MathDB

Let

\mathcal{C}

be the family of circumferences in

\mathbb{R}^2

that satisfy the following properties:
(i) if

C_n

is the circumference with center

(n,1/2)

and radius

1/2

, then

C_n\in \mathcal{C}

, for all

n\in \mathbb{Z}

. (ii) if

C

and

C'

, both in

\mathcal{C}

, are externally tangent, then the circunference externally tangent to

C

and

C'

and tanget to

x

-axis also belongs to

\mathcal{C}

. (iii)

\mathcal{C}

is the least family which these properties.
Determine the set of the real numbers which are obtained