2006 AMC 12A Problem 23

Problem 23 of 25HarderAlgebra

Given a finite sequence

S = (a_1, a_2, \ldots, a_n)

of

n

real numbers, let

A(S)

be the sequence

\small \left(\frac{a_1 + a_2}{2}, \frac{a_2 + a_3}{2}, \ldots, \frac{a_{n-1} + a_n}{2}\right)

of

n - 1

real numbers. Define

A^1(S) = A(S)

and, for each integer

m,

2 \le m \le n - 1,

define

A^m(S) = A(A^{m-1}(S)).

Suppose

x \gt 0,

and let

S = (1, x, x^2, \ldots, x^{100}).

If

A^{100}(S) = (\frac{1}{2^{50}}),

then what is

x?

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Solution

Each application of

A

averages adjacent terms, so after

100

steps the single remaining term is

\begin{gathered} \frac{1}{2^{100}} \sum_{m=0}^{100} \binom{100}{m} x^m \\ = \frac{(1 + x)^{100}}{2^{100}}. \end{gathered}

Setting this equal to

\dfrac{1}{2^{50}}

gives

(1 + x)^{100} = 2^{50},

so

1 + x = 2^{\frac{1}{2}} = \sqrt{2}.

Since

x \gt 0,

we get

x = \sqrt{2} - 1.

Thus, the correct answer is B.

More practice

Concepts: binomial theorem · recursion

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.