MathDB
Problems
Contests
National and Regional Contests
Indonesia Contests
Indonesia Juniors
2016 Indonesia Juniors
2016 Indonesia Juniors
Part of
Indonesia Juniors
Subcontests
(2)
day 2
1
Hide problems
Indonesia Juniors 2016 day 2 OSN SMP
p1. Given
f
(
x
)
=
1
+
x
1
−
x
f(x)=\frac{1+x}{1-x}
f
(
x
)
=
1
−
x
1
+
x
, for
x
≠
1
x \ne 1
x
=
1
. Defined
p
@
q
=
p
+
q
1
+
p
q
p @ q = \frac{p+q}{1+pq}
p
@
q
=
1
+
pq
p
+
q
for all positive rational numbers
p
p
p
and
q
q
q
. Note the sequence with
a
1
,
a
2
,
a
3
,
.
.
.
a_1,a_2,a_3,...
a
1
,
a
2
,
a
3
,
...
with
a
1
=
2
@
3
a_1=2 @3
a
1
=
2@3
,
a
n
=
a
n
−
1
@
(
n
+
2
)
a_{n}=a_{n-1}@ (n+2)
a
n
=
a
n
−
1
@
(
n
+
2
)
for
n
≥
2
n \ge 2
n
≥
2
. Determine
f
(
a
233
)
f(a_{233})
f
(
a
233
)
and
a
233
a_{233}
a
233
p2. It is known that
a
a
a
and
b
b
b
are positive integers with
a
>
b
>
2
a > b > 2
a
>
b
>
2
. Is
2
a
+
1
2
b
−
1
\frac{2^a+1}{2^b-1}
2
b
−
1
2
a
+
1
an integer? Write down your reasons. p3. Given a cube
A
B
C
D
.
E
F
G
H
ABCD.EFGH
A
BC
D
.
EFG
H
with side length
1
1
1
dm. There is a square
P
Q
R
S
PQRS
PQRS
on the diagonal plane
A
B
G
H
ABGH
A
BG
H
with points
P
P
P
on
H
G
HG
H
G
and
Q
Q
Q
on
A
H
AH
A
H
as shown in the figure below. Point
T
T
T
is the center point of the square
P
Q
R
S
PQRS
PQRS
. The line
H
T
HT
H
T
is extended so that it intersects the diagonal line
B
G
BG
BG
at
N
N
N
. Point
M
M
M
is the projection of
N
N
N
on
B
C
BC
BC
. Determine the volume of the truncated prism
D
C
M
.
H
G
N
DCM.HGN
D
CM
.
H
GN
. https://cdn.artofproblemsolving.com/attachments/f/6/22c26f2c7c66293ad7065a3c8ce3ac2ffd938b.png 4. Nine pairs of husband and wife want to take pictures in a three-line position with the background of the Palembang Ampera Bridge. There are
4
4
4
people in the front row,
6
6
6
people in the middle row, and
8
8
8
people in the back row. They agreed that every married couple must be in the same row, and every two people next to each other must be a married couple or of the same sex. Specify the number of different possible arrangements of positions. p5. p5. A hotel provides four types of rooms with capacity, rate, and number of rooms as presented in the following table. type of room, capacity of persons/ room, day / rate (Rp.), / number of rooms https://cdn.artofproblemsolving.com/attachments/3/c/e9e1ed86887e692f9d66349a82eaaffc730b46.jpg A group of four families wanted to stay overnight at the hotel. Each family consists of husband and wife and their unmarried children. The number of family members by gender is presented in the following table.family / man / woman/ total https://cdn.artofproblemsolving.com/attachments/4/6/5961b130c13723dc9fa4e34b43be30c31ee635.jpg The group leader enforces the following provisions. I. Each husband and wife must share a room and may not share a room with other married couples. II. Men and women may not share the same room unless they are from the same family. III. At least one room is occupied by all family representatives (“representative room”) IV. Each family occupies at most
3
3
3
types of rooms. V. No rooms are occupied by more than one family except representative rooms. You are asked to arrange a room for the group so that the total cost of lodging is as low as possible. Provide two possible alternative room arrangements for each family and determine the total cost.
day 1
1
Hide problems
Indonesia Juniors 2016 day 1 OSN SMP
p1. Find all real numbers that satisfy the equation
(
1
+
x
2
+
x
4
+
.
.
.
.
+
x
2014
)
(
x
2016
+
1
)
=
2016
x
2015
(1 + x^2 + x^4 + .... + x^{2014})(x^{2016} + 1) = 2016x^{2015}
(
1
+
x
2
+
x
4
+
....
+
x
2014
)
(
x
2016
+
1
)
=
2016
x
2015
p2. Let
A
A
A
be an integer and
A
=
2
+
20
+
201
+
2016
+
20162
+
.
.
.
+
20162016...2016
⏟
40
d
i
g
i
t
s
A = 2 + 20 + 201 + 2016 + 20162 + ... + \underbrace{20162016...2016}_{40\,\, digits}
A
=
2
+
20
+
201
+
2016
+
20162
+
...
+
40
d
i
g
i
t
s
20162016...2016
Find the last seven digits of
A
A
A
, in order from millions to units. p3. In triangle
A
B
C
ABC
A
BC
, points
P
P
P
and
Q
Q
Q
are on sides of
B
C
BC
BC
so that the length of
B
P
BP
BP
is equal to
C
Q
CQ
CQ
,
∠
B
A
P
=
∠
C
A
Q
\angle BAP = \angle CAQ
∠
B
A
P
=
∠
C
A
Q
and
∠
A
P
B
\angle APB
∠
A
PB
is acute. Is triangle
A
B
C
ABC
A
BC
isosceles? Write down your reasons. p4. Ayu is about to open the suitcase but she forgets the key. The suitcase code consists of nine digits, namely four
0
0
0
s (zero) and five
1
1
1
s. Ayu remembers that no four consecutive numbers are the same. How many codes might have to try to make sure the suitcase is open? p5. Fulan keeps
100
100
100
turkeys with the weight of the
i
i
i
-th turkey, being
x
i
x_i
x
i
for
i
∈
{
1
,
2
,
3
,
.
.
.
,
100
}
i\in\{1, 2, 3, ... , 100\}
i
∈
{
1
,
2
,
3
,
...
,
100
}
. The weight of the
i
i
i
-th turkey in grams is assumed to follow the function
x
i
(
t
)
=
S
i
t
+
200
−
i
x_i(t) = S_it + 200 - i
x
i
(
t
)
=
S
i
t
+
200
−
i
where
t
t
t
represents the time in days and
S
i
S_i
S
i
is the
i
i
i
-th term of an arithmetic sequence where the first term is a positive number
a
a
a
with a difference of
b
=
1
5
b =\frac15
b
=
5
1
. It is known that the average data on the weight of the hundred turkeys at
t
=
a
t = a
t
=
a
is
150.5
150.5
150.5
grams. Calculate the median weight of the turkey at time
t
=
20
t = 20
t
=
20
days.