MathDB
Problems
Contests
National and Regional Contests
Malaysia Contests
Malaysian IMO Training Camp
2023 Malaysian IMO Training Camp
2023 Malaysian IMO Training Camp
Part of
Malaysian IMO Training Camp
Subcontests
(8)
8
1
Hide problems
Union of two neighbourhoods
Given two positive integers
m
m
m
and
n
n
n
, find the largest
k
k
k
in terms of
m
m
m
and
n
n
n
such that the following condition holds: Any tree graph
G
G
G
with
k
k
k
vertices has two (possibly equal) vertices
u
u
u
and
v
v
v
such that for any other vertex
w
w
w
in
G
G
G
, either there is a path of length at most
m
m
m
from
u
u
u
to
w
w
w
, or there is a path of length at most
n
n
n
from
v
v
v
to
w
w
w
.Proposed by Ivan Chan Kai Chin
7
1
Hide problems
0, P(0), P(P(0)), ... eventually constant mod n
Find all polynomials with integer coefficients
P
P
P
such that for all positive integers
n
n
n
, the sequence
0
,
P
(
0
)
,
P
(
P
(
0
)
)
,
⋯
0, P(0), P(P(0)), \cdots
0
,
P
(
0
)
,
P
(
P
(
0
))
,
⋯
is eventually constant modulo
n
n
n
. Proposed by Ivan Chan Kai Chin
6
2
Hide problems
Versatile networks
Suppose there are
n
n
n
points on the plane, no three of which are collinear. Draw
n
−
1
n-1
n
−
1
non-intersecting segments (except possibly at endpoints) between pairs of points, such that it is possible to travel between any two points by travelling along the segments. Such a configuration of points and segments is called a network. Given a network, we may assign labels from
1
1
1
to
n
−
1
n-1
n
−
1
to each segment such that each segment gets a different label. Define a spin as the following operation:
∙
\bullet
∙
Choose a point
v
v
v
and rotate the labels of its adjacent segments clockwise. Formally, let
e
1
,
e
2
,
⋯
,
e
k
e_1,e_2,\cdots,e_k
e
1
,
e
2
,
⋯
,
e
k
be the segments which contain
v
v
v
as an endpoint, sorted in clockwise order (it does not matter which segment we choose as
e
1
e_1
e
1
). Then, the label of
e
i
+
1
e_{i+1}
e
i
+
1
is replaced with the label of
e
i
e_{i}
e
i
simultaneously for all
1
≤
i
≤
k
1 \le i \le k
1
≤
i
≤
k
. (where
e
k
+
1
=
e
1
e_{k+1}=e_{1}
e
k
+
1
=
e
1
)A network is nontrivial if there exists at least
2
2
2
points with at least
2
2
2
adjacent segments each. A network is versatile if any labeling of its segments can be obtained from any initial labeling using a finite amount of spins. Find all integers
n
≥
5
n \ge 5
n
≥
5
such that any nontrivial network with
n
n
n
points is versatile.Proposed by Yeoh Zi Song
Three circles meet at a point
Given a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
with circumcenter
O
O
O
, let the circle
(
A
O
D
)
(AOD)
(
A
O
D
)
intersect the segments
A
B
AB
A
B
,
A
C
AC
A
C
,
D
B
DB
D
B
,
D
C
DC
D
C
at
P
P
P
,
Q
Q
Q
,
R
R
R
,
S
S
S
respectively. Suppose
X
X
X
is the reflection of
D
D
D
about
P
Q
PQ
PQ
and
Y
Y
Y
is the reflection of
A
A
A
about
R
S
RS
RS
. Prove that the circles
(
A
O
D
)
(AOD)
(
A
O
D
)
,
(
B
P
X
)
(BPX)
(
BPX
)
,
(
C
S
Y
)
(CSY)
(
CS
Y
)
meet at a common point.Proposed by Leia Mayssa & Ivan Chan Kai Chin
