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International Contests
Iran-Singapore-Taiwan Friendly Math Competition
2022 Iran-Taiwan Friendly Math Competition
2022 Iran-Taiwan Friendly Math Competition
Part of
Iran-Singapore-Taiwan Friendly Math Competition
Subcontests
(6)
4
1
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Boring Geometry: Easy Exercise of Method of Moving Point
Given an acute triangle
A
B
C
ABC
A
BC
, let
P
P
P
be an arbitrary point on segment
B
C
BC
BC
. A line passing through
P
P
P
and perpendicular to
A
C
AC
A
C
intersects
A
B
AB
A
B
at
P
b
P_b
P
b
. A line passing through
P
P
P
and perpendicular to
A
B
AB
A
B
intersects
A
C
AC
A
C
at
P
c
P_c
P
c
. Prove that the circumcircle of triangle
A
P
b
P
c
AP_bP_c
A
P
b
P
c
passes through a fixed point other than
A
A
A
when
P
P
P
varies on segment
B
C
BC
BC
.Proposed by ltf0501
1
1
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aops compiled \lfloor :)
Let
k
⩾
2
k\geqslant 2
k
⩾
2
be an integer, and
a
,
b
a,b
a
,
b
be real numbers. prove that
a
−
b
a-b
a
−
b
is an integer divisible by
k
k
k
if and only if for every positive integer
n
n
n
⌊
a
n
⌋
≡
⌊
b
n
⌋
(
m
o
d
k
)
\lfloor an \rfloor \equiv \lfloor bn \rfloor \ (mod \ k)
⌊
an
⌋
≡
⌊
bn
⌋
(
m
o
d
k
)
Proposed by Navid Safaei
5
1
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Classic-like Combo Problem
Let
S
S
S
be the set of lattice points whose both coordinates are positive integers no larger than
2022
2022
2022
. i.e.,
S
=
{
(
x
,
y
)
∣
x
,
y
∈
N
,
1
≤
x
,
y
≤
2022
}
S=\{(x, y) \mid x, y\in \mathbb{N}, \, 1\leq x, y\leq 2022\}
S
=
{(
x
,
y
)
∣
x
,
y
∈
N
,
1
≤
x
,
y
≤
2022
}
. We put a card with one gold side and one black side on each point in
S
S
S
. We call a rectangle "good" if:(i) All of its sides are parallel to the axes and have positive integer coordinates no larger than
2022
2022
2022
. (ii) The cards on its top-left and bottom-right corners are showing gold, and the cards on its top-right and bottom-left corners are showing black.Each "move" consists of choosing a good rectangle and flipping all cards simultaneously on its four corners. Find the maximum possible number of moves one can perform, or show that one can perform infinitely many moves.Proposed by CSJL
6
1
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Euclidian algorithm-ish function
Find all completely multipiclative functions
f
:
Z
→
Z
⩾
0
f:\mathbb{Z}\rightarrow \mathbb{Z}_{\geqslant 0}
f
:
Z
→
Z
⩾
0
such that for any
a
,
b
∈
Z
a,b\in \mathbb{Z}
a
,
b
∈
Z
and
b
≠
0
b\neq 0
b
=
0
, there exist integers
q
,
r
q,r
q
,
r
such that
a
=
b
q
+
r
a=bq+r
a
=
b
q
+
r
and
f
(
r
)
<
f
(
b
)
f(r)<f(b)
f
(
r
)
<
f
(
b
)
Proposed by Navid Safaei
2
1
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Function using maximum
Find all functions
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
such that:
∙
\bullet
∙
f
(
x
)
<
2
f(x)<2
f
(
x
)
<
2
for all
x
∈
(
0
,
1
)
x\in (0,1)
x
∈
(
0
,
1
)
;
∙
\bullet
∙
for all real numbers
x
,
y
x,y
x
,
y
we have:
m
a
x
{
f
(
x
+
y
)
,
f
(
x
−
y
)
}
=
f
(
x
)
+
f
(
y
)
max\{f(x+y),f(x-y)\}=f(x)+f(y)
ma
x
{
f
(
x
+
y
)
,
f
(
x
−
y
)}
=
f
(
x
)
+
f
(
y
)
Proposed by Navid Safaei
3
1
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Parallel Lines with Incenter Properties
Let
A
B
C
ABC
A
BC
be a scalene triangle with
I
I
I
be its incenter. The incircle touches
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at
D
D
D
,
E
E
E
,
F
F
F
, respectively.
Y
Y
Y
,
Z
Z
Z
are the midpoints of
D
F
DF
D
F
,
D
E
DE
D
E
respectively, and
S
S
S
,
V
V
V
are the intersections of lines
Y
Z
YZ
Y
Z
and
B
C
BC
BC
,
A
D
AD
A
D
, respectively.
T
T
T
is the second intersection of
⊙
(
A
B
C
)
\odot(ABC)
⊙
(
A
BC
)
and
A
S
AS
A
S
.
K
K
K
is the foot from
I
I
I
to
A
T
AT
A
T
. Prove that
K
V
KV
K
V
is parallel to
D
T
DT
D
T
. Proposed by ltf0501