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Problems
Contests
National and Regional Contests
Taiwan Contests
Taiwan APMO Prelininary
2021 Taiwan APMO Preliminary First Round
2021 Taiwan APMO Preliminary First Round
Part of
Taiwan APMO Prelininary
Subcontests
(7)
7
1
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Chessbord good cells
Let
n
n
n
be a fixed positive integer. We have a
n
×
n
n\times n
n
×
n
chessboard. We call a pair of cells good if they share a common vertex (May be common edge or common vertex). How many good pairs are there on this chessboard?
6
1
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2AB=\overline{AB}
Find all positive integers
A
,
B
A,B
A
,
B
satisfying the following properties: (i)
A
A
A
and
B
B
B
has same digit in decimal. (ii)
2
⋅
A
⋅
B
=
A
B
‾
2\cdot A\cdot B=\overline{AB}
2
⋅
A
⋅
B
=
A
B
(Here
⋅
\cdot
⋅
denotes multiplication,
A
B
‾
\overline{AB}
A
B
denotes we write
A
A
A
and
B
B
B
in turn. For example, if
A
=
12
,
B
=
34
A=12,B=34
A
=
12
,
B
=
34
, then
A
B
‾
=
1234
\overline{AB}=1234
A
B
=
1234
)
5
1
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external bisector
△
A
B
C
\triangle ABC
△
A
BC
,
∠
A
=
2
3
∘
,
∠
B
=
4
6
∘
\angle A=23^{\circ},\angle B=46^{\circ}
∠
A
=
2
3
∘
,
∠
B
=
4
6
∘
. Let
Γ
\Gamma
Γ
be a circle with center
C
C
C
, radius
A
C
AC
A
C
. Let the external angle bisector of
∠
B
\angle B
∠
B
intersects
Γ
\Gamma
Γ
at
M
,
N
M,N
M
,
N
. Find
∠
M
A
N
\angle MAN
∠
M
A
N
.
4
1
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m^6=1 mod n
Let
n
n
n
be a positive integer. All numbers
m
m
m
which are coprime to
n
n
n
all satisfy
m
6
≡
1
(
m
o
d
n
)
m^6\equiv 1\pmod n
m
6
≡
1
(
mod
n
)
. Find the maximum possible value of
n
n
n
.
3
1
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Skull,coin possibility
Let a board game has
10
10
10
cards:
3
3
3
skull cards,
5
5
5
coin cards and
2
2
2
blank cards. We put these
10
10
10
cards downward and shuffle them and take cards one by one from the top. Once
3
3
3
skull cards or coin cards appears we stop. What is the possibility of it stops because there appears
3
3
3
skull cards?
2
1
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Incenter, parrllel
(a) Let the incenter of
△
A
B
C
\triangle ABC
△
A
BC
be
I
I
I
. We connect
I
I
I
other
3
3
3
vertices and divide
△
A
B
C
\triangle ABC
△
A
BC
into
3
3
3
small triangles which has area
2
,
3
2,3
2
,
3
and
4
4
4
. Find the area of the inscribed circle of
△
A
B
C
\triangle ABC
△
A
BC
. (b) Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram. Point
E
,
F
E,F
E
,
F
is on
A
B
,
B
C
AB,BC
A
B
,
BC
respectively. If
[
A
E
D
]
=
7
,
[
E
B
F
]
=
3
,
[
C
D
F
]
=
6
[AED]=7,[EBF]=3,[CDF]=6
[
A
E
D
]
=
7
,
[
EBF
]
=
3
,
[
C
D
F
]
=
6
, then find
[
D
E
F
]
.
[DEF].
[
D
EF
]
.
(Here
[
X
Y
Z
]
[XYZ]
[
X
Y
Z
]
denotes the area of
X
Y
Z
XYZ
X
Y
Z
)
1
1
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Min of (|b|+|c|)/a
Let the three roots of
x
3
+
a
x
+
1
=
0
x^3+ax+1=0
x
3
+
a
x
+
1
=
0
be
α
,
β
,
γ
\alpha,\beta,\gamma
α
,
β
,
γ
where
a
a
a
is a positive real number. Let the three roots of
x
3
+
b
x
2
+
c
x
−
1
=
0
x^3+bx^2+cx-1=0
x
3
+
b
x
2
+
c
x
−
1
=
0
be
α
β
,
β
γ
,
γ
α
\frac{\alpha}{\beta},\frac{\beta}{\gamma},\frac{\gamma}{\alpha}
β
α
,
γ
β
,
α
γ
. Find the minimum value of
∣
b
∣
+
∣
c
∣
a
\dfrac{|b|+|c|}{a}
a
∣
b
∣
+
∣
c
∣
.