written on the sides of the cube.