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Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
1997 China Team Selection Test
1997 China Team Selection Test
Part of
China Team Selection Test
Subcontests
(3)
3
2
Hide problems
3 bottles A, B, C can contain at most 1997, 97, 17 pieces
There are 1997 pieces of medicine. Three bottles
A
,
B
,
C
A, B, C
A
,
B
,
C
can contain at most 1997, 97, 19 pieces of medicine respectively. At first, all 1997 pieces are placed in bottle
A
A
A
, and the three bottles are closed. Each piece of medicine can be split into 100 part. When a bottle is opened, all pieces of medicine in that bottle lose a part each. A man wishes to consume all the medicine. However, he can only open each of the bottles at most once each day, consume one piece of medicine, move some pieces between the bottles, and close them. At least how many parts will be lost by the time he finishes consuming all the medicine?
China integral sequence built by three rules
Prove that there exists
m
∈
N
m \in \mathbb{N}
m
∈
N
such that there exists an integral sequence
{
a
n
}
\lbrace a_n \rbrace
{
a
n
}
which satisfies: I.
a
0
=
1
,
a
1
=
337
a_0 = 1, a_1 = 337
a
0
=
1
,
a
1
=
337
; II.
(
a
n
+
1
a
n
−
1
−
a
n
2
)
+
3
4
(
a
n
+
1
+
a
n
−
1
−
2
a
n
)
=
m
,
∀
(a_{n + 1} a_{n - 1} - a_n^2) + \frac{3}{4}(a_{n + 1} + a_{n - 1} - 2a_n) = m, \forall
(
a
n
+
1
a
n
−
1
−
a
n
2
)
+
4
3
(
a
n
+
1
+
a
n
−
1
−
2
a
n
)
=
m
,
∀
n
≥
1
n \geq 1
n
≥
1
; III.
1
6
(
a
n
+
1
)
(
2
a
n
+
1
)
\frac{1}{6}(a_n + 1)(2a_n + 1)
6
1
(
a
n
+
1
)
(
2
a
n
+
1
)
is a perfect square
∀
\forall
∀
n
≥
1
n \geq 1
n
≥
1
.
2
2
Hide problems
China TST 1997 football teams
There are
n
n
n
football teams in a round-robin competition where every 2 teams meet once. The winner of each match receives 3 points while the loser receives 0 points. In the case of a draw, both teams receive 1 point each. Let
k
k
k
be as follows: 2 \leq k \leq n \minus{} 1. At least how many points must a certain team get in the competition so as to ensure that there are at most k \minus{} 1 teams whose scores are not less than that particular team's score?
China TST 1997 5-element subsets
Let
n
n
n
be a natural number greater than 6.
X
X
X
is a set such that
∣
X
∣
=
n
|X| = n
∣
X
∣
=
n
.
A
1
,
A
2
,
…
,
A
m
A_1, A_2, \ldots, A_m
A
1
,
A
2
,
…
,
A
m
are distinct 5-element subsets of
X
X
X
. If
m
>
n
(
n
−
1
)
(
n
−
2
)
(
n
−
3
)
(
4
n
−
15
)
600
m > \frac{n(n - 1)(n - 2)(n - 3)(4n - 15)}{600}
m
>
600
n
(
n
−
1
)
(
n
−
2
)
(
n
−
3
)
(
4
n
−
15
)
, prove that there exists
A
i
1
,
A
i
2
,
…
,
A
i
6
A_{i_1}, A_{i_2}, \ldots, A_{i_6}
A
i
1
,
A
i
2
,
…
,
A
i
6
(
1
≤
i
1
<
i
2
<
⋯
,
i
6
≤
m
)
(1 \leq i_1 < i_2 < \cdots, i_6 \leq m)
(
1
≤
i
1
<
i
2
<
⋯
,
i
6
≤
m
)
, such that
⋃
k
=
1
6
A
i
k
=
6
\bigcup_{k = 1}^6 A_{i_k} = 6
⋃
k
=
1
6
A
i
k
=
6
.
1
2
Hide problems
China TST polynomial inequality involving bionomials
Find all real-coefficient polynomials
f
(
x
)
f(x)
f
(
x
)
which satisfy the following conditions:i.
f
(
x
)
=
a
0
x
2
n
+
a
2
x
2
n
−
2
+
⋯
+
a
2
n
−
2
x
2
+
a
2
n
,
a
0
>
0
f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2} x^2 + a_{2n}, a_0 > 0
f
(
x
)
=
a
0
x
2
n
+
a
2
x
2
n
−
2
+
⋯
+
a
2
n
−
2
x
2
+
a
2
n
,
a
0
>
0
; ii.
∑
j
=
0
n
a
2
j
a
2
n
−
2
j
≤
(
2
n
n
)
a
0
a
2
n
\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left( \begin{array}{c} 2n\\ n\end{array} \right) a_0 a_{2n}
∑
j
=
0
n
a
2
j
a
2
n
−
2
j
≤
(
2
n
n
)
a
0
a
2
n
; iii. All the roots of
f
(
x
)
f(x)
f
(
x
)
are imaginary numbers with no real part.
Extend sides and find the locus of a point
Given a real number
λ
>
1
\lambda > 1
λ
>
1
, let
P
P
P
be a point on the arc
B
A
C
BAC
B
A
C
of the circumcircle of
△
A
B
C
\bigtriangleup ABC
△
A
BC
. Extend
B
P
BP
BP
and
C
P
CP
CP
to
U
U
U
and
V
V
V
respectively such that
B
U
=
λ
B
A
BU = \lambda BA
B
U
=
λ
B
A
,
C
V
=
λ
C
A
CV = \lambda CA
C
V
=
λ
C
A
. Then extend
U
V
UV
U
V
to
Q
Q
Q
such that
U
Q
=
λ
U
V
UQ = \lambda UV
U
Q
=
λ
U
V
. Find the locus of point
Q
Q
Q
.