MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
1999 China Second Round Olympiad
1999 China Second Round Olympiad
Part of
(China) National High School Mathematics League
Subcontests
(3)
3
1
Hide problems
Balance and counterweights
n
n
n
is a given positive integer, such that it’s possible to weigh out the mass of any product weighing
1
,
2
,
3
,
⋯
,
n
g
1,2,3,\cdots ,ng
1
,
2
,
3
,
⋯
,
n
g
with a counter balance without sliding poise and
k
k
k
counterweights, which weigh
x
i
g
(
i
=
1
,
2
,
⋯
,
k
)
,
x_ig(i=1,2,\cdots ,k),
x
i
g
(
i
=
1
,
2
,
⋯
,
k
)
,
respectively, where
x
i
∈
N
∗
x_i\in \mathbb{N}^*
x
i
∈
N
∗
for any
i
∈
{
1
,
2
,
⋯
,
k
}
i \in \{ 1,2,\cdots ,k\}
i
∈
{
1
,
2
,
⋯
,
k
}
and
x
1
≤
x
2
≤
⋯
≤
x
k
.
x_1\leq x_2\leq\cdots \leq x_k.
x
1
≤
x
2
≤
⋯
≤
x
k
.
(
1
)
(1)
(
1
)
Let
f
(
n
)
f(n)
f
(
n
)
be the least possible number of
k
k
k
. Find
f
(
n
)
f(n)
f
(
n
)
in terms of
n
.
n.
n
.
(
2
)
(2)
(
2
)
Find all possible number of
n
,
n,
n
,
such that sequence
x
1
,
x
2
,
⋯
,
x
f
(
n
)
x_1,x_2,\cdots ,x_{f(n)}
x
1
,
x
2
,
⋯
,
x
f
(
n
)
is uniquely determined.
1
1
Hide problems
1999 China Second Round Geometry
In convex quadrilateral
A
B
C
D
,
∠
B
A
C
=
∠
C
A
D
.
ABCD, \angle BAC=\angle CAD.
A
BC
D
,
∠
B
A
C
=
∠
C
A
D
.
E
E
E
lies on segment
C
D
CD
C
D
, and
B
E
BE
BE
and
A
C
AC
A
C
intersect at
F
,
F,
F
,
D
F
DF
D
F
and
B
C
BC
BC
intersect at
G
.
G.
G
.
Prove that
∠
G
A
C
=
∠
E
A
C
.
\angle GAC=\angle EAC.
∠
G
A
C
=
∠
E
A
C
.
2
1
Hide problems
Find the norm of complex number
Let
a
a
a
,
b
b
b
,
c
c
c
be real numbers. Let
z
1
z_{1}
z
1
,
z
2
z_{2}
z
2
,
z
3
z_{3}
z
3
be complex numbers such that
∣
z
k
∣
=
1
|z_{k}|=1
∣
z
k
∣
=
1
(
k
=
1
,
2
,
3
)
(k=1,2,3)
(
k
=
1
,
2
,
3
)
~
and
~
z
1
z
2
+
z
2
z
3
+
z
3
z
1
=
1
\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}=1
z
2
z
1
+
z
3
z
2
+
z
1
z
3
=
1
Find
∣
a
z
1
+
b
z
2
+
c
z
3
∣
|az_{1}+bz_{2}+cz_{3}|
∣
a
z
1
+
b
z
2
+
c
z
3
∣
.