MathDB
Problems
Contests
National and Regional Contests
Malaysia Contests
Malaysian IMO Training Camp
BIMO 2021
BIMO 2021
Part of
Malaysian IMO Training Camp
Subcontests
(3)
3
1
Hide problems
A, Q, R are colinear
Let
A
B
C
ABC
A
BC
be an actue triangle with
A
B
<
A
C
AB<AC
A
B
<
A
C
. Let
Γ
\Gamma
Γ
be its circumcircle,
I
I
I
its incenter and
P
P
P
is a point on
Γ
\Gamma
Γ
such that
∠
A
P
I
=
9
0
∘
\angle API=90^{\circ}
∠
A
P
I
=
9
0
∘
. Let
Q
Q
Q
be a point on
Γ
\Gamma
Γ
such that
Q
B
⋅
tan
∠
B
=
Q
C
⋅
tan
∠
C
QB\cdot\tan \angle B=QC\cdot \tan \angle C
QB
⋅
tan
∠
B
=
QC
⋅
tan
∠
C
Consider a point
R
R
R
such that
P
R
PR
PR
is tangent to
Γ
\Gamma
Γ
and
B
R
=
C
R
BR=CR
BR
=
CR
. Prove that the points
A
,
Q
,
R
A, Q, R
A
,
Q
,
R
are colinear.
2
2
Hide problems
Empty a pile of stones
There are
k
k
k
piles of stones with
2020
2020
2020
stones in each pile. Amber can choose any two non-empty piles of stones, and Barbara can take one stone from one of the two chosen piles and puts it into the other pile. Amber wins if she can eventually make an empty pile. What is the least
k
k
k
such that Amber can always win?
KM perpendicular to EF
Let
A
B
C
ABC
A
BC
be a triangle with incircle centered at
I
I
I
, tangent to sides
A
C
AC
A
C
and
A
B
AB
A
B
at
E
E
E
and
F
F
F
respectively. Let
N
N
N
be the midpoint of major arc
B
A
C
BAC
B
A
C
. Let
I
N
IN
I
N
intersect
E
F
EF
EF
at
K
K
K
, and
M
M
M
be the midpoint of
B
C
BC
BC
. Prove that
K
M
⊥
E
F
KM\perp EF
K
M
⊥
EF
.
1
2
Hide problems
non-trivial divisors add up to n-1
Given a natural number
n
n
n
, call a divisor
d
d
d
of
n
n
n
to be
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
n
o
n
t
r
i
v
i
a
l
<
/
s
p
a
n
>
<span class='latex-italic'>nontrivial</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
n
o
n
t
r
i
v
ia
l
<
/
s
p
an
>
if
d
>
1
d>1
d
>
1
. A natural number
n
n
n
is
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
o
o
d
<
/
s
p
a
n
>
<span class='latex-italic'>good</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
oo
d
<
/
s
p
an
>
if one or more distinct nontrivial divisors of
n
n
n
sum up to
n
−
1
n-1
n
−
1
. Prove that every natural number
n
n
n
has a multiple that is good.
f(x^2+f(y))=f(f(y)-x^2)+f(xy)
Find all continuous functions
f
:
R
→
R
f : \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
such that for all real numbers
x
,
y
x, y
x
,
y
f
(
x
2
+
f
(
y
)
)
=
f
(
f
(
y
)
−
x
2
)
+
f
(
x
y
)
f(x^2+f(y))=f(f(y)-x^2)+f(xy)
f
(
x
2
+
f
(
y
))
=
f
(
f
(
y
)
−
x
2
)
+
f
(
x
y
)
[Extra: Can you solve this without continuity?]