MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
1999 China Second Round Olympiad
3
3
Part of
1999 China Second Round Olympiad
Problems
(1)
Balance and counterweights
Source: 1999 China Second Round Olympiad P3
8/28/2019
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.
combinatorics