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Contests
National and Regional Contests
Indonesia Contests
Indonesia Juniors
2013 Indonesia Juniors
2013 Indonesia Juniors
Part of
Indonesia Juniors
Subcontests
(2)
day 2
1
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Indonesia Juniors 2013 day 2 OSN SMP
p1. Is there any natural number n such that
n
2
+
5
n
+
1
n^2 + 5n + 1
n
2
+
5
n
+
1
is divisible by
49
49
49
? Explain. p2. It is known that the parabola
y
=
a
x
2
+
b
x
+
c
y = ax^2 + bx + c
y
=
a
x
2
+
b
x
+
c
passes through the points
(
−
3
,
4
)
(-3,4)
(
−
3
,
4
)
and
(
3
,
16
)
(3,16)
(
3
,
16
)
, and does not cut the
x
x
x
-axis. Find all possible abscissa values for the vertex point of the parabola. p3. It is known that
T
.
A
B
C
T.ABC
T
.
A
BC
is a regular triangular pyramid with side lengths of
2
2
2
cm. The points
P
,
Q
,
R
P, Q, R
P
,
Q
,
R
, and
S
S
S
are the centroids of triangles
A
B
C
ABC
A
BC
,
T
A
B
TAB
T
A
B
,
T
B
C
TBC
TBC
and
T
C
A
TCA
TC
A
, respectively . Determine the volume of the triangular pyramid
P
.
Q
R
S
P.QRS
P
.
QRS
. p4. At an event invited
13
13
13
special guests consisting of
8
8
8
people men and
5
5
5
women. Especially for all those special guests provided
13
13
13
seats in a special row. If it is not expected two women sitting next to each other, determine the number of sitting positions possible for all those special guests. p5. A table of size
n
n
n
rows and
n
n
n
columns will be filled with numbers
1
1
1
or
−
1
-1
−
1
so that the product of all the numbers in each row and the product of all the numbers in each column is
−
1
-1
−
1
. How many different ways to fill the table?
day 1
1
Hide problems
Indonesia Juniors 2013 day 1 OSN SMP
p1. It is known that
f
f
f
is a function such that
f
(
x
)
+
2
f
(
1
x
)
=
3
x
f(x)+2f\left(\frac{1}{x}\right)=3x
f
(
x
)
+
2
f
(
x
1
)
=
3
x
for every
x
≠
0
x\ne 0
x
=
0
. Find the value of
x
x
x
that satisfies
f
(
x
)
=
f
(
−
x
)
f(x) = f(-x)
f
(
x
)
=
f
(
−
x
)
. p2. It is known that ABC is an acute triangle whose vertices lie at circle centered at point
O
O
O
. Point
P
P
P
lies on side
B
C
BC
BC
so that
A
P
AP
A
P
is the altitude of triangle ABC. If
∠
A
B
C
+
3
0
o
≤
∠
A
C
B
\angle ABC + 30^o \le \angle ACB
∠
A
BC
+
3
0
o
≤
∠
A
CB
, prove that
∠
C
O
P
+
∠
C
A
B
<
9
0
o
\angle COP + \angle CAB < 90^o
∠
COP
+
∠
C
A
B
<
9
0
o
. p3. Find all natural numbers
a
,
b
a, b
a
,
b
, and
c
c
c
that are greater than
1
1
1
and different, and fulfills the property that
a
b
c
abc
ab
c
divides evenly
b
c
+
a
c
+
a
b
+
2
bc + ac + ab + 2
b
c
+
a
c
+
ab
+
2
. p4. Let
A
,
B
A, B
A
,
B
, and
P
P
P
be the nails planted on the board
A
B
P
ABP
A
BP
. The length of
A
P
=
a
AP = a
A
P
=
a
units and
B
P
=
b
BP = b
BP
=
b
units. The board
A
B
P
ABP
A
BP
is placed on the paths
x
1
x
2
x_1x_2
x
1
x
2
and
y
1
y
2
y_1y_2
y
1
y
2
so that
A
A
A
only moves freely along path
x
1
x
2
x_1x_2
x
1
x
2
and only moves freely along the path
y
1
y
2
y_1y_2
y
1
y
2
as in following image. Let
x
x
x
be the distance from point
P
P
P
to the path
y
1
y
2
y_1y_2
y
1
y
2
and y is with respect to the path
x
1
x
2
x_1x_2
x
1
x
2
. Show that the equation for the path of the point
P
P
P
is
x
2
b
2
+
y
2
a
2
=
1
\frac{x^2}{b^2}+\frac{y^2}{a^2}=1
b
2
x
2
+
a
2
y
2
=
1
. https://cdn.artofproblemsolving.com/attachments/4/6/d88c337370e8c3bc5a1833bc9588d3fb047bd0.png p5. There are three boxes
A
,
B
A, B
A
,
B
, and
C
C
C
each containing
3
3
3
colored white balls and
2
2
2
red balls. Next, take three ball with the following rules: 1. Step 1 Take one ball from box
A
A
A
. 2. Step 2
∙
\bullet
∙
If the ball drawn from box
A
A
A
in step 1 is white, then the ball is put into box
B
B
B
. Next from box
B
B
B
one ball is drawn, if it is a white ball, then the ball is put into box
C
C
C
, whereas if the one drawn is red ball, then the ball is put in box
A
A
A
.
∙
\bullet
∙
If the ball drawn from box
A
A
A
in step 1 is red, then the ball is put into box
C
C
C
. Next from box
C
C
C
one ball is taken. If what is drawn is a white ball then the ball is put into box
A
A
A
, whereas if the ball drawn is red, the ball is placed in box
B
B
B
. 3. Step 3 Take one ball each from squares
A
,
B
A, B
A
,
B
, and
C
C
C
. What is the probability that all the balls drawn in step 3 are colored red?