MathDB
Problems
Contests
National and Regional Contests
Malaysia Contests
Malaysian IMO Training Camp
BIMO 2022
BIMO 2022
Part of
Malaysian IMO Training Camp
Subcontests
(7)
6
1
Hide problems
FX and BZ meet at w
Given a triangle
A
B
C
ABC
A
BC
with
A
B
=
A
C
AB=AC
A
B
=
A
C
and circumcenter
O
O
O
. Let
D
D
D
and
E
E
E
be midpoints of
A
C
AC
A
C
and
A
B
AB
A
B
respectively, and let
D
E
DE
D
E
intersect
A
O
AO
A
O
at
F
F
F
. Denote
ω
\omega
ω
to be the circle
(
B
O
E
)
(BOE)
(
BOE
)
. Let
B
D
BD
B
D
intersect
ω
\omega
ω
again at
X
X
X
and let
A
X
AX
A
X
intersect
ω
\omega
ω
again at
Y
Y
Y
. Suppose the line parallel to
A
B
AB
A
B
passing through
O
O
O
meets
C
Y
CY
C
Y
at
Z
Z
Z
. Prove that the lines
F
X
FX
FX
and
B
Z
BZ
BZ
meet at
ω
\omega
ω
.Proposed by Ivan Chan Kai Chin
5
1
Hide problems
rad(f(a)f(b)+f(b)f(c)+f(c)f(a))=rad(ab+bc+ca)
Find all functions
f
:
Z
→
Z
f : \mathbb{Z}\rightarrow \mathbb{Z}
f
:
Z
→
Z
such that for all prime
p
p
p
the following condition holds:
p
∣
a
b
+
b
c
+
c
a
⟺
p
∣
f
(
a
)
f
(
b
)
+
f
(
b
)
f
(
c
)
+
f
(
c
)
f
(
a
)
p \mid ab + bc + ca \iff p \mid f(a)f(b) + f(b)f(c) + f(c)f(a)
p
∣
ab
+
b
c
+
c
a
⟺
p
∣
f
(
a
)
f
(
b
)
+
f
(
b
)
f
(
c
)
+
f
(
c
)
f
(
a
)
Proposed by Anzo Teh Zhao Yang
3
1
Hide problems
PQ parallel RS
Let
ω
\omega
ω
be the circumcircle of an actue triangle
A
B
C
ABC
A
BC
and let
H
H
H
be the feet of aliitude from
A
A
A
to
B
C
BC
BC
. Let
M
M
M
and
N
N
N
be the midpoints of the sides
A
C
AC
A
C
and
A
B
AB
A
B
. The lines
B
M
BM
BM
and
C
N
CN
CN
intersect each other at
G
G
G
and intersect
ω
\omega
ω
at
P
P
P
and
Q
Q
Q
respectively. The circles
(
H
M
G
)
(HMG)
(
H
MG
)
and
(
H
N
G
)
(HNG)
(
H
NG
)
intersect the segments
H
P
HP
H
P
and
H
Q
HQ
H
Q
again at
R
R
R
and
S
S
S
respectively. Prove that
P
Q
∥
R
S
PQ\parallel RS
PQ
∥
RS
.
4
2
Hide problems
Sum of P(x_i) are equal
Given a polynomial
P
∈
Z
[
X
]
P\in \mathbb{Z}[X]
P
∈
Z
[
X
]
of degree
k
k
k
, show that there always exist
2
d
2d
2
d
distinct integers
x
1
,
x
2
,
⋯
x
2
d
x_1, x_2, \cdots x_{2d}
x
1
,
x
2
,
⋯
x
2
d
such that
P
(
x
1
)
+
P
(
x
2
)
+
⋯
P
(
x
d
)
=
P
(
x
d
+
1
)
+
P
(
x
d
+
2
)
+
⋯
+
P
(
x
2
d
)
P(x_1)+P(x_2)+\cdots P(x_{d})=P(x_{d+1})+P(x_{d+2})+\cdots + P(x_{2d})
P
(
x
1
)
+
P
(
x
2
)
+
⋯
P
(
x
d
)
=
P
(
x
d
+
1
)
+
P
(
x
d
+
2
)
+
⋯
+
P
(
x
2
d
)
for some
d
≤
k
+
1
d\le k+1
d
≤
k
+
1
.[Extra: Is this still true if
d
≤
k
d\le k
d
≤
k
? (Of course false for linear polynomials, but what about higher degree?)]
Minimum degree in 2 variables
Given a positive integer
n
n
n
, suppose that
P
(
x
,
y
)
P(x,y)
P
(
x
,
y
)
is a real polynomial such that
P
(
x
,
y
)
=
1
1
+
x
+
y
for all
x
,
y
∈
{
0
,
1
,
2
,
…
,
n
}
P(x,y)=\frac{1}{1+x+y} \hspace{0.5cm} \text{for all $x,y\in\{0,1,2,\dots,n\}$}
P
(
x
,
y
)
=
1
+
x
+
y
1
for all
x
,
y
∈
{
0
,
1
,
2
,
…
,
n
}
What is the minimum degree of
P
P
P
?Proposed by Loke Zhi Kin
Open
1
Hide problems
f^k(n)=P(n)
Given
k
≥
2
k\ge 2
k
≥
2
, for which polynomials
P
∈
Z
[
X
]
P\in \mathbb{Z}[X]
P
∈
Z
[
X
]
does there exist a function
h
:
N
→
N
h:\mathbb{N}\rightarrow\mathbb{N}
h
:
N
→
N
with
h
(
k
)
(
n
)
=
P
(
n
)
h^{(k)}(n)=P(n)
h
(
k
)
(
n
)
=
P
(
n
)
?
2
5
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1
7
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