MathDB
Problems
Contests
National and Regional Contests
Malaysia Contests
Malaysian IMO Training Camp
2024 Malaysian IMO Training Camp
2024 Malaysian IMO Training Camp
Part of
Malaysian IMO Training Camp
Subcontests
(8)
8
1
Hide problems
Proving three lines concurrent
Given a triangle
A
B
C
ABC
A
BC
, let
I
I
I
be the incenter, and
J
J
J
be the
A
A
A
-excenter. A line
ℓ
\ell
ℓ
through
A
A
A
perpendicular to
B
C
BC
BC
intersect the lines
B
I
BI
B
I
,
C
I
CI
C
I
,
B
J
BJ
B
J
,
C
J
CJ
C
J
at
P
P
P
,
Q
Q
Q
,
R
R
R
,
S
S
S
respectively. Suppose the angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
meet
B
C
BC
BC
at
K
K
K
, and
L
L
L
is a point such that
A
L
AL
A
L
is a diameter in
(
A
B
C
)
(ABC)
(
A
BC
)
.Prove that the line
K
L
KL
K
L
,
ℓ
\ell
ℓ
, and the line through the centers of circles
(
I
P
Q
)
(IPQ)
(
I
PQ
)
and
(
J
R
S
)
(JRS)
(
J
RS
)
, are concurrent.Proposed by Chuah Jia Herng & Ivan Chan Kai Chin
7
1
Hide problems
max v_p are equal in consecutive values
Let
P
P
P
be the set of all primes. Given any positive integer
n
n
n
, define
f
(
n
)
=
max
p
∈
P
v
p
(
n
)
\displaystyle f(n) = \max_{p \in P}v_p(n)
f
(
n
)
=
p
∈
P
max
v
p
(
n
)
Prove that for any positive integer
k
≥
2
k\ge 2
k
≥
2
, there exists infinitely many positive integers
m
m
m
such that
f
(
m
+
1
)
=
f
(
m
+
2
)
=
⋯
=
f
(
m
+
k
)
f(m+1) = f(m+2) = \cdots = f(m+k)
f
(
m
+
1
)
=
f
(
m
+
2
)
=
⋯
=
f
(
m
+
k
)
Proposed by Ivan Chan Guan Yu
6
2
Hide problems
Three murderous tangent circles
Let
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
,
ω
3
\omega_3
ω
3
are three externally tangent circles, with
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
tangent at
A
A
A
. Choose points
B
B
B
and
C
C
C
on
ω
1
\omega_1
ω
1
so that lines
A
B
AB
A
B
and
A
C
AC
A
C
are tangent to
ω
3
\omega_3
ω
3
. Suppose the line
B
C
BC
BC
intersect
ω
3
\omega_3
ω
3
at two distinct points, and
X
X
X
is the intersection further away to
B
B
B
and
C
C
C
than the other one.Prove that one of the tangent lines of
ω
2
\omega_2
ω
2
passing through
X
X
X
, is also tangent to an excircle of triangle
A
B
C
ABC
A
BC
.Proposed by Ivan Chan Kai Chin
Pushing triominoes without undo-ing
Let
n
n
n
be a positive integer, and Megavan has a
(
3
n
+
1
)
×
(
3
n
+
1
)
(3n+1)\times (3n+1)
(
3
n
+
1
)
×
(
3
n
+
1
)
board. All squares, except one, are tiled by non-overlapping
1
×
3
1\times 3
1
×
3
triominoes. In each step, he can choose a triomino that is untouched in the step right before it, and then shift this triomino horizontally or vertically by one square, as long as the triominoes remain non-overlapping after this move. Show that there exist some
k
k
k
, such that after
k
k
k
moves Megavan can no longer make any valid moves irregardless of the initial configuration, and find the smallest possible
k
k
k
for each
n
n
n
.(Note: While he cannot undo a move immediately before the current step, he may still choose to move a triomino that has already been moved at least two steps before.)Proposed by Ivan Chan Kai Chin
4
2
Hide problems
Walking along a graph
Zscoder has an simple undirected graph
G
G
G
with
n
≥
3
n\ge 3
n
≥
3
vertices. Navi labels a positive integer to each vertex, and places a token at one of the vertex. This vertex is now marked red. In each turn, Zscoder plays with following rule:
∙
\bullet
∙
If the token is currently at vertex
v
v
v
with label
t
t
t
, then he can move the token along the edges in
G
G
G
(possibly repeating some edges) exactly
t
t
t
times. After these
t
t
t
moves, he marks the current vertex red where the token is at if it is unmarked, or does nothing otherwise, then finishes the turn.Zscoder claims that he can mark all vertices in
G
G
G
red after finite number of turns, regardless of Navi's labels and starting vertex. What is the minimum number of edges must
G
G
G
have, in terms of
n
n
n
?Proposed by Yeoh Zi Song
Almost similar to continuity
Fix a real polynomial
P
P
P
with degree at least
1
1
1
, and a real number
c
c
c
. Prove that there exist a real number
k
k
k
such that for all reals
a
a
a
and
b
b
b
, P(a)+P(b)=c \Rightarrow |a+b|
Proposed by Wong Jer Ren
5
2
Hide problems
Good residue sets
Let
n
n
n
be an odd integer and
m
=
ϕ
(
n
)
m=\phi(n)
m
=
ϕ
(
n
)
be the Euler's totient function. Call a set of residues
T
=
{
a
1
,
⋯
,
a
k
}
(
m
o
d
n
)
T=\{a_1, \cdots, a_k\} \pmod n
T
=
{
a
1
,
⋯
,
a
k
}
(
mod
n
)
to be good if
gcd
(
a
i
,
n
)
>
1
\gcd(a_i, n) > 1
g
cd
(
a
i
,
n
)
>
1
∀
i
\forall i
∀
i
, and
gcd
(
a
i
,
a
j
)
=
1
,
∀
i
≠
j
\gcd(a_i, a_j) = 1, \forall i \neq j
g
cd
(
a
i
,
a
j
)
=
1
,
∀
i
=
j
. Define the set
S
n
S_n
S
n
consisting of the residues
∑
i
=
1
k
a
i
m
(
m
o
d
n
)
\sum_{i=1}^k a_i ^m\pmod{n}
i
=
1
∑
k
a
i
m
(
mod
n
)
over all possible residue sets
T
=
{
a
1
,
⋯
,
a
k
}
T=\{a_1,\cdots,a_k\}
T
=
{
a
1
,
⋯
,
a
k
}
that is good. Determine
∣
S
n
∣
|S_n|
∣
S
n
∣
.Proposed by Anzo Teh Zhao Yang
(a^2+1)(b^2+1)((a+b)^2+1) being a square
Do there exist infinitely many positive integers
a
,
b
a, b
a
,
b
such that
(
a
2
+
1
)
(
b
2
+
1
)
(
(
a
+
b
)
2
+
1
)
(a^2+1)(b^2+1)((a+b)^2+1)
(
a
2
+
1
)
(
b
2
+
1
)
((
a
+
b
)
2
+
1
)
is a perfect square?Proposed Ivan Chan Guan Yu
3
2
Hide problems
x^3+2023xy+y^3 is CRS mod p?
