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Indonesia Contests
Indonesia Regional
2017 Indonesia Regional
2017 Indonesia Regional
Part of
Indonesia Regional
Subcontests
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2
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Indonesia Regional MO 2017 Part B
p1. For each unit square on a
5
×
9
5 \times 9
5
×
9
board write the number
1
1
1
or
0
0
0
. Then calculate the sum of all the numbers in each column and also in each row so are obtained
14
14
14
numbers. Suppose H is a set containing those numbers. Determine the maximum number of members of
H
H
H
.p2. The natural number
k
>
2
k> 2
k
>
2
is said to be beautiful if for every natural number
n
≥
4
n \ge 4
n
≥
4
with
5
n
+
1
5n + 1
5
n
+
1
a perfect square number, can be found real numbers
a
1
,
a
2
,
.
.
.
,
a
k
a_1, a_2,..., a_k
a
1
,
a
2
,
...
,
a
k
such that
n
+
1
=
a
1
2
+
a
2
2
+
.
.
.
+
a
k
2
.
n+1= a_1^2+ a_2^2+...+a_k^2.
n
+
1
=
a
1
2
+
a
2
2
+
...
+
a
k
2
.
Find the smallest beautiful number[url=https://artofproblemsolving.com/community/c6h2370986p19378672]p3. Given triangle
A
B
C
ABC
A
BC
, the three altitudes intersect at point
H
H
H
. Determine all points
X
X
X
on the side
B
C
BC
BC
so that the symmetric of
H
H
H
wrt point
X
X
X
lies on the circumcircle of triangle
A
B
C
ABC
A
BC
.[url=https://artofproblemsolving.com/community/c6h2685865p23301414]p4. Let
a
a
a
,
b
b
b
, and
c
c
c
be real numbers whose absolute value is not greater than
1
1
1
. Prove that
∣
a
−
b
∣
+
∣
b
−
c
∣
+
∣
c
−
a
∣
≤
2
+
2
\sqrt{|a-b|}+\sqrt{|b-c|}+\sqrt{|c-a|}\le 2+\sqrt2
∣
a
−
b
∣
+
∣
b
−
c
∣
+
∣
c
−
a
∣
≤
2
+
2
p5. On a
2017
×
n
2017\times n
2017
×
n
chessboard , Ani and Banu play a game. First player choose a square and then color it red. Next player selects a square from an area that has not been colored red and then color it in red. The selected square can be any size but must accurately cover a number of square units on a chessboard. Then the two players take turns doing the same thing. One player is said to have won, if the next player can no longer continue the game. If Ani gets the first turn, determine all values of
n
≥
2017
n\ge 2017
n
≥
2017
until Ani has a strategy to win the game.
Indonesia Regional MO 2017 Part A
Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2017 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2685870p23301432]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. Two nonzero real numbers
a
a
a
and
b
b
b
satisfy
a
b
=
a
−
b
ab = a - b
ab
=
a
−
b
. The possible value of
a
b
+
b
a
−
a
b
\frac{a}{b}+\frac{b}{a}- ab
b
a
+
a
b
−
ab
is ... p2. Community leaders somewhere in RW, apart from Mr. RW and Mrs. RW, there are
5
5
5
women and
6
6
6
men. Kelurahan asked
6
6
6
people to attend a seminar at the city level.
6
6
6
people were chosen as RW delegates, with a composition of
3
3
3
women and
3
3
3
men, one of whom was Mr. RW. The number of ways to choose the delegate is ... p3. Given a triangle
A
B
C
ABC
A
BC
with
A
B
=
13
AB = 13
A
B
=
13
,
A
C
=
15
AC = 15
A
C
=
15
, and the length of the altitude on BC is
12
12
12
. The sum of all possible lengths of
B
C
BC
BC
is ... p4. The two-digit prime number
p
=
a
b
‾
p =\overline{ab}
p
=
ab
that satisfies
b
a
‾
\overline{ba}
ba
is also prime is ... p5. Suppose
f
f
f
is a real function that satisfies
f
(
x
3
)
=
x
2
+
2
x
+
3
f \left(\frac{x}{3}\right) = x^2 +2x+3
f
(
3
x
)
=
x
2
+
2
x
+
3
. The sum of all
z
z
z
values that satisfy
f
(
3
z
)
=
12
f(3z) = 12
f
(
3
z
)
=
12
is ... p6. Ita chooses 5 numbers from
{
1
,
2
,
3
,
4
,
5
,
6
,
7
}
\{1,2,3,4,5, 6, 7\}
{
1
,
2
,
3
,
4
,
5
,
6
,
7
}
and tells Budi the product of the five numbers. Then Ita asked if Budi knew that the sum of the five numbers was an odd or even number. Budi replied that he couldn't be sure. The value of the product of five numbers owned by Ita is ... p7. Let
A
B
C
D
ABCD
A
BC
D
be a square with side length
2017
2017
2017
. Point
E
E
E
lies on the segment
C
D
CD
C
D
so
C
E
F
G
CEFG
CEFG
is a square with side length
1702
1702
1702
.
