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Contests
National and Regional Contests
Taiwan Contests
Taiwan APMO Prelininary
2020 Taiwan APMO Preliminary
2020 Taiwan APMO Preliminary
Part of
Taiwan APMO Prelininary
Subcontests
(7)
P5
1
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2020 Taiwan APMO Preliminary Problem 5
Let
S
S
S
is the set of permutation of {1,2,3,4,5,6,7,8} (1)For all
σ
=
σ
1
σ
2
.
.
.
σ
8
∈
S
\sigma=\sigma_1\sigma_2...\sigma_8\in S
σ
=
σ
1
σ
2
...
σ
8
∈
S
Evaluate the sum of S=
σ
1
σ
2
+
σ
3
σ
4
+
σ
5
σ
6
+
σ
7
σ
8
\sigma_1\sigma_2+\sigma_3\sigma_4+\sigma_5\sigma_6+\sigma_7\sigma_8
σ
1
σ
2
+
σ
3
σ
4
+
σ
5
σ
6
+
σ
7
σ
8
. Then for all elements in
S
S
S
,what is the arithmetic mean of S? (Notice
S
S
S
and S are different.) (2)In
S
S
S
, how many permutations are there which satisfies "For all
k
=
1
,
2
,
.
.
.
,
7
k=1,2,...,7
k
=
1
,
2
,
...
,
7
,the digit after k is not (k+1)"?
P7
1
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2020 Taiwan APMO Preliminary Problem 7
[
X
Y
Z
XYZ
X
Y
Z
] denotes the area of
△
X
Y
Z
\triangle XYZ
△
X
Y
Z
We have a
△
A
B
C
\triangle ABC
△
A
BC
,
B
C
=
6
,
C
A
=
7
,
A
B
=
8
BC=6,CA=7,AB=8
BC
=
6
,
C
A
=
7
,
A
B
=
8
(1)If
O
O
O
is the circumcenter of
△
A
B
C
\triangle ABC
△
A
BC
, find [
O
B
C
OBC
OBC
]:[
O
C
A
OCA
OC
A
]:[
O
A
B
OAB
O
A
B
] (2)If
H
H
H
is the orthocenter of
△
A
B
C
\triangle ABC
△
A
BC
, find [
H
B
C
HBC
H
BC
]:[
H
C
A
HCA
H
C
A
]:[
H
A
B
HAB
H
A
B
]
P6
1
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2020 Taiwan APMO Preliminary Problem 6
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive reals. Find the minimum value of
13
a
+
13
b
+
2
c
2
a
+
2
b
+
24
a
−
b
+
13
c
2
b
+
2
c
+
(
−
a
+
24
b
+
13
c
)
2
c
+
2
a
\dfrac{13a+13b+2c}{2a+2b}+\dfrac{24a-b+13c}{2b+2c}+\dfrac{(-a+24b+13c)}{2c+2a}
2
a
+
2
b
13
a
+
13
b
+
2
c
+
2
b
+
2
c
24
a
−
b
+
13
c
+
2
c
+
2
a
(
−
a
+
24
b
+
13
c
)
. (1)What is the minimum value? (2)If the minimum value occurs when
(
a
,
b
,
c
)
=
(
a
0
,
b
0
,
c
0
)
(a,b,c)=(a_0,b_0,c_0)
(
a
,
b
,
c
)
=
(
a
0
,
b
0
,
c
0
)
,then find
b
0
a
0
+
c
0
b
0
\frac{b_0}{a_0}+\frac{c_0}{b_0}
a
0
b
0
+
b
0
c
0
.
P4
1
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2020 Taiwan APMO Preliminary Problem 4
Let
(
a
,
b
)
=
(
a
n
,
a
n
+
1
)
,
∀
n
∈
N
(a,b)=(a_n,a_{n+1}),\forall n\in\mathbb{N}
(
a
,
b
)
=
(
a
n
,
a
n
+
1
)
,
∀
n
∈
N
all be positive interger solutions that satisfies
1
≤
a
≤
b
1\leq a\leq b
1
≤
a
≤
b
and
a
2
+
b
2
+
a
+
b
+
1
a
b
∈
N
\dfrac{a^2+b^2+a+b+1}{ab}\in\mathbb{N}
ab
a
2
+
b
2
+
a
+
b
+
1
∈
N
And the value of
a
n
a_n
a
n
is only determined by the following recurrence relation:
a
n
+
2
=
p
a
n
+
1
+
q
a
n
+
r
a_{n+2} = pa_{n+1} + qa_n + r
a
n
+
2
=
p
a
n
+
1
+
q
a
n
+
r
Find
(
p
,
q
,
r
)
(p,q,r)
(
p
,
q
,
r
)
.
P3
1
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2020 Taiwan APMO Preliminary Problem 3
Let
M
M
M
is a four digit positive interger. Write
M
M
M
backwards and get a new number
N
N
N
.(e.g
M
=
1234
M=1234
M
=
1234
then
N
=
4321
N=4321
N
=
4321
) Let
C
C
C
is the sum of every digit of
M
M
M
. If
M
,
N
,
C
M,N,C
M
,
N
,
C
satisfies (i)
d
=
gcd
(
M
−
C
,
N
−
C
)
d=\gcd(M-C,N-C)
d
=
g
cd
(
M
−
C
,
N
−
C
)
and
d
<
10
d<10
d
<
10
(ii)
M
−
C
d
=
⌊
N
2
+
1
⌋
\dfrac{M-C}{d}=\lfloor\dfrac{N}{2}+1\rfloor
d
M
−
C
=
⌊
2
N
+
1
⌋
(1)Find
d
d
d
. (2)If there are "m(s)"
M
M
M
satisfies (i) and (ii), and the largest
M
M
M
=
M
m
a
x
M_{max}
M
ma
x
. Find
(
m
,
M
m
a
x
)
(m,M_{max})
(
m
,
M
ma
x
)
P2
1
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2020 Taiwan APMO Preliminary Problem 2
A and B two people are throwing n fair coins.X and Y are the times they get heads. If throwing coins are mutually independent events, (1)When n=5, what is the possibility of X=Y? (2)When n=6, what is the possibility of X=Y+1?
P1
1
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2020 Taiwan APMO Preliminary Problem 1
Let
△
A
B
C
\triangle ABC
△
A
BC
satisfies
cos
A
:
cos
B
:
cos
C
=
1
:
1
:
2
\cos A:\cos B:\cos C=1:1:2
cos
A
:
cos
B
:
cos
C
=
1
:
1
:
2
, then
sin
A
=
t
\sin A=\sqrt{t}
sin
A
=
t
(
s
∈
N
,
t
∈
Q
+
s\in\mathbb{N},t\in\mathbb{Q^+}
s
∈
N
,
t
∈
Q
+
and
t
t
t
is an irreducible fraction). Find
s
+
t
s+t
s
+
t
.