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National and Regional Contests
Indonesia Contests
Indonesia Regional
2024 Indonesia Regional
2024 Indonesia Regional
Part of
Indonesia Regional
Subcontests
(5)
1
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2024 Indonesia Regional MO Short Answer Section
2024 Indonesia Regional MO Short Answer Section There are 8 problems, time allowed is 60 minutes. Answers are always in integer form.1. It is known that
a
b
‾
\overline{ab}
ab
and
c
d
‾
\overline{cd}
c
d
are both two-digit numbers whose product is
777
777
777
. If
a
b
‾
<
c
d
‾
\overline{ab}<\overline{cd}
ab
<
c
d
, find the value of
a
+
b
a+b
a
+
b
. 2. Let
f
f
f
and
g
g
g
be linear functions that satisfy the equation
f
(
x
+
g
(
y
)
)
=
7
x
+
2
y
+
11
for every real number
x
,
y
f(x+g(y)) = 7x+2y+11 \text{ for every real number } x,y
f
(
x
+
g
(
y
))
=
7
x
+
2
y
+
11
for every real number
x
,
y
If
g
(
7
)
=
3
g(7)=3
g
(
7
)
=
3
, find the value of
g
(
−
11
+
f
(
4
)
)
g(-11+f(4))
g
(
−
11
+
f
(
4
))
. Note: A linear function is a function of the form
h
(
x
)
=
a
x
+
b
h(x)=ax+b
h
(
x
)
=
a
x
+
b
with real constants
a
,
b
a,b
a
,
b
. 3. Given a triangle
A
B
C
ABC
A
BC
with side lengths
A
B
=
15
,
A
C
=
13
,
B
C
=
4
AB=15, AC=13, BC=4
A
B
=
15
,
A
C
=
13
,
BC
=
4
. There exists an equilateral triangle
P
Q
R
PQR
PQR
with
P
,
Q
,
and
R
P,Q,\text{ and } R
P
,
Q
,
and
R
lying on sides
B
C
,
C
A
,
and
A
B
BC,CA, \text{ and } AB
BC
,
C
A
,
and
A
B
respectively such that
P
Q
PQ
PQ
is parallel to
A
B
AB
A
B
. The value
P
Q
A
B
\dfrac{PQ}{AB}
A
B
PQ
can be expressed in the form
a
b
+
c
d
\dfrac{a }{b+c\sqrt{d} }
b
+
c
d
a
with
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
such that
a
a
a
is a positive integer,
d
d
d
is squarefree, and
GCD
(
a
,
b
,
c
)
=
1
\text{GCD}(a,b,c)=1
GCD
(
a
,
b
,
c
)
=
1
. Find value of
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
. 4. Each tile on a board of size
2023
×
3
2023 \times 3
2023
×
3
will be colored either black or white, such that each
2
×
2
2\times 2
2
×
2
sub-board has an odd number of black tiles and an odd number of white tiles. Suppose the number of possible ways of such coloring is
A
A
A
. Find the remainder of
A
A
A
when divided by
1000
1000
1000
. 5. Find the number of positive integers
a
<
209
a<209
a
<
209
such that
GCD
(
a
,
209
)
=
1
\text{GCD}(a,209)=1
GCD
(
a
,
209
)
=
1
and
a
2
−
1
a^2-1
a
2
−
1
is not a multiple of
209
209
209
. 6. In a square
A
B
C
D
ABCD
A
BC
D
with side length
2
+
6
\sqrt{2}+\sqrt{6}
2
+
6
,
X
X
X
lies on the diagonal
A
C
AC
A
C
such that
A
X
>
X
C
AX>XC
A
X
>
XC
. The internal bisector of angle
A
X
B
AXB
A
XB
intersects side
A
B
AB
A
B
at
U
U
U
. The internal bisector of angle
C
X
D
CXD
CX
D
intersects side
C
D
CD
C
D
at
V
V
V
. If
∠
U
X
V
=
15
0
∘
\angle UXV = 150^{\circ}
∠
U
X
V
=
15
0
∘
, find the value of
⌊
3
×
U
V
2
⌋
\lfloor 3 \times UV^2 \rfloor
⌊
3
×
U
V
2
⌋
. Note: the notation
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
represents the largest integer that is less than or equal to
x
x
x
. 7. Given the set
S
=
{
1
,
2
,
…
,
18
}
S = \{1,2,\ldots,18\}
S
=
{
1
,
2
,
…
,
18
}
. Let
N
N
N
be the number of ordered pairs
(
A
,
B
)
(A,B)
(
A
,
B
)
of subsets
A
,
B
⊆
S
A,B\subseteq S
A
,
B
⊆
S
such that
∣
A
∩
B
∣
≤
2
| A \cap B | \le 2
∣
A
∩
B
∣
≤
2
. Find the value of
N
3
16
\dfrac{N}{3^{16} }
3
16
N
. Note:
∣
X
∣
|X|
∣
X
∣
is defined as the number of elements in the set
X
X
X
. 8. Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real numbers that satisfy the inequality:
∣
a
x
2
+
b
x
+
c
∣
≤
(
18
x
−
5
)
2
for all real numbers
x
|ax^2+bx+c|\le (18x-5)^2 \text{ for all real numbers } x
∣
a
x
2
+
b
x
+
c
∣
≤
(
18
x
−
5
)
2
for all real numbers
x
Find the smallest possible value of
a
+
2
b
+
5
c
a+2b+5c
a
+
2
b
+
5
c
.
