MathDB
Problems
Contests
National and Regional Contests
Singapore Contests
Singapore MO Open
2019 Singapore MO Open
2019 Singapore MO Open
Part of
Singapore MO Open
Subcontests
(3)
2
1
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smo 2019 open 2nd round q2
find all functions
f
:
Z
→
Z
f : \mathbb{Z} \to \mathbb{Z}
f
:
Z
→
Z
such that
f
(
−
f
(
x
)
−
f
(
y
)
)
=
1
−
x
−
y
f(-f(x)-f(y)) = 1-x-y
f
(
−
f
(
x
)
−
f
(
y
))
=
1
−
x
−
y
\forall x,y \in \mathbb{Z}
1
1
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equal angles wanted, circumcircle, perpendiculars and orthocenter related
In the acute-angled triangle
A
B
C
ABC
A
BC
with circumcircle
ω
\omega
ω
and orthocenter
H
H
H
, points
D
D
D
and
E
E
E
are the feet of the perpendiculars from
A
A
A
onto
B
C
BC
BC
and from
B
B
B
onto
A
C
AC
A
C
respecively. Let
P
P
P
be a point on the minor arc
B
C
BC
BC
of
ω
\omega
ω
. Points
M
M
M
and
N
N
N
are the feet of the perpendiculars from
P
P
P
onto lines
B
C
BC
BC
and
A
C
AC
A
C
respectively. Let
P
H
PH
P
H
and
M
N
MN
MN
intersect at
R
R
R
. Prove that
∠
D
M
R
=
∠
M
D
R
\angle DMR=\angle MDR
∠
D
MR
=
∠
M
D
R
.
3
1
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Robot tries to get to (0,0) or (1,0)
A robot is placed at point
P
P
P
on the
x
x
x
-axis but different from
(
0
,
0
)
(0,0)
(
0
,
0
)
and
(
1
,
0
)
(1,0)
(
1
,
0
)
and can only move along the axis either to the left or to the right. Two players play the following game. Player
A
A
A
gives a distance and
B
B
B
gives a direction and the robot will move the indicated distance along the indicated direction. Player
A
A
A
aims to move the robot to either
(
0
,
0
)
(0,0)
(
0
,
0
)
or
(
1
,
0
)
(1,0)
(
1
,
0
)
. Player
B
B
B
's aim is to stop
A
A
A
from achieving his aim. For which
P
P
P
can
A
A
A
win?