MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2020 China Team Selection Test
2020 China Team Selection Test
Part of
China Team Selection Test
Subcontests
(6)
2
1
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BF_|_BC iff circles omega_1 m $\omega_2 are symmetric wrt L
Given an isosceles triangle
△
A
B
C
\triangle ABC
△
A
BC
,
A
B
=
A
C
AB=AC
A
B
=
A
C
. A line passes through
M
M
M
, the midpoint of
B
C
BC
BC
, and intersects segment
A
B
AB
A
B
and ray
C
A
CA
C
A
at
D
D
D
and
E
E
E
, respectively. Let
F
F
F
be a point of
M
E
ME
ME
such that
E
F
=
D
M
EF=DM
EF
=
D
M
, and
K
K
K
be a point on
M
D
MD
M
D
. Let
Γ
1
\Gamma_1
Γ
1
be the circle passes through
B
,
D
,
K
B,D,K
B
,
D
,
K
and
Γ
2
\Gamma_2
Γ
2
be the circle passes through
C
,
E
,
K
C,E,K
C
,
E
,
K
.
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
intersect again at
L
≠
K
L \neq K
L
=
K
. Let
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
be the circumcircle of
△
L
D
E
\triangle LDE
△
L
D
E
and
△
L
K
M
\triangle LKM
△
L
K
M
. Prove that, if
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
are symmetric wrt
L
L
L
, then
B
F
BF
BF
is perpendicular to
B
C
BC
BC
.
1
1
Hide problems
Linear combinations of n-th roots of unity cannot vanish so many
Let
ω
\omega
ω
be a
n
n
n
-th primitive root of unity. Given complex numbers
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots,a_n
a
1
,
a
2
,
⋯
,
a
n
, and
p
p
p
of them are non-zero. Let
b
k
=
∑
i
=
1
n
a
i
ω
k
i
b_k=\sum_{i=1}^n a_i \omega^{ki}
b
k
=
i
=
1
∑
n
a
i
ω
ki
for
k
=
1
,
2
,
⋯
,
n
k=1,2,\cdots, n
k
=
1
,
2
,
⋯
,
n
. Prove that if
p
>
0
p>0
p
>
0
, then at least
n
p
\tfrac{n}{p}
p
n
numbers in
b
1
,
b
2
,
⋯
,
b
n
b_1,b_2,\cdots,b_n
b
1
,
b
2
,
⋯
,
b
n
are non-zero.
4
1
Hide problems
Bounding Diophantine Equation
Show that the following equation has finitely many solutions
(
t
,
A
,
x
,
y
,
z
)
(t,A,x,y,z)
(
t
,
A
,
x
,
y
,
z
)
in positive integers
t
(
1
−
A
−
2
)
(
1
−
x
−
2
)
(
1
−
y
−
2
)
(
1
−
z
−
2
)
=
(
1
+
x
−
1
)
(
1
+
y
−
1
)
(
1
+
z
−
1
)
\sqrt{t(1-A^{-2})(1-x^{-2})(1-y^{-2})(1-z^{-2})}=(1+x^{-1})(1+y^{-1})(1+z^{-1})
t
(
1
−
A
−
2
)
(
1
−
x
−
2
)
(
1
−
y
−
2
)
(
1
−
z
−
2
)
=
(
1
+
x
−
1
)
(
1
+
y
−
1
)
(
1
+
z
−
1
)
5
1
Hide problems
Easy inequality with integer variable
Let
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots,a_n
a
1
,
a
2
,
⋯
,
a
n
be a permutation of
1
,
2
,
⋯
,
n
1,2,\cdots,n
1
,
2
,
⋯
,
n
. Among all possible permutations, find the minimum of
∑
i
=
1
n
min
{
a
i
,
2
i
−
1
}
.
\sum_{i=1}^n \min \{ a_i,2i-1 \}.
i
=
1
∑
n
min
{
a
i
,
2
i
−
1
}
.
6
1
Hide problems
Separating vertices, retaining connectivity
Given a simple, connected graph with
n
n
n
vertices and
m
m
m
edges. Prove that one can find at least
m
m
m
ways separating the set of vertices into two parts, such that the induced subgraphs on both parts are connected.
3
1
Hide problems
Involuted sum with NT structure
For a non-empty finite set
A
A
A
of positive integers, let
lcm
(
A
)
\text{lcm}(A)
lcm
(
A
)
denote the least common multiple of elements in
A
A
A
, and let
d
(
A
)
d(A)
d
(
A
)
denote the number of prime factors of
lcm
(
A
)
\text{lcm}(A)
lcm
(
A
)
(counting multiplicity). Given a finite set
S
S
S
of positive integers, and
f
S
(
x
)
=
∑
∅
≠
A
⊂
S
(
−
1
)
∣
A
∣
x
d
(
A
)
lcm
(
A
)
.
f_S(x)=\sum_{\emptyset \neq A \subset S} \frac{(-1)^{|A|} x^{d(A)}}{\text{lcm}(A)}.
f
S
(
x
)
=
∅
=
A
⊂
S
∑
lcm
(
A
)
(
−
1
)
∣
A
∣
x
d
(
A
)
.
Prove that, if
0
≤
x
≤
2
0 \le x \le 2
0
≤
x
≤
2
, then
−
1
≤
f
S
(
x
)
≤
0
-1 \le f_S(x) \le 0
−
1
≤
f
S
(
x
)
≤
0
.