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 4 of 4: Attain the bound

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In plain words

Halve 1994 times and invert once to return to the start.

H^{1994}(2^{997})=2^{-997},\quad I(2^{-997})=2^{997}

Detailed analysis

Starting with $x_0=2^{997}$ , apply $H$ for $1994$ steps and then $I$ . This gives $H^{1994}(2^{997})=2^{-997}$ and $I(2^{-997})=2^{997}$ , so the sequence closes and the upper bound is attained. The maximum is $2^{997}$ .