4
1
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2027 divides
Find the number of positive integer pairs
1
⩽
a
,
b
⩽
2027
1\leqslant a,b \leqslant 2027
1
⩽
a
,
b
⩽
2027
that satisfy
2027
∣
a
6
+
b
5
+
b
2
2027 \mid a^6+b^5+b^2
2027
∣
a
6
+
b
5
+
b
2
(Note: For integers
a
a
a
and
b
b
b
, the notation
a
∣
b
a \mid b
a
∣
b
means that there is an integer
c
c
c
such that
a
c
=
b
ac=b
a
c
=
b
)Proposed by Valentio Iverson, Indonesia
3
1
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Geometry Indonesia Regional
Given a triangle
A
B
C
ABC
A
BC
, points
X
,
Y
,
X,Y,
X
,
Y
,
and
Z
Z
Z
are the midpoints of
B
C
,
C
A
,
BC,CA,
BC
,
C
A
,
and
A
B
AB
A
B
respectively. The perpendicular bisector of
A
B
AB
A
B
intersects line
X
Y
XY
X
Y
and line
A
C
AC
A
C
at
Z
1
Z_1
Z
1
and
Z
2
Z_2
Z
2
respectively. The perpendicular bisector of
A
C
AC
A
C
intersects line
X
Z
XZ
XZ
and line
A
B
AB
A
B
at
Y
1
Y_1
Y
1
and
Y
2
Y_2
Y
2
respectively. Let
K
K
K
be a point such that
K
Z
1
=
K
Z
2
KZ_1 = KZ_2
K
Z
1
=
K
Z
2
and
K
Y
1
=
K
Y
2
KY_1 = KY_2
K
Y
1
=
K
Y
2
. Prove that
K
B
=
K
C
KB=KC
K
B
=
K
C
.
2
1
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2024 ways to place domino Indonesia Regional
Given an
n
×
n
n \times n
n
×
n
board which is divided into
n
2
n^2
n
2
squares of size
1
×
1
1 \times 1
1
×
1
, all of which are white. Then, Aqua selects several squares from this board and colors them black. Ruby then places exactly one
1
×
2
1\times 2
1
×
2
domino on the board, so that the domino covers exactly two squares on the board. Ruby can rotate the domino into a
2
×
1
2\times 1
2
×
1
domino. After Aqua colors, it turns out there are exactly
2024
2024
2024
ways for Ruby to place a domino on the board so that it covers exactly
1
1
1
black square and
1
1
1
white square.Determine the smallest possible value of
n
n
n
so that Aqua and Ruby can do this.Proposed by Muhammad Afifurrahman, Indonesia
1
1
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Indonesia Regional Inequality
Given a real number
C
⩽
2
C\leqslant 2
C
⩽
2
. Prove that for every positive real number
x
,
y
x,y
x
,
y
with
x
y
=
1
xy=1
x
y
=
1
, the following inequality holds:
x
2
+
y
2
2
+
C
x
+
y
⩾
1
+
C
2
\sqrt{\frac{x^2+y^2}{2}} + \frac{C}{x+y} \geqslant 1 + \frac{C}{2}
2
x
2
+
y
2
+
x
+
y
C
⩾
1
+
2
C
Proposed by Fajar Yuliawan, Indonesia