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Indonesia Regional
2018 Indonesia Regional
2018 Indonesia Regional
Part of
Indonesia Regional
Subcontests
(1)
2
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Indonesia Regional MO 2018 Part B
p1. A number of
n
n
n
students sit around a round table. It is known that there are as many male students as female students. If the number of pairs of
2
2
2
people sitting next to each other is counted, it turns out that the ratio between adjacent pairs of the same sex and adjacent pairs of the opposite sex is
3
:
2
3:2
3
:
2
. Find the smallest possible
n
n
n
. p2. Let
a
,
b
a, b
a
,
b
, and
c
c
c
be positive integers so that
c
=
a
+
b
a
−
1
b
.
c=a+\frac{b}{a}-\frac{1}{b}.
c
=
a
+
a
b
−
b
1
.
Prove that
c
c
c
is the square of an integer.[url=https://artofproblemsolving.com/community/c6h2370981p19378649]p3. Let
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
be two different circles with the radius of same length and centers at points
O
1
O_1
O
1
and
O
2
O_2
O
2
, respectively. Circles
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
are tangent at point
P
P
P
. The line
ℓ
\ell
ℓ
passing through
O
1
O_1
O
1
is tangent to
Γ
2
\Gamma_2
Γ
2
at point
A
A
A
. The line
ℓ
\ell
ℓ
intersects
Γ
1
\Gamma_1
Γ
1
at point
X
X
X
with
X
X
X
between
A
A
A
and
O
1
O_1
O
1
. Let
M
M
M
be the midpoint of
A
X
AX
A
X
and
Y
Y
Y
the intersection of
P
M
PM
PM
and
Γ
2
\Gamma_2
Γ
2
with
Y
≠
P
Y\ne P
Y
=
P
. Prove that
X
Y
XY
X
Y
is parallel to
O
1
O
2
O_1O_2
O
1
O
2
.[url=https://artofproblemsolving.com/community/c4h2686086p23303294]p4. Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers with
1
a
+
1
b
+
1
c
=
3
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3
a
1
+
b
1
+
c
1
=
3
. Prove that
a
+
b
+
c
+
4
1
+
(
a
b
c
)
2
/
3
≥
5
a+b+c+\frac{4}{1+(abc)^{2/3}}\ge 5
a
+
b
+
c
+
1
+
(
ab
c
)
2/3
4
≥
5
p5. On a chessboard measuring
200
×
200
200 \times 200
200
×
200
square units are placed red or blue marbles so that each unit square has at most
1
1
1
marble. Two marbles are said to be in a row if they are in the same row or column. It is known that for every red marble there are exactly
5
5
5
blue marbles in a row and for every blue marble there are exactly
5
5
5
red marbles in a row. Determine the maximum number of marbles possible on the chessboard.
Indonesia Regional MO 2018 Part A
Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2018 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2686089p23303313]hereTime: 90 minutes
∙
\bullet
∙
Write only the answers to the questions given.
∙
\bullet
∙
Some questions can have more than one correct answer. You are asked to provide the most correct or exact answer to a question like this. Scores will only be given to the giver of the most correct or most exact answer.
∙
\bullet
∙
Each question is worth 1 (one) point.
∙
∙
\bullet \bullet
∙
∙
to be more exact:
⊳
\rhd
⊳
in years 2002-08 time was 90' for part A and 120' for part B
⊳
\rhd
⊳
since years 2009 time is 210' for part A and B totally
⊳
\rhd
⊳
each problem in part A is 1 point, in part B is 7 points p1. The number of ordered pairs of integers
(
a
,
b
)
(a, b)
(
a
,
b
)
so that
a
2
+
b
2
=
a
+
b
a^2 + b^2 = a + b
a
2
+
b
2
=
a
+
b
is .. p2. Given a trapezoid
A
B
C
D
ABCD
A
BC
D
, with
A
D
AD
A
D
parallel to
B
C
BC
BC
. It is known that
B
D
=
1
BD = 1
B
D
=
1
,
∠
D
B
A
=
2
3
o
\angle DBA = 23^o
∠
D
B
A
=
2
3
o
, and
∠
B
D
C
=
4
6
o
\angle BDC = 46^o
∠
B
D
C
=
4
6
o
. If the ratio
B
C
:
A
D
=
9
:
5
BC: AD = 9:5
BC
:
A
D
=
9
:
5
, then the length of the side
C
D
CD
C
D
is ... p3. Suppose
a
>
0
a > 0
a
>
0
and
0
<
r
1
<
r
2
<
1
0 < r_1 < r_2 < 1
0
<
r
1
<
r
2
<
1
so that
a
+
a
r
1
+
a
r
1
2
+
.
