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Contests
National and Regional Contests
Indonesia Contests
Indonesia Regional
2023 Indonesia Regional
2023 Indonesia Regional
Part of
Indonesia Regional
Subcontests
(6)
1
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2023 Indonesia Regional (Short Answer Section)
The 2023 Indonesia Regional MO was held on 5th of June 2023. There are 8 short form problems and 5 essays.The 8 short form problems all have nonnegative integer answers. The time is 60 minutes.1. Given two non constant arithmetic sequences
a
1
,
a
2
,
…
a_1,a_2,\ldots
a
1
,
a
2
,
…
dan
b
1
,
b
2
,
…
b_1,b_2,\ldots
b
1
,
b
2
,
…
. If
a
228
=
b
15
a_{228} = b_{15}
a
228
=
b
15
and
a
8
=
b
5
a_8 = b_5
a
8
=
b
5
, find
b
4
−
b
3
a
2
−
a
1
\dfrac{b_4-b_3}{a_2-a_1}
a
2
−
a
1
b
4
−
b
3
. 2. Find the number of positive integers
n
≤
221
n\leq 221
n
≤
221
such that
1
2
+
2
2
+
…
+
n
2
1
+
2
+
…
+
n
\frac{1^2+2^2+\ldots+n^2}{1+2+\ldots+n}
1
+
2
+
…
+
n
1
2
+
2
2
+
…
+
n
2
is an integer.3. How many different lines on a cartesian coordinate system has equation
a
x
+
b
y
=
0
ax+by=0
a
x
+
b
y
=
0
with
a
,
b
∈
{
0
,
1
,
2
,
3
,
6
,
7
}
a,b \in \{0,1,2,3,6,7\}
a
,
b
∈
{
0
,
1
,
2
,
3
,
6
,
7
}
? Note:
a
a
a
and
b
b
b
are not necessarily distinct 4. Given a rectangle
A
B
C
D
ABCD
A
BC
D
and equilateral triangles
B
C
P
BCP
BCP
and
C
D
Q
CDQ
C
D
Q
in the figure below. If
A
B
=
8
AB = 8
A
B
=
8
and
A
D
=
10
AD = 10
A
D
=
10
, the sum of the area
A
C
P
ACP
A
CP
and
A
C
Q
ACQ
A
CQ
is
m
3
+
n
m\sqrt{3}+n
m
3
+
n
, for rational numbers
m
,
n
m,n
m
,
n
. Find
m
+
n
m+n
m
+
n
. 5. Find the number of three element subsets of
S
=
{
1
,
5
,
6
,
7
,
9
,
10
,
11
,
13
,
15
,
20
,
27
,
45
}
S = \{1,5,6,7,9,10,11,13,15,20,27,45 \}
S
=
{
1
,
5
,
6
,
7
,
9
,
10
,
11
,
13
,
15
,
20
,
27
,
45
}
such that the multiplication of those three elements is divisible by
18
18
18
.6. In the figure below, if
A
P
=
22
AP=22
A
P
=
22
,
C
Q
=
14
CQ=14
CQ
=
14
,
R
E
=
35
RE=35
RE
=
35
, with
P
Q
R
PQR
PQR
being an equilateral triangle, find the value of
B
P
+
Q
D
+
R
F
BP+QD+RF
BP
+
Q
D
+
RF
. 7. Find the sum of all positive integers
n
n
n
such that
2
n
−
12
+
2
n
+
40
\sqrt{2n-12}+\sqrt{2n+40}
2
n
−
12
+
2
n
+
40
is a positive integer. 8. Given positive real numbers
a
a
a
and
b
b
b
that satisfies \begin{align*} \frac{1}{a}+\frac{1}{b} &\leq 2\sqrt{\frac{3}{7}}\\ (a-b)^2 &= \frac{9}{49}(ab)^3 \end{align*} Find the maximum value of
a
2
+
b
2
a^2+b^2
a
2
+
b
2
.
1
1
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Longest line to cut a square into a 20:23 ratio
Let
A
B
C
D
ABCD
A
BC
D
be a square with side length
43
43
43
and points
X
X
X
and
Y
Y
Y
lies on sides
A
D
AD
A
D
and
B
C
BC
BC
respectively such that the ratio of the area of
A
B
Y
X
ABYX
A
B
Y
X
to the area of
C
D
X
Y
CDXY
C
D
X
Y
is
20
:
23
20 : 23
20
:
23
. Find the maximum possible length of
X
Y
XY
X
Y
.
2
1
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Perfect Cubes
Let
K
K
K
be a positive integer such that there exist a triple of positive integers
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
such that
x
3
+
K
y
,
y
3
+
K
z
,
and
z
3
+
K
x
x^3+Ky , y^3 + Kz, \text{and } z^3 + Kx
x
3
+
Ky
,
y
3
+
Kz
,
and
z
3
+
K
x
are all perfect cubes. (a) Prove that
K
≠
2
K \ne 2
K
=
2
and
K
≠
4
K \ne 4
K
=
4
(b) Find the minimum value of
K
K
K
that satisfies.
4
1
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find irrational numbers such that these two things are rational
Find all irrational real numbers
α
\alpha
α
such that
α
3
−
15
α
and
α
4
−
56
α
\alpha^3 - 15 \alpha \text{ and } \alpha^4 - 56 \alpha
α
3
−
15
α
and
α
4
−
56
α
are both rational numbers.
5
1
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Indonesia Regional Part II P5
Given
△
A
B
C
\triangle ABC
△
A
BC
and points
D
D
D
and
E
E
E
at the line
B
C
BC
BC
, furthermore there are points
X
X
X
and
Y
Y
Y
inside
△
A
B
C
\triangle ABC
△
A
BC
. Let
P
P
P
be the intersection of line
A
D
AD
A
D
and
X
E
XE
XE
, and
Q
Q
Q
be the intersection of line
A
E
AE
A
E
and
Y
D
YD
Y
D
. If there exist a circle that passes through
X
,
Y
,
D
,
E
X, Y, D, E
X
,
Y
,
D
,
E
, and
∠
B
X
E
+
∠
B
C
A
=
∠
C
Y
D
+
∠
C
B
A
=
18
0
∘
\angle BXE + \angle BCA = \angle CYD + \angle CBA = 180^{\circ}
∠
BXE
+
∠
BC
A
=
∠
C
Y
D
+
∠
CB
A
=
18
0
∘
Prove that the line
B
P
BP
BP
,
C
Q
CQ
CQ
, and the perpendicular bisector of
B
C
BC
BC
intersect at one point.
3
1
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Find the maximum value of B
Find the maximum value of an integer
B
B
B
such that for every 9 distinct natural number with the sum of
2023
2023
2023
, there must exist a sum of 4 of the number that is greater than or equal to
B
B
B