F
F
F
and
G
G
G
lie outside
A
B
C
D
ABCD
A
BC
D
. If the circumcircle of the triangle
A
C
F
ACF
A
CF
intersects
B
C
BC
BC
again at point
H
H
H
, then the length of
C
H
CH
C
H
is ... p8. The number of pairs of natural numbers
(
x
,
y
)
(x, y)
(
x
,
y
)
that satisfy the equation
x
+
y
=
x
+
y
+
x
y
x + y = \sqrt{x} + \sqrt{y} + \sqrt{xy}
x
+
y
=
x
+
y
+
x
y
is ... p9. Let
x
x
x
and
y
y
y
be real numbers that satisfy the equation
x
2
y
2
+
4
x
2
+
y
2
+
1
=
6
x
y
x^2y^2 + 4x^2 + y^2 + 1 = 6xy
x
2
y
2
+
4
x
2
+
y
2
+
1
=
6
x
y
. If
M
M
M
and
m
m
m
represent the largest and smallest possible values of
x
−
y
x - y
x
−
y
, respectively, then the value of
M
−
m
M - m
M
−
m
is ... p10. Given a 2017 lamp equipped with a switch to turn the lights on and off. At first all the lights were off. Every minute Ani has to press exactly 5 switches. Every time the switch is pressed, the light that was extinguished becomes on and the light that was lit becomes extinguished. To turn on all the lights Ani requires at least ... minutes. p11. Given a positive real number
k
k
k
. In a triangle
A
B
C
ABC
A
BC
the points
D
,
E
D, E
D
,
E
, and
F
F
F
lie on sides
B
C
,
C
A
BC, CA
BC
,
C
A
, and
A
B
AB
A
B
respectively so that
B
D
D
C
=
C
E
E
A
=
A
F
F
B
=
k
\frac{BD}{DC}=\frac{CE}{EA}=\frac{AF}{FB}=k
D
C
B
D
=
E
A
CE
=
FB
A
F
=
k
If
[
A
B
C
]
[ABC]
[
A
BC
]
and
[
D
E
F
]
[DEF]
[
D
EF
]
represent the area of triangles
A
B
C
ABC
A
BC
and
D
E
F
DEF
D
EF
, respectively, then
[
A
B
C
]
[
D
E
F
]
=
\frac{[ABC]}{[DEF]}=
[
D
EF
]
[
A
BC
]
=
... p12. For any natural number
k
k
k
, let
I
k
=
10...064
I_k= 10... 064
I
k
=
10...064
with
k
k
k
times
0
0
0
between 1 and
6
6
6
. If
N
(
k
)
N(k)
N
(
k
)
represents the number of factors of
2
2
2
in the prime factorization of
I
k
I_k
I
k
, then the maximum value for
N
(
k
)
N(k)
N
(
k
)
is ... p13. If
x
,
y
x, y
x
,
y
, and
z
z
z
are positive real numbers that satisfy
x
+
1
y
=
4
,
y
+
1
z
=
1
,
z
+
1
z
=
7
3
x+\frac{1}{y}=4, y+\frac{1}{z}=1, z+\frac{1}{z}=\frac{7}{3}
x
+
y
1
=
4
,
y
+
z
1
=
1
,
z
+
z
1
=
3
7
then the value of
x
y
z
xyz
x
yz
is ... p14. Ten students have different heights. The gym teacher wanted them to line up sideways, on the condition that no student was flanked by two other students who were taller than him. The number of ways to make such a sequence is ... p15. Given a triangle
A
B
C
ABC
A
BC
with
ω
\omega
ω
