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 2 of 4: Encode the operations

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In plain words

Halving and inversion become affine operations on an exponent.

t=\log_2x,\quad H:t\mapsto t-1,\quad I:t\mapsto-t

Detailed analysis

Put $t=\log_2x$ . Then $H:t\mapsto t-1$ and $I:t\mapsto-t$ . A word with $r$ inversions has the form $t\mapsto(-1)^rt+c$ , where $c$ is an integer whose absolute value is at most the number of halvings.