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Problems
Contests
National and Regional Contests
Indonesia Contests
Indonesia Juniors
2010 Indonesia Juniors
2010 Indonesia Juniors
Part of
Indonesia Juniors
Subcontests
(2)
day 2
1
Hide problems
Indonesia Juniors 2010 day 2 OSN SMP - cat chasing mouse for p5
p1. If
x
+
y
+
z
=
2
x + y + z = 2
x
+
y
+
z
=
2
, show that
1
x
y
+
z
−
1
+
1
y
z
+
x
−
1
+
1
x
z
+
y
−
1
=
−
1
(
x
−
1
)
(
y
−
1
)
(
z
−
1
)
\frac{1}{xy+z-1}+\frac{1}{yz+x-1}+\frac{1}{xz+y-1}=\frac{-1}{(x-1)(y-1)(z-1)}
x
y
+
z
−
1
1
+
yz
+
x
−
1
1
+
x
z
+
y
−
1
1
=
(
x
−
1
)
(
y
−
1
)
(
z
−
1
)
−
1
. p2. Determine the simplest form of
3
1
!
+
2
!
+
3
!
+
4
2
!
+
3
!
+
4
!
+
5
3
!
+
4
!
+
5
!
+
.
.
.
+
100
98
!
+
99
!
+
100
!
\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+\frac{5}{3!+4!+5!}+...+\frac{100}{98!+99!+100!}
1
!
+
2
!
+
3
!
3
+
2
!
+
3
!
+
4
!
4
+
3
!
+
4
!
+
5
!
5
+
...
+
98
!
+
99
!
+
100
!
100
p3. It is known that
A
B
C
D
ABCD
A
BC
D
and
D
E
F
G
DEFG
D
EFG
are two parallelograms. Point
E
E
E
lies on
A
B
AB
A
B
and point
C
C
C
lie on
F
G
FG
FG
. The area of
A
B
C
D
ABCD
A
BC
D
is
20
20
20
units.
H
H
H
is the point on
D
G
DG
D
G
so that
E
H
EH
E
H
is perpendicular to
D
G
DG
D
G
. If the length of
D
G
DG
D
G
is
5
5
5
units, determine the length of
E
H
EH
E
H
. https://cdn.artofproblemsolving.com/attachments/b/e/42453bf6768129ed84fbdc81ab7235e780b0e1.png p4. Each room in the following picture will be painted so that every two rooms which is directly connected to the door is given a different color. If
10
10
10
different colors are provided and
4
4
4
of them can not be used close together for two rooms that are directly connected with a door, determine how many different ways to color the
4
4
4
rooms. https://cdn.artofproblemsolving.com/attachments/4/a/e80a464a282b3fe3cdadde832b2fd38b51a41a.png 5. The floor of a hall is rectangular
A
B
C
D
ABCD
A
BC
D
with
A
B
=
30
AB = 30
A
B
=
30
meters and
B
C
=
15
BC = 15
BC
=
15
meters. A cat is in position
A
A
A
. Seeing the cat, the mouse in the midpoint of
A
B
AB
A
B
ran and tried to escape from cat. The mouse runs from its place to point
C
C
C
at a speed of
3
3
3
meters/second. The trajectory is a straight line. Watching the mice run away, at the same time from point
A
A
A
the cat is chasing with a speed of
5
5
5
meters/second. If the cat's path is also a straight line and the mouse caught before in
C
C
C
, determine an equation that can be used for determine the position and time the mouse was caught by the cat.
day 1
1
Hide problems
Indonesia Juniors 2010 day 1 OSN SMP
p1. A fraction is called Toba-
n
n
n
if the fraction has a numerator of
1
1
1
and the denominator of
n
n
n
. If
A
A
A
is the sum of all the fractions of Toba-
101
101
101
, Toba-
102
102
102
, Toba-
103
103
103
, to Toba-
200
200
200
, show that
7
12
<
A
<
5
6
\frac{7}{12} <A <\frac56
12
7
<
A
<
6
5
. p2. If
a
,
b
a, b
a
,
b
, and
c
c
c
satisfy the system of equations
a
b
a
+
b
=
1
2
\frac{ab}{a+b}=\frac12
a
+
b
ab
=
2
1
b
c
b
+
c
=
1
3
\frac{bc}{b+c}=\frac13
b
+
c
b
c
=
3
1
a
c
a
+
c
=
1
7
\frac{ac}{a+c}=\frac17
a
+
c
a
c
=
7
1
Determine the value of
(
a
−
c
)
b
(a- c)^b
(
a
−
c
)
b
. p3. Given triangle
A
B
C
ABC
A
BC
. If point
M
M
M
is located at the midpoint of
A
C
AC
A
C
, point
N
N
N
is located at the midpoint of
B
C
BC
BC
, and the point
P
P
P
is any point on
A
B
AB
A
B
. Determine the area of the quadrilateral
P
M
C
N
PMCN
PMCN
. https://cdn.artofproblemsolving.com/attachments/4/d/175e2d55f889b9dd2d8f89b8bae6c986d87911.png p4. Given the rule of motion of a particle on a flat plane
x
y
xy
x
y
as following:
N
:
(
m
,
n
)
→
(
m
+
1
,
n
+
1
)
N: (m, n)\to (m + 1, n + 1)
N
:
(
m
,
n
)
→
(
m
+
1
,
n
+
1
)
T
:
(
m
,
n
)
→
(
m
+
1
,
n
−
1
)
T: (m, n)\to (m + 1, n - 1)
T
:
(
m
,
n
)
→
(
m
+
1
,
n
−
1
)
, where
m
m
m
and
n
n
n
are integers. How many different tracks are there from
(
0
,
3
)
(0, 3)
(
0
,
3
)
to
(
7
,
2
)
(7, 2)
(
7
,
2
)
by using the above rules ? p5. Andra and Dedi played “SUPER-AS”. The rules of this game as following. Players take turns picking marbles from a can containing
30
30
30
marbles. For each take, the player can take the least a minimum of
1
1
1
and a maximum of
6
6
6
marbles. The player who picks up the the last marbels is declared the winner. If Andra starts the game by taking
3
3
3
marbles first, determine how many marbles should be taken by Dedi and what is the next strategy to take so that Dedi can be the winner.