as the outer circle. The chord
A
D
AD
A
D
is the bisector of the angle
B
A
C
BAC
B
A
C
that intersects
B
C
BC
BC
at point
L
L
L
. The chord
D
K
DK
DK
is perpendicular to
A
C
AC
A
C
and intersects
A
C
AC
A
C
at point
M
M
M
. If
B
L
L
C
=
1
2
\frac{BL}{LC}=\frac{1}{2}
L
C
B
L
=
2
1
then the value of
A
M
M
C
=
\frac{AM}{MC}=
MC
A
M
=
is..... p16. The original four-digit number
n
n
n
is completely divided by
7
7
7
. The original number
k
k
k
, obtained by writing the n-digits from back to front, is also completely divided by
7
7
7
. In addition, it is known that
n
n
n
and
k
k
k
have the same remainder when divided by
37
37
37
. If
k
>
n
k> n
k
>
n
, then the sum of all
n
n
n
that satisfy is ..... p17. Given
{
a
i
}
i
≥
1
\{a_i\}_{i\ge 1}
{
a
i
}
i
≥
1
real numbers with
a
1
=
20
,
17
a_1 = 20,17
a
1
=
20
,
17
. If
a
1
,
a
2
,
.
.
.
,
a
11
a_1,a_2,...,a_{11}
a
1
,
a
2
,
...
,
a
11
and
⌊
a
1
⌋
,
⌊
a
2
⌋
,
.
.
.
,
⌊
a
10
⌋
\lfloor a_1 \rfloor , \lfloor a_2\rfloor , ..., \lfloor a_{10} \rfloor
⌊
a
1
⌋
,
⌊
a
2
⌋
,
...
,
⌊
a
10
⌋
, are each an arithmetic sequence; whereas
⌊
a
1
⌋
,
⌊
a
2
⌋
,
.
.
.
,
⌊
a
11
⌋
\lfloor a_1 \rfloor , \lfloor a_2\rfloor , ..., \lfloor a_{11} \rfloor
⌊
a
1
⌋
,
⌊
a
2
⌋
,
...
,
⌊
a
11
⌋
is not arithmetic sequence, then the minimum value of
a
2
−
a
1
−
⌊
a
2
−
a
1
⌋
a_2 - a_1 -\lfloor a_2-a_1 \rfloor
a
2
−
a
1
−
⌊
a
2
−
a
1
⌋
is ... p18. In a Snack Center there are four shops each selling three type of food. There are
n
n
n
people who each buy exactly one food at each store. For every three shoppers there is at least one store where all three types of food are bought. The maximum possible value of
n
n
n
is ... 19. Given is the regular hexagon
A
B
C
D
E
F
G
ABCDEFG
A
BC
D
EFG
. The distances from
A
A
A
on the lines
B
C
BC
BC
,
B
E
BE
BE
,
C
F
CF
CF
, and
E
F
EF
EF
respectively are
a
,
b
,
c
a, b, c
a
,
b
,
c
, and
d
d
d
. The value of
a
d
b
c
\frac{ad}{bc}
b
c
a
d
is ... 20. It is known
f
(
x
)
f (x)
f
(
x
)
is a polynomial of degree
n
n
n
with integer coefficients satisfying
f
(
0
)
=
39
,
f
(
x
1
)
=
f
(
x
2
)
=
f
(
x
3
)
=
.
.
.
=
f
(
x
n
)
=
2017
f(0) = 39, \,\,\, f(x_1) = f(x_2) = f(x_3) = ... = f (x_n) = 2017
f
(
0
)
=
39
,
f
(
x
1
)
=
f
(
x
2
)
=
f
(
x
3
)
=
...
=
f
(
x
n
)
=
2017
, with
x
1
,
x
2
,
x
3
,
.
.
.
,
x
n
x_1, x_2, x_3, ..., x_n
x
1
,
x
2
,
x
3
,
...
,
x
n
are all different. The largest possible number of
n
n
n
is .....