.
.
a + ar_1 + ar_1^2 + ...
a
+
a
r
1
+
a
r
1
2
+
...
and
a
+
a
r
2
+
a
r
2
2
+
.
.
.
a + ar_2 + ar_2^2+...
a
+
a
r
2
+
a
r
2
2
+
...
are two infinite geometric series with the sum of
r
1
r_1
r
1
and
r
2
r_2
r
2
, respectively. The value of
r
1
+
r
2
r_1 + r_2
r
1
+
r
2
is .... p4. It is known that
S
=
{
10
,
11
,
12
,
.
.
.
,
N
}
S = \{10,11,12,...,N\}
S
=
{
10
,
11
,
12
,
...
,
N
}
. An element in
S
S
S
is said to be trubus if the sum of its digits is the cube of a natural number. If
S
S
S
has exactly
12
12
12
trubus, then the largest possible value of
N
N
N
is.... p5. The smallest natural number
n
n
n
such that
(
2
n
)
!
(
n
!
)
2
\frac{(2n)!}{(n!)^2}
(
n
!
)
2
(
2
n
)!
to be completely divided by
30
30
30
is .. p6. Given an isosceles triangle
A
B
C
ABC
A
BC
with
M
M
M
midpoint of
B
C
BC
BC
. Let
K
K
K
be the centroid of triangle
A
B
M
ABM
A
BM
. Point
N
N
N
lies on the side
A
C
AC
A
C
so that the area of the quadrilateral
K
M
C
N
KMCN
K
MCN
is half of the area of triangle
A
B
C
ABC
A
BC
. The value of
A
N
N
C
\frac{AN}{NC}
NC
A
N
is ... p7. In a box there are
n
n
n
red marbles and
m
m
m
blue marbles. Take
5
5
5
marbles at a time. If the probability that
3
3
3
red marbles and
2
2
2
blue marbles are drawn is
25
77
\frac{25}{77}
77
25
, then the smallest possible value of
m
2
+
n
2
m^2 + n^2
m
2
+
n
2
is ... p8. Let P(x) be a non-constant polynomial with a non-negative integer coefficient that satisfies
P
(
10
)
=
2018
P(10) = 2018
P
(
10
)
=
2018
. Let
m
m
m
and
M
M
M
be the minimum and maximum possible values of
P
(
l
)
P(l)
P
(
l
)
, respectively). The value of
m
+
M
m + M
m
+
M
is ... p9. A province consists of nine cities named
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
1, 2, 3, 4, 5, 6, 7, 8, 9
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
. From city
a
a
a
there is a direct road to city
b
b
b
if and only if
a
b
‾
\overline{ab}
ab
and
b
a
‾
\overline{ba}
ba
are two-digit numbers that are divisible by
3
3
3
. Two distinct cities
a
1
a_1
a
1
and
a
n
a_n
a
n
are said to be connected if there is a sequence of cities
a
1
,
a
2
,
.
.
.
,
a
n
−
1
,
a
n
a_1,a_2,...,a_{n-1},a_n
a
1
,
a
2
,
...
,
a
n
−
1
,
a
n
, so that there is a direct path from
a
i
a_i
a
i
to
a
i
+
i
a_{i+ i}
a
i
+
i
for each
i
=
1
,
.
.
.
,
n
−
1
i =1, ... , n -1
i
=
1
,
...
