MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
1991 China Team Selection Test
1991 China Team Selection Test
Part of
China Team Selection Test
Subcontests
(3)
3
2
Hide problems
number of good circles is n
5
5
5
points are given in the plane, any three non-collinear and any four non-concyclic. If three points determine a circle that has one of the remaining points inside it and the other one outside it, then the circle is said to be good. Let the number of good circles be
n
n
n
; find all possible values of
n
n
n
.
edges of a polyhedron are painted with red or yellow
All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called excentric. The excentricity of a vertex
A
A
A
, namely
S
A
S_A
S
A
, is defined as the number of excentric angles it has. Prove that there exist two vertices
B
B
B
and
C
C
C
such that
S
B
+
S
C
≤
4
S_B + S_C \leq 4
S
B
+
S
C
≤
4
.
2
2
Hide problems
f(n+2) = 23 * f(n+1) + f(n)
Let
f
f
f
be a function
f
:
N
∪
{
0
}
↦
N
,
f: \mathbb{N} \cup \{0\} \mapsto \mathbb{N},
f
:
N
∪
{
0
}
↦
N
,
and satisfies the following conditions: (1)
f
(
0
)
=
0
,
f
(
1
)
=
1
,
f(0) = 0, f(1) = 1,
f
(
0
)
=
0
,
f
(
1
)
=
1
,
(2)
f
(
n
+
2
)
=
23
⋅
f
(
n
+
1
)
+
f
(
n
)
,
n
=
0
,
1
,
…
.
f(n+2) = 23 \cdot f(n+1) + f(n), n = 0,1, \ldots.
f
(
n
+
2
)
=
23
⋅
f
(
n
+
1
)
+
f
(
n
)
,
n
=
0
,
1
,
…
.
Prove that for any
m
∈
N
m \in \mathbb{N}
m
∈
N
, there exist a
d
∈
N
d \in \mathbb{N}
d
∈
N
such that
m
∣
f
(
f
(
n
)
)
⇔
d
∣
n
.
m | f(f(n)) \Leftrightarrow d | n.
m
∣
f
(
f
(
n
))
⇔
d
∣
n
.
We write numbers on points
For
i
=
1
,
2
,
…
,
1991
i = 1,2, \ldots, 1991
i
=
1
,
2
,
…
,
1991
, we choose
n
i
n_i
n
i
points and write number
i
i
i
on them (each point has only written one number on it). A set of chords are drawn such that:(i) They are pairwise non-intersecting. (ii) The endpoints of each chord have distinct numbers.If for all possible assignments of numbers the operation can always be done, find the necessary and sufficient condition the numbers
n
1
,
n
2
,
…
,
n
1991
n_1, n_2, \ldots, n_{1991}
n
1
,
n
2
,
…
,
n
1991
must satisfy for this to be possible.
1
2
Hide problems
Polynomial inequality
Let real coefficient polynomial
f
(
x
)
=
x
n
+
a
1
⋅
x
n
−
1
+
…
+
a
n
f(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n
f
(
x
)
=
x
n
+
a
1
⋅
x
n
−
1
+
…
+
a
n
has real roots
b
1
,
b
2
,
…
,
b
n
b_1, b_2, \ldots, b_n
b
1
,
b
2
,
…
,
b
n
,
n
≥
2
,
n \geq 2,
n
≥
2
,
prove that
∀
x
≥
m
a
x
{
b
1
,
b
2
,
…
,
b
n
}
\forall x \geq max\{b_1, b_2, \ldots, b_n\}
∀
x
≥
ma
x
{
b
1
,
b
2
,
…
,
b
n
}
, we have
f
(
x
+
1
)
≥
2
⋅
n
2
1
x
−
b
1
+
1
x
−
b
2
+
…
+
1
x
−
b
n
.
f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.
f
(
x
+
1
)
≥
x
−
b
1
1
+
x
−
b
2
1
+
…
+
x
−
b
n
1
2
⋅
n
2
.
circumference of radius 1
We choose 5 points
A
1
,
A
2
,
…
,
A
5
A_1, A_2, \ldots, A_5
A
1
,
A
2
,
…
,
A
5
on a circumference of radius 1 and centre
O
.
O.
O
.
P
P
P
is a point inside the circle. Denote
Q
i
Q_i
Q
i
as the intersection of
A
i
A
i
+
2
A_iA_{i+2}
A
i
A
i
+
2
and
A
i
+
1
P
A_{i+1}P
A
i
+
1
P
, where
A
7
=
A
2
A_7 = A_2
A
7
=
A
2
and
A
6
=
A
1
.
A_6 = A_1.
A
6
=
A
1
.
Let
O
Q
i
=
d
i
,
i
=
1
,
2
,
…
,
5.
OQ_i = d_i, i = 1,2, \ldots, 5.
O
Q
i
=
d
i
,
i
=
1
,
2
,
…
,
5.
Find the product
∏
i
=
1
5
A
i
Q
i
\prod^5_{i=1} A_iQ_i
∏
i
=
1
5
A
i
Q
i
in terms of
d
i
.
d_i.
d
i
.