MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
1998 China Team Selection Test
1998 China Team Selection Test
Part of
China Team Selection Test
Subcontests
(3)
3
2
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Inequality involving sines and cosines
For a fixed
θ
∈
[
0
,
π
2
]
\theta \in \lbrack 0, \frac{\pi}{2} \rbrack
θ
∈
[
0
,
2
π
]
, find the smallest
a
∈
R
+
a \in \mathbb{R}^{+}
a
∈
R
+
which satisfies the following conditions: I.
a
cos
θ
+
a
sin
θ
>
1
\frac{\sqrt a}{\cos \theta} + \frac{\sqrt a}{\sin \theta} > 1
c
o
s
θ
a
+
s
i
n
θ
a
>
1
. II. There exists
x
∈
[
1
−
a
sin
θ
,
a
cos
θ
]
x \in \lbrack 1 - \frac{\sqrt a}{\sin \theta}, \frac{\sqrt a}{\cos \theta} \rbrack
x
∈
[
1
−
s
i
n
θ
a
,
c
o
s
θ
a
]
such that
[
(
1
−
x
)
sin
θ
−
a
−
x
2
cos
2
θ
]
2
+
[
x
cos
θ
−
a
−
(
1
−
x
)
2
sin
2
θ
]
2
≤
a
\lbrack (1 - x)\sin \theta - \sqrt{a - x^2 \cos^{2} \theta} \rbrack^{2} + \lbrack x\cos \theta - \sqrt{a - (1 - x)^2 \sin^{2} \theta} \rbrack^{2} \leq a
[(
1
−
x
)
sin
θ
−
a
−
x
2
cos
2
θ
]
2
+
[
x
cos
θ
−
a
−
(
1
−
x
)
2
sin
2
θ
]
2
≤
a
.
China TST 1998 division probelm
For any
h
=
2
r
h = 2^{r}
h
=
2
r
(
r
r
r
is a non-negative integer), find all
k
∈
N
k \in \mathbb{N}
k
∈
N
which satisfy the following condition: There exists an odd natural number
m
>
1
m > 1
m
>
1
and
n
∈
N
n \in \mathbb{N}
n
∈
N
, such that
k
∣
m
h
−
1
,
m
∣
n
m
h
−
1
k
+
1
k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1
k
∣
m
h
−
1
,
m
∣
n
k
m
h
−
1
+
1
.
2
2
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More than 5 teams are playing football
n
≥
5
n \geq 5
n
≥
5
football teams participate in a round-robin tournament. For every game played, the winner receives 3 points, the loser receives 0 points, and in the event of a draw, both teams receive 1 point. The third-from-bottom team has fewer points than all the teams ranked before it, and more points than the last 2 teams; it won more games than all the teams before it, but fewer games than the 2 teams behind it. Find the smallest possible
n
n
n
.
Find sum of distances on plane surface
Let
n
n
n
be a natural number greater than 2.
l
l
l
is a line on a plane. There are
n
n
n
distinct points
P
1
P_1
P
1
,
P
2
P_2
P
2
, …,
P
n
P_n
P
n
on
l
l
l
. Let the product of distances between
P
i
P_i
P
i
and the other
n
−
1
n-1
n
−
1
points be
d
i
d_i
d
i
(
i
=
1
,
2
,
i = 1, 2,
i
=
1
,
2
,
…,
n
n
n
). There exists a point
Q
Q
Q
, which does not lie on
l
l
l
, on the plane. Let the distance from
Q
Q
Q
to
P
i
P_i
P
i
be
C
i
C_i
C
i
(
i
=
1
,
2
,
i = 1, 2,
i
=
1
,
2
,
…,
n
n
n
). Find
S
n
=
∑
i
=
1
n
(
−
1
)
n
−
i
c
i
2
d
i
S_n = \sum_{i = 1}^{n} (-1)^{n-i} \frac{c_i^2}{d_i}
S
n
=
∑
i
=
1
n
(
−
1
)
n
−
i
d
i
c
i
2
.
1
2
Hide problems
Binomial coefficients forming arithmetic progression
Find
k
∈
N
k \in \mathbb{N}
k
∈
N
such thata.) For any
n
∈
N
n \in \mathbb{N}
n
∈
N
, there does not exist
j
∈
Z
j \in \mathbb{Z}
j
∈
Z
which satisfies the conditions
0
≤
j
≤
n
−
k
+
1
0 \leq j \leq n - k + 1
0
≤
j
≤
n
−
k
+
1
and
(
n
j
)
,
(
n
j
+
1
)
,
…
,
(
n
j
+
k
−
1
)
\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots, \left( \begin{array}{c} n\\ j + k - 1\end{array} \right)
(
n
j
)
,
(
n
j
+
1
)
,
…
,
(
n
j
+
k
−
1
)
forms an arithmetic progression.b.) There exists
n
∈
N
n \in \mathbb{N}
n
∈
N
such that there exists
j
j
j
which satisfies
0
≤
j
≤
n
−
k
+
2
0 \leq j \leq n - k + 2
0
≤
j
≤
n
−
k
+
2
, and
(
n
j
)
,
(
n
j
+
1
)
,
…
,
(
n
j
+
k
−
2
)
\left( \begin{array}{c} n\\ j\end{array} \right), \left( \begin{array}{c} n\\ j + 1\end{array} \right), \ldots , \left( \begin{array}{c} n\\ j + k - 2\end{array} \right)
(
n
j
)
,
(
n
j
+
1
)
,
…
,
(
n
j
+
k
−
2
)
forms an arithmetic progression.Find all
n
n
n
which satisfies part b.)
Orthocenter, circumcenter and incenter
In acute-angled
△
A
B
C
\bigtriangleup ABC
△
A
BC
,
H
H
H
is the orthocenter,
O
O
O
is the circumcenter and
I
I
I
is the incenter. Given that
∠
C
>
∠
B
>
∠
A
\angle C > \angle B > \angle A
∠
C
>
∠
B
>
∠
A
, prove that
I
I
I
lies within
△
B
O
H
\bigtriangleup BOH
△
BO
H
.