MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2001 China Team Selection Test
2001 China Team Selection Test
Part of
China Team Selection Test
Subcontests
(3)
3
2
Hide problems
f[x^k + f(y)] = y +[f(x)]^k
For a given natural number
k
>
1
k > 1
k
>
1
, find all functions
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that for all
x
,
y
∈
R
x, y \in \mathbb{R}
x
,
y
∈
R
,
f
[
x
k
+
f
(
y
)
]
=
y
+
[
f
(
x
)
]
k
f[x^k + f(y)] = y +[f(x)]^k
f
[
x
k
+
f
(
y
)]
=
y
+
[
f
(
x
)
]
k
.
Minimize |x^3 - ax^2 - bx - c|
Let
F
=
max
1
≤
x
≤
3
∣
x
3
−
a
x
2
−
b
x
−
c
∣
F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|
F
=
max
1
≤
x
≤
3
∣
x
3
−
a
x
2
−
b
x
−
c
∣
. When
a
a
a
,
b
b
b
,
c
c
c
run over all the real numbers, find the smallest possible value of
F
F
F
.
2
2
Hide problems
China TST 2001 natural number sequence
a
a
a
and
b
b
b
are natural numbers such that
b
>
a
>
1
b > a > 1
b
>
a
>
1
, and
a
a
a
does not divide
b
b
b
. The sequence of natural numbers
{
b
n
}
n
=
1
∞
\{b_n\}_{n=1}^\infty
{
b
n
}
n
=
1
∞
satisfies
b
n
+
1
≥
2
b
n
∀
n
∈
N
b_{n + 1} \geq 2b_n \forall n \in \mathbb{N}
b
n
+
1
≥
2
b
n
∀
n
∈
N
. Does there exist a sequence
{
a
n
}
n
=
1
∞
\{a_n\}_{n=1}^\infty
{
a
n
}
n
=
1
∞
of natural numbers such that for all
n
∈
N
n \in \mathbb{N}
n
∈
N
,
a
n
+
1
−
a
n
∈
{
a
,
b
}
a_{n + 1} - a_n \in \{a, b\}
a
n
+
1
−
a
n
∈
{
a
,
b
}
, and for all
m
,
l
∈
N
m, l \in \mathbb{N}
m
,
l
∈
N
(
m
m
m
may be equal to
l
l
l
),
a
m
+
a
l
∉
{
b
n
}
n
=
1
∞
a_m + a_l \not\in \{b_n\}_{n=1}^\infty
a
m
+
a
l
∈
{
b
n
}
n
=
1
∞
?
China TST 2001 circumcenter and incenter
In the equilateral
△
A
B
C
\bigtriangleup ABC
△
A
BC
,
D
D
D
is a point on side
B
C
BC
BC
.
O
1
O_1
O
1
and
I
1
I_1
I
1
are the circumcenter and incenter of
△
A
B
D
\bigtriangleup ABD
△
A
B
D
respectively, and
O
2
O_2
O
2
and
I
2
I_2
I
2
are the circumcenter and incenter of
△
A
D
C
\bigtriangleup ADC
△
A
D
C
respectively.
O
1
I
1
O_1I_1
O
1
I
1
intersects
O
2
I
2
O_2I_2
O
2
I
2
at
P
P
P
. Find the locus of point
P
P
P
as
D
D
D
moves along
B
C
BC
BC
.
1
2
Hide problems
China TST 2001 interior points of convex quadrilateral
E
E
E
and
F
F
F
are interior points of convex quadrilateral
A
B
C
D
ABCD
A
BC
D
such that
A
E
=
B
E
AE = BE
A
E
=
BE
,
C
E
=
D
E
CE = DE
CE
=
D
E
,
∠
A
E
B
=
∠
C
E
D
\angle AEB = \angle CED
∠
A
EB
=
∠
CE
D
,
A
F
=
D
F
AF = DF
A
F
=
D
F
,
B
F
=
C
F
BF = CF
BF
=
CF
,
∠
A
F
D
=
∠
B
F
C
\angle AFD = \angle BFC
∠
A
F
D
=
∠
BFC
. Prove that
∠
A
F
D
+
∠
A
E
B
=
π
\angle AFD + \angle AEB = \pi
∠
A
F
D
+
∠
A
EB
=
π
.
Minimum value of fraction sum product
For a given natural number
n
>
3
n > 3
n
>
3
, the real numbers
x
1
,
x
2
,
…
,
x
n
,
x
n
+
1
,
x
n
+
2
x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2}
x
1
,
x
2
,
…
,
x
n
,
x
n
+
1
,
x
n
+
2
satisfy the conditions
0
<
x
1
<
x
2
<
⋯
<
x
n
<
x
n
+
1
<
x
n
+
2
0 < x_1 < x_2 < \cdots < x_n < x_{n + 1} < x_{n + 2}
0
<
x
1
<
x
2
<
⋯
<
x
n
<
x
n
+
1
<
x
n
+
2
. Find the minimum possible value of
(
∑
i
=
1
n
x
i
+
1
x
i
)
(
∑
j
=
1
n
x
j
+
2
x
j
+
1
)
(
∑
k
=
1
n
x
k
+
1
x
k
+
2
x
k
+
1
2
+
x
k
x
k
+
2
)
(
∑
l
=
1
n
x
l
+
1
2
+
x
l
x
l
+
2
x
l
x
l
+
1
)
\frac{(\sum _{i=1}^n \frac{x_{i + 1}}{x_i})(\sum _{j=1}^n \frac{x_{j + 2}}{x_{j + 1}})}{(\sum _{k=1}^n \frac{x_{k + 1} x_{k + 2}}{x_{k + 1}^2 + x_k x_{k + 2}})(\sum _{l=1}^n \frac{x_{l + 1}^2 + x_l x_{l + 2}}{x_l x_{l + 1}})}
(
∑
k
=
1
n
x
k
+
1
2
+
x
k
x
k
+
2
x
k
+
1
x
k
+
2
)
(
∑
l
=
1
n
x
l
x
l
+
1
x
l
+
1
2
+
x
l
x
l
+
2
)
(
∑
i
=
1
n
x
i
x
i
+
1
)
(
∑
j
=
1
n
x
j
+
1
x
j
+
2
)
and find all
(
n
+
2
)
(n + 2)
(
n
+
2
)
-tuplets of real numbers
(
x
1
,
x
2
,
…
,
x
n
,
x
n
+
1
,
x
n
+
2
)
(x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2})
(
x
1
,
x
2
,
…
,
x
n
,
x
n
+
1
,
x
n
+
2
)
which gives this value.