Problem 4

Find the maximum value of

x_0

for which there exists a sequence

x_0,x_1,\ldots,x_{1995}

of positive reals with

x_0=x_{1995}

such that for

i=1,\ldots,1995

x_{i-1}+\frac2{x_{i-1}}=2x_i+\frac1{x_i}

Step 3 of 4: Bound a closed odd word

Previous step Next step

In plain words

An even number of inversions would require an even number of halvings, impossible here.

r\text{ is odd},\quad 2t_0=c\le1994

Detailed analysis

There are $1995$ operations. If $r$ were even, closure $t_{1995}=t_0$ would require the alternating sum of halving-run lengths to be zero, forcing their total (the number of halvings) to be even; but $1995-r$ is odd. Hence $r$ is odd. Closure now gives $2t_0=c$ . Since at most $1994$ operations are halvings, $c\le1994$ , so $t_0\le997$ and $x_0\le2^{997}$ .