,
n
−
1
. The number of cities connected to city
4
4
4
is ... p10. Given
37
37
37
points as shown in the figure so that every two neighboring points are one unit apart. From each of the three distinct points a red triangle is drawn. The number of possible sidelengths of an equilateral red triangle is ... https://cdn.artofproblemsolving.com/attachments/e/1/75f1cb9051226e09fc52673762276564a32210.png p11. A positive integer
k
k
k
is taken at random with
k
≤
2018
k \le 2018
k
≤
2018
. The probability that
k
1009
k^{1009}
k
1009
has a remainder of
2
2
2
when divided by
2018
2018
2018
is .... p12. Given the non-negative real numbers
a
,
b
,
c
,
d
,
e
a, b, c, d, e
a
,
b
,
c
,
d
,
e
where
a
b
+
b
c
+
c
d
+
d
e
=
2018
ab + bc + cd + de = 2018
ab
+
b
c
+
c
d
+
d
e
=
2018
. The minimum value of
a
+
b
+
c
+
d
+
e
a + b + c + d + e
a
+
b
+
c
+
d
+
e
is ... p13. The number of subsets (including empty sets) of
X
=
{
1
,
2
,
3
,
.
.
.
,
2017
,
2018
}
X = \{1,2,3,... ,2017,2018\}
X
=
{
1
,
2
,
3
,
...
,
2017
,
2018
}
which does not have two elements
x
x
x
and
y
y
y
so that
x
y
=
2018
xy = 2018
x
y
=
2018
there are
m
2
n
m2^n
m
2
n
with
m
m
m
odd. The value of
m
+
n
m + n
m
+
n
is ... p14. Let
S
=
{
1
,
2
,
.
.
.
,
n
}
S = \{ 1 ,2 ,..., n\}
S
=
{
1
,
2
,
...
,
n
}
. It is known that there are exactly
1001
1001
1001
pairs
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
with
a
,
b
,
c
,
d
∈
S
a, b, c, d \in S
a
,
b
,
c
,
d
∈
S
and
a
<
b
<
c
<
d
a < b < c < d
a
<
b
<
c
<
d
so that
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
are arithmetic sequences. The value of
n
n
n
is .... p15. The number of natural numbers
n
n
n
such that
n
4
−
5
n
3
−
5
n
2
+
4
n
+
10
n^4 - 5n^3- 5n^2 + 4n + 10
n
4
−
5
n
3
−
5
n
2
+
4
n
+
10
is a prime number is... p16. Point
M
M
M
lies on the circumcircle of regular pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
. The greatest value that
M
B
+
M
E
M
A
+
M
C
+
M
D
\frac{MB + ME}{MA + MC + MD}
M
A
+
MC
+
M
D
MB
+
ME
might be is ... p17. For
x
,
y
x, y
x
,
y
non-zero real numbers, the sum of the maximum and minimum values of
x
y
−
4
y
2
x
2
+
4
y
2
\frac{xy - 4y^2}{x^2 + 4y^2}
x
2
+
4
y
2
x
y
−
4
y
2
is ... p18. An alien race has a unique language consisting of only two letters
X
X
X
and
Z
Z
Z
. In this language, each word consists of at least one letter and no more than
11
11
11
letters. For every two words, if the first and second words are written side by side then the result is not a word. For example, if
X
X
Z
XXZ
XXZ
and
Z
Z
Z
Z
X
ZZZZX
ZZZZX
are words, then
X
X
Z
Z
Z
Z
Z
X
XXZZZZZX
XXZZZZZX
is not a word. The maximum number of words in this language is ... 19. An acute triangle
A
B
C
ABC
A
BC
has integer sidelengths. It is known that
A
C
=
B
D
AC = BD
A
C
=
B
D
where
D
D
D
is a point on the line
B
C
BC
BC
such that
A
D
AD
A
D
is perpendicular to
B
C
BC
BC
. The smallest possible length of side
B
C
BC
BC
is ... p20. The largest natural number
n
n
n
such that
50
⌊
x
⌋
−
⌊
x
⌊
x
⌋
⌋
=
100
n
−
27
⌈
x
⌉
50\lfloor x \rfloor - \lfloor x \lfloor x \rfloor \rfloor = 100n - 27 \lceil x \rceil
50
⌊
x
⌋
−
⌊
x
⌊
x
⌋⌋
=
100
n
−
27
⌈
x
⌉
has a real solution
x
x
x
is ...