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Problems
Contests
National and Regional Contests
Malaysia Contests
Malaysian APMO Camp Selection Test
2024 Malaysian APMO Camp Selection Test
2024 Malaysian APMO Camp Selection Test
Part of
Malaysian APMO Camp Selection Test
Subcontests
(5)
5
1
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Difficult geometry
Let
A
B
C
ABC
A
BC
be a scalene triangle and
D
D
D
be the feet of altitude from
A
A
A
to
B
C
BC
BC
. Let
I
1
I_1
I
1
,
I
2
I_2
I
2
be incenters of triangles
A
B
D
ABD
A
B
D
and
A
C
D
ACD
A
C
D
respectively, and let
H
1
H_1
H
1
,
H
2
H_2
H
2
be orthocenters of triangles
A
B
I
1
ABI_1
A
B
I
1
and
A
C
I
2
ACI_2
A
C
I
2
respectively. The circles
(
A
I
1
H
1
)
(AI_1H_1)
(
A
I
1
H
1
)
and
(
A
I
2
H
2
)
(AI_2H_2)
(
A
I
2
H
2
)
meet again at
X
X
X
. The lines
A
H
1
AH_1
A
H
1
and
X
I
1
XI_1
X
I
1
meet at
Y
Y
Y
, and the lines
A
H
2
AH_2
A
H
2
and
X
I
2
XI_2
X
I
2
meet at
Z
Z
Z
. Suppose the external common tangents of circles
(
B
I
1
H
1
)
(BI_1H_1)
(
B
I
1
H
1
)
and
(
C
I
2
H
2
)
(CI_2H_2)
(
C
I
2
H
2
)
meet at
U
U
U
. Prove that
U
Y
=
U
Z
UY=UZ
U
Y
=
U
Z
. Proposed by Ivan Chan Kai Chin
4
1
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Snakes on a chessboard
Ivan has a
n
×
n
n \times n
n
×
n
board. He colors some of the squares black such that every black square has exactly two neighbouring square that are also black. Let
d
n
d_n
d
n
be the maximum number of black squares possible, prove that there exist some real constants
a
a
a
,
b
b
b
,
c
≥
0
c\ge 0
c
≥
0
such that;
a
n
2
−
b
n
≤
d
n
≤
a
n
2
+
c
n
.
an^2-bn\le d_n\le an^2+cn.
a
n
2
−
bn
≤
d
n
≤
a
n
2
+
c
n
.
Proposed by Ivan Chan Kai Chin
3
1
Hide problems
f(x-f(y))=f(f(y))+f(x-2y)
Find all functions
f
:
Z
→
Z
f:\mathbb{Z}\rightarrow \mathbb{Z}
f
:
Z
→
Z
such that for all integers
x
x
x
,
y
y
y
,
f
(
x
−
f
(
y
)
)
=
f
(
f
(
y
)
)
+
f
(
x
−
2
y
)
f(x-f(y))=f(f(y))+f(x-2y)
f
(
x
−
f
(
y
))
=
f
(
f
(
y
))
+
f
(
x
−
2
y
)
Proposed by Ivan Chan Kai Chin
2
1
Hide problems
Fixed point exists except when...?
Let
k
>
1
k>1
k
>
1
. Fix a circle
ω
\omega
ω
with center
O
O
O
and radius
r
r
r
, and fix a point
A
A
A
with
O
A
=
k
r
OA=kr
O
A
=
k
r
. Let
A
B
AB
A
B
,
A
C
AC
A
C
be tangents to
ω
\omega
ω
. Choose a variable point
P
P
P
on the minor arc
B
C
BC
BC
in
ω
\omega
ω
. Lines
A
B
AB
A
B
and
C
P
CP
CP
intersect at
X
X
X
and lines
A
C
AC
A
C
and
B
P
BP
BP
intersect at
Y
Y
Y
. The circles
(
B
P
X
)
(BPX)
(
BPX
)
and
(
C
P
Y
)
(CPY)
(
CP
Y
)
meet at another point
Z
Z
Z
. Prove that the line
P
Z
PZ
PZ
always passes through a fixed point except for one value of
k
>
1
k>1
k
>
1
, and determine this value.Proposed by Ivan Chan Kai Chin
1
1
Hide problems
Eventually double of the previous term
Let
a
1
<
a
2
<
⋯
a_1<a_2< \cdots
a
1
<
a
2
<
⋯
be a strictly increasing sequence of positive integers. Suppose there exist
N
N
N
such that for all
n
>
N
n>N
n
>
N
,
a
n
+
1
∣
a
1
+
a
2
+
⋯
+
a
n
a_{n+1}\mid a_1+a_2+\cdots+a_n
a
n
+
1
∣
a
1
+
a
2
+
⋯
+
a
n
Prove that there exist
M
M
M
such that
a
m
+
1
=
2
a
m
a_{m+1}=2a_m
a
m
+
1
=
2
a
m
for all
m
>
M
m>M
m
>
M
.Proposed by Ivan Chan Kai Chin