5
2
Hide problems
Inversion of a special circle
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral, with circumcircle
ω
\omega
ω
and circumcenter
O
O
O
. Let
A
B
AB
A
B
intersect
C
D
CD
C
D
at
E
E
E
,
A
D
AD
A
D
intersect
B
C
BC
BC
at
F
F
F
, and
A
C
AC
A
C
intersect
B
D
BD
B
D
at
G
G
G
. The points
A
1
,
B
1
,
C
1
,
D
1
A_1, B_1, C_1, D_1
A
1
,
B
1
,
C
1
,
D
1
are chosen on rays
G
A
GA
G
A
,
G
B
GB
GB
,
G
C
GC
GC
,
G
D
GD
G
D
such that:
∙
\bullet
∙
G
A
1
G
A
=
G
B
1
G
B
=
G
C
1
G
C
=
G
D
1
G
D
\displaystyle \frac{GA_1}{GA} = \frac{GB_1}{GB} = \frac{GC_1}{GC} = \frac{GD_1}{GD}
G
A
G
A
1
=
GB
G
B
1
=
GC
G
C
1
=
G
D
G
D
1
∙
\bullet
∙
The points
A
1
,
B
1
,
C
1
,
D
1
,
O
A_1, B_1, C_1, D_1, O
A
1
,
B
1
,
C
1
,
D
1
,
O
lie on a circle.Let
A
1
B
1
A_1B_1
A
1
B
1
intersect
C
1
D
1
C_1D_1
C
1
D
1
at
K
K
K
, and
A
1
D
1
A_1D_1
A
1
D
1
intersect
B
1
C
1
B_1C_1
B
1
C
1
at
L
L
L
. Prove that the image of the circle
(
A
1
B
1
C
1
D
1
)
(A_1B_1C_1D_1)
(
A
1
B
1
C
1
D
1
)
under inversion about
ω
\omega
ω
is a line passing through the midpoints of
K
E
KE
K
E
and
L
F
LF
L
F
. Proposed by Anzo Teh Zhao Yang & Ivan Chan Kai Chin
{x_i-a}+{x_{i+1}-b} at most 1/2024
Find the maximal value of
c
>
0
c>0
c
>
0
such that for any
n
≥
1
n\ge 1
n
≥
1
, and for any
n
n
n
real numbers
x
1
,
⋯
,
x
n
x_1, \cdots, x_n
x
1
,
⋯
,
x
n
there exists real numbers
a
,
b
a ,b
a
,
b
such that
{
x
i
−
a
}
+
{
x
i
+
1
−
b
}
≤
1
2024
\{x_i-a\}+\{x_{i+1}-b\}\le \frac{1}{2024}
{
x
i
−
a
}
+
{
x
i
+
1
−
b
}
≤
2024
1
for at least
c
n
cn
c
n
indices
i
i
i
. Here,
x
n
+
1
=
x
1
x_{n+1}=x_1
x
n
+
1
=
x
1
and
{
x
}
\{x\}
{
x
}
denotes the fractional part of
x
x
x
.Proposed by Wong Jer Ren
4
2
Hide problems
a^2+b^2+c^2 divides a!+b!+c!
Do there exist infinitely many triples of positive integers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
such that
a
a
a
,
b
b
b
,
c
c
c
are pairwise coprime, and
a
!
+
b
!
+
c
!
a! + b! + c!
a
!
+
b
!
+
c
!
is divisible by
a
2
+
b
2
+
c
2
a^2 + b^2 + c^2
a
2
+
b
2
+
c
2
?Proposed by Anzo Teh Zhao Yang
Bounds on number of divisors
Find the largest constant
c
>
0
c>0
c
>
0
such that for every positive integer
n
≥
2
n\ge 2
n
≥
2
, there always exist a positive divisor
d
d
d
of
n
n
n
such that
d
≤
n
and
τ
(
d
)
≥
c
τ
(
n
)
d\le \sqrt{n}\hspace{0.5cm} \text{and} \hspace{0.5cm} \tau(d)\ge c\sqrt{\tau(n)}
d
≤
n
and
τ
(
d
)
≥
c
τ
(
n
)
where
τ
(
n
)
\tau(n)
τ
(
n
)
is the number of divisors of
n
n
n
. Proposed by Mohd. Suhaimi Ramly
3
2
Hide problems
Three circles meet at a point
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
≠
A
C
AB\neq AC
A
B
=
A
C
. Let
D
,
E
,
F
D, E, F
D
,
E
,
F
be the midpoints of the sides
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
respectively, and
M
,
N
M, N
M
,
N
be the midpoints of minor arc
B
C
BC
BC
not containing
A
A
A
and major arc
B
A
C
BAC
B
A
C
respectively. Suppose
W
,
X
,
Y
,
Z
W, X, Y, Z
W
,
X
,
Y
,
Z
are the incenter,
D
D
D
-excenter,
E
E
E
-excenter, and
F
F
F
-excenter of triangle
D
E
F
DEF
D
EF