Find all primes
p
p
p
such that for any integer
k
k
k
, there exist two integers
x
x
x
and
y
y
y
such that
x
3
+
2023
x
y
+
y
3
≡
k
(
m
o
d
p
)
x^3+2023xy+y^3 \equiv k \pmod p
x
3
+
2023
x
y
+
y
3
≡
k
(
mod
p
)
Proposed by Tristan Chaang Tze Shen
Giving candies
Given
n
n
n
students in the plane such that the
n
(
n
−
1
)
2
\frac{n(n-1)}{2}
2
n
(
n
−
1
)
distances are pairwise distinct. Each student gives a candy each to the
k
k
k
students closest to him. Given that each student receives the same amount of candies, determine all possible values of
n
n
n
in terms of
k
k
k
.Proposed by Wong Jer Ren
2
2
Hide problems
Sum of certain images of P is still an image
Let
k
k
k
be a positive integer. Find all collection of integers
(
a
1
,
a
2
,
⋯
,
a
k
)
(a_1, a_2,\cdots, a_k)
(
a
1
,
a
2
,
⋯
,
a
k
)
such that there exist a non-linear polynomial
P
P
P
with integer coefficients, so that for all positive integers
n
n
n
there exist a positive integer
m
m
m
satisfying:
P
(
n
+
a
1
)
+
P
(
n
+
a
2
)
+
.
.
.
+
P
(
n
+
a
k
)
=
P
(
m
)
P(n+a_1)+P(n+a_2)+...+P(n+a_k)=P(m)
P
(
n
+
a
1
)
+
P
(
n
+
a
2
)
+
...
+
P
(
n
+
a
k
)
=
P
(
m
)
Proposed by Ivan Chan Kai Chin
Common Sequences
A finite sequence of decimal digits from
{
0
,
1
,
⋯
,
9
}
\{0,1,\cdots, 9\}
{
0
,
1
,
⋯
,
9
}
is said to be common if for each sufficiently large positive integer
n
n
n
, there exists a positive integer
m
m
m
such that the expansion of
n
n
n
in base
m
m
m
ends with this sequence of digits.For example,
0
0
0
is common because for any large
n
n
n
, the expansion of
n
n
n
in base
n
n
n
is
10
10
10
, whereas
00
00
00
is not common because for any squarefree
n
n
n
, the expansion of
n
n
n
in any base cannot end with
00
00
00
.Determine all common sequences.Proposed by Wong Jer Ren
1
2
Hide problems
Cyclic implies parallel?
Let
A
B
C
ABC
A
BC
be an acute triangle with orthocenter
H
H
H
, and let
B
E
BE
BE
and
C
F
CF
CF
be the altitudes of the triangle. Choose two points
P
P
P
and
Q
Q
Q
on rays
B
H
BH
B
H
and
C
H
CH
C
H
respectively, such that:
∙
\bullet
∙
P
Q
PQ
PQ
is parallel to
B
C
BC
BC
;
∙
\bullet
∙
The quadrilateral
A
P
H
Q
APHQ
A
P
H
Q
is cyclic. Suppose the circumcircles of triangles
A
P
F
APF
A
PF
and
A
Q
E
AQE
A
QE
meet again at
X
≠
A
X\neq A
X
=
A
. Prove that
A
X
AX
A
X
is parallel to
B
C
BC
BC
.Proposed by Ivan Chan Kai Chin
Proving lengths are equal
A cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
has diameter
A
C
AC
A
C
with circumcircle
ω
\omega
ω
. Let
K
K
K
be the foot of the perpendicular from
C
C
C
to
B
D
BD
B
D
, and the tangent to
ω
\omega
ω
at
A
A
A
meets
B
D
BD
B
D
at
T
T
T
. Let the line
A
K
AK
A
K
meets
ω
\omega
ω
at
X
X
X
and choose a point
Y
Y
Y
on line
A
K
AK
A
K
such that
∠
T
Y
A
=
9
0
∘
\angle TYA=90^{\circ}
∠
T
Y
A
=
9
0
∘
. Prove that
A
Y
=
K
X
AY=KX
A
Y
=
K
X
.Proposed by Anzo Teh Zhao Yang