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 1 of 4: Factor the recurrence

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In plain words

Every transition has exactly two possible positive values.

x_i=1/x_{i-1}\text{ or }x_i=x_{i-1}/2

Detailed analysis

For fixed $x_{i-1}$ , the equation is quadratic in $x_i$ ; its two positive roots are $x_i=1/x_{i-1}$ and $x_i=x_{i-1}/2$ . Thus every sequence is a word in inversion $I(x)=1/x$ and halving $H(x)=x/2$ .