b*k
发帖数: 27 | 1 Let n >= 2 be a positive integer. Initially, there are n fleas on a horizontal
line, not all at the same point.
For a positive real number r, define a moves as follows:
choose any two fleas, at points A and B, with A to the left of B;
let the flea ar A jump to the point C on the line to the right of B
with BC/AB = r.
Determine all values of r such that, for any point M on the line and any
initial positions of the n fleas, there is a finite sequence of moves that
will take all the fleas to the po | C******a
发帖数: 115 | 2 对于一个n元实数集合X，定义f(X)=n*max(X)-sum(X)。
下面证明如果r>=1/(n-1)，则满足条件。
在这种情况下，对开始的任意n个数，进行这样的变化：
以最大的数为跳板，把最小的数跳到最右面，即成为新的最大。
初始时的集合记为X_0，跳过以后的集合记作X_1。
则max(X_1)-max(X_0)=r*l_0, sum(X_1)-sum(X_0)=(1+r)*l_0。
其中l_0=max(X_0)-min(X_0)>=f(X_0)/n>0。
那么f(X_1)-f(X_0)=n*r*l_0-(1+r)*l_0=((n-1)*r-1)*l_0>=0。
继续类似地做下去，则f(X_i)是不减正数列，
max(X_{i+1})-max(X_i)=r*l_i>=r*f(X_i)/n>=f(X_0)/(n*(n-1))。
于是max(X_i)是以不小于一个给定正数的步伐递增。
经过有限步后，必然有max(X)>M，然后再经过n-1步，则都大于了M。
下面证明如果r<1/(n-1)，则不满足条件。
考察从初始情况开始的任意一次跳动。要证明存在C(n,r)>0，使得
max
【在 b*k 的大作中提到】
: Let n >= 2 be a positive integer. Initially, there are n fleas on a horizontal
: line, not all at the same point.
: For a positive real number r, define a moves as follows:
: choose any two fleas, at points A and B, with A to the left of B;
: let the flea ar A jump to the point C on the line to the right of B
: with BC/AB = r.
: Determine all values of r such that, for any point M on the line and any
: initial positions of the n fleas, there is a finite sequence of moves that
: will take all the fleas to the po
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