respectively. Prove that the circumcircles of the triangles
A
B
C
ABC
A
BC
,
W
N
X
WNX
W
NX
,
Y
M
Z
YMZ
Y
MZ
meet at a common point.Proposed by Ivan Chan Kai Chin
Strange recurrence
A sequence of reals
a
1
,
a
2
,
⋯
a_1, a_2, \cdots
a
1
,
a
2
,
⋯
satisfies for all
m
>
1
m>1
m
>
1
,
a
m
+
1
a
m
−
1
=
a
m
2
−
a
1
2
a_{m+1}a_{m-1}=a_m^2-a_1^2
a
m
+
1
a
m
−
1
=
a
m
2
−
a
1
2
Prove that for all
m
>
n
>
1
m>n>1
m
>
n
>
1
, the sequence satisfies the equation
a
m
+
n
a
m
−
n
=
a
m
2
−
a
n
2
a_{m+n}a_{m-n}=a_m^2-a_n^2
a
m
+
n
a
m
−
n
=
a
m
2
−
a
n
2
Proposed by Ivan Chan Kai Chin
2
2
Hide problems
Maximizing score of permutations
Let
a
1
,
a
2
,
⋯
,
a
n
a_1, a_2, \cdots, a_n
a
1
,
a
2
,
⋯
,
a
n
be a sequence of real numbers with
a
1
+
a
2
+
⋯
+
a
n
=
0
a_1+a_2+\cdots+a_n=0
a
1
+
a
2
+
⋯
+
a
n
=
0
. Define the score
S
(
σ
)
S(\sigma)
S
(
σ
)
of a permutation
σ
=
(
b
1
,
⋯
b
n
)
\sigma=(b_1, \cdots b_n)
σ
=
(
b
1
,
⋯
b
n
)
of
(
a
1
,
⋯
a
n
)
(a_1, \cdots a_n)
(
a
1
,
⋯
a
n
)
to be the minima of the sum
(
x
1
−
b
1
)
2
+
⋯
+
(
x
n
−
b
n
)
2
(x_1-b_1)^2+\cdots+(x_n-b_n)^2
(
x
1
−
b
1
)
2
+
⋯
+
(
x
n
−
b
n
)
2
over all real numbers
x
1
≤
⋯
≤
x
n
x_1\le \cdots \le x_n
x
1
≤
⋯
≤
x
n
.Prove that
S
(
σ
)
S(\sigma)
S
(
σ
)
attains the maxima over all permutations
σ
\sigma
σ
, if and only if for all
1
≤
k
≤
n
1\le k\le n
1
≤
k
≤
n
,
b
1
+
b
2
+
⋯
+
b
k
≥
0.
b_1+b_2+\cdots+b_k\ge 0.
b
1
+
b
2
+
⋯
+
b
k
≥
0.
Proposed by Anzo Teh Zhao Yang
Reflect sides across altitude
Let
A
B
C
ABC
A
BC
be a triangle with orthocenter
H
H
H
. Let
ℓ
b
,
ℓ
c
\ell_b, \ell_c
ℓ
b
,
ℓ
c
be the reflection of lines
A
B
AB
A
B
and
A
C
AC
A
C
about
A
H
AH
A
H
respectively. Suppose
ℓ
b
\ell_b
ℓ
b
intersect
C
H
CH
C
H
at
P
P
P
, and
ℓ
c
\ell_c
ℓ
c
intersect
B
H
BH
B
H
at
Q
Q
Q
. Prove that
A
H
,
P
Q
,
B
C
AH, PQ, BC
A
H
,
PQ
,
BC
are concurrent. Proposed by Ivan Chan Kai Chin
1
2
Hide problems
Matcha Sweep Game
Let
P
P
P
be a cyclic polygon with circumcenter
O
O
O
that does not lie on any diagonal, and let
S
S
S
be the set of points on 2D plane containing
P
P
P
and
O
O
O
. The
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
M
a
t
c
h
a
S
w
e
e
p
G
a
m
e
<
/
s
p
a
n
>
<span class='latex-italic'>Matcha Sweep Game</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
M
a
t
c
ha
Sw
ee
pG
am
e
<
/
s
p
an
>
is a game between two players
A
A
A
and
B
B
B
, with
A
A
A
going first, such that each choosing a nonempty subset
T
T
T
of points in
S
S
S
that has not been previously chosen, and such that if
T
T
T
has at least
3
3
3
vertices then
T
T
T
forms a convex polygon. The game ends with all points have been chosen, with the player picking the last point wins. For which polygons
P
P
P
can
A
A
A
guarantee a win? Proposed by Anzo Teh Zhao Yang
Avoid L-triominos
Ivan has a
m
×
n
m \times n
m
×
n
board, and he color some squares black, so that no three black squares form a L-triomino up to rotations and reflections. What is the maximal number of black squares that Ivan can color?Proposed by Ivan Chan Kai Chin