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Problems
Contests
National and Regional Contests
Indonesia Contests
Indonesia MO
2019 Indonesia MO
2019 Indonesia MO
Part of
Indonesia MO
Subcontests
(8)
4
1
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INAMO 2019 P4 - Triangle equivalence
Let us define a
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
t
r
i
a
n
g
l
e
e
q
u
i
v
a
l
e
n
c
e
<
/
s
p
a
n
>
<span class='latex-italic'>triangle equivalence</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
t
r
ian
g
l
ee
q
u
i
v
a
l
e
n
ce
<
/
s
p
an
>
a group of numbers that can be arranged as shown
a
+
b
=
c
a+b=c
a
+
b
=
c
d
+
e
+
f
=
g
+
h
d+e+f=g+h
d
+
e
+
f
=
g
+
h
i
+
j
+
k
+
l
=
m
+
n
+
o
i+j+k+l=m+n+o
i
+
j
+
k
+
l
=
m
+
n
+
o
and so on...Where at the
j
j
j
-th row, the left hand side has
j
+
1
j+1
j
+
1
terms and the right hand side has
j
j
j
terms.Now, we are given the first
N
2
N^2
N
2
positive integers, where
N
N
N
is a positive integer. Suppose we eliminate any one number that has the same parity with
N
N
N
.Prove that the remaining
N
2
−
1
N^2-1
N
2
−
1
numbers can be formed into a
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
t
r
i
a
n
g
l
e
e
q
u
i
v
a
l
e
n
c
e
<
/
s
p
a
n
>
<span class='latex-italic'>triangle equivalence</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
t
r
ian
g
l
ee
q
u
i
v
a
l
e
n
ce
<
/
s
p
an
>
.For example, if
10
10
10
is eliminated from the first
16
16
16
numbers, the remaining numbers can be arranged into a
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
t
r
i
a
n
g
l
e
e
q
u
i
v
a
l
e
n
c
e
<
/
s
p
a
n
>
<span class='latex-italic'>triangle equivalence</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
t
r
ian
g
l
ee
q
u
i
v
a
l
e
n
ce
<
/
s
p
an
>
as shown.
1
+
3
=
4
1+3=4
1
+
3
=
4
2
+
5
+
8
=
6
+
9
2+5+8=6+9
2
+
5
+
8
=
6
+
9
7
+
11
+
12
+
14
=
13
+
15
+
16
7+11+12+14=13+15+16
7
+
11
+
12
+
14
=
13
+
15
+
16
8
1
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Annoying C
Let
n
>
1
n > 1
n
>
1
be a positive integer and
a
1
,
a
2
,
…
,
a
2
n
∈
{
−
n
,
−
n
+
1
,
…
,
n
−
1
,
n
}
a_1, a_2, \dots, a_{2n} \in \{ -n, -n + 1, \dots, n - 1, n \}
a
1
,
a
2
,
…
,
a
2
n
∈
{
−
n
,
−
n
+
1
,
…
,
n
−
1
,
n
}
. Suppose
a
1
+
a
2
+
a
3
+
⋯
+
a
2
n
=
n
+
1
a_1 + a_2 + a_3 + \dots + a_{2n} = n + 1
a
1
+
a
2
+
a
3
+
⋯
+
a
2
n
=
n
+
1
Prove that some of
a
1
,
a
2
,
…
,
a
2
n
a_1, a_2, \dots, a_{2n}
a
1
,
a
2
,
…
,
a
2
n
have sum 0.
6
1
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INAMO 2019 P6 - Ez Geo
Given a circle with center
O
O
O
, such that
A
A
A
is not on the circumcircle. Let
B
B
B
be the reflection of
A
A
A
with respect to
O
O
O
. Now let
P
P
P
be a point on the circumcircle. The line perpendicular to
A
P
AP
A
P
through
P
P
P
intersects the circle at
Q
Q
Q
. Prove that
A
P
×
B
Q
AP \times BQ
A
P
×
BQ
remains constant as
P
P
P
varies.
7
1
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INAMO 2019 P7
Determine all solutions of
x
+
y
2
=
p
m
x + y^2 = p^m
x
+
y
2
=
p
m
x
2
+
y
=
p
n
x^2 + y = p^n
x
2
+
y
=
p
n
For
x
,
y
,
m
,
n
x,y,m,n
x
,
y
,
m
,
n
positive integers and
p
p
p
being a prime.
5
1
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INAMO 2019 P5
Given that
a
a
a
and
b
b
b
are real numbers such that for infinitely many positive integers
m
m
m
and
n
n
n
,
⌊
a
n
+
b
⌋
≥
⌊
a
+
b
n
⌋
\lfloor an + b \rfloor \ge \lfloor a + bn \rfloor
⌊
an
+
b
⌋
≥
⌊
a
+
bn
⌋
⌊
a
+
b
m
⌋
≥
⌊
a
m
+
b
⌋
\lfloor a + bm \rfloor \ge \lfloor am + b \rfloor
⌊
a
+
bm
⌋
≥
⌊
am
+
b
⌋
Prove that
a
=
b
a = b
a
=
b
.
2
1
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INAMO 2019 P2
Given
19
19
19
red boxes and
200
200
200
blue boxes filled with balls. None of which is empty. Suppose that every red boxes have a maximum of
200
200
200
balls and every blue boxes have a maximum of
19
19
19
balls. Suppose that the sum of all balls in the red boxes is less than the sum of all the balls in the blue boxes. Prove that there exists a subset of the red boxes and a subset of the blue boxes such that their sum is the same.
3
1
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INAMO 2019 P3
Given that
A
B
C
D
ABCD
A
BC
D
is a rectangle such that
A
D
>
A
B
AD > AB
A
D
>
A
B
, where
E
E
E
is on
A
D
AD
A
D
such that
B
E
⊥
A
C
BE \perp AC
BE
⊥
A
C
. Let
M
M
M
be the intersection of
A
C
AC
A
C
and
B
E
BE
BE
. Let the circumcircle of
△
A
B
E
\triangle ABE
△
A
BE
intersects
A
C
AC
A
C
and
B
C
BC
BC
at
N
N
N
and
F
F
F
. Moreover, let the circumcircle of
△
D
N
E
\triangle DNE
△
D
NE
intersects
C
D
CD
C
D
at
G
G
G
. Suppose
F
G
FG
FG
intersects
A
B
AB
A
B
at
P
P
P
. Prove that
P
M
=
P
N
PM = PN
PM
=
PN
.
1
1
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INAMO 2019 Problem 01
Given that
n
n
n
and
r
r
r
are positive integers. Suppose that
1
+
2
+
⋯
+
(
n
−
1
)
=
(
n
+
1
)
+
(
n
+
2
)
+
⋯
+
(
n
+
r
)
1 + 2 + \dots + (n - 1) = (n + 1) + (n + 2) + \dots + (n + r)
1
+
2
+
⋯
+
(
n
−
1
)
=
(
n
+
1
)
+
(
n
+
2
)
+
⋯
+
(
n
+
r
)
Prove that
n
n
n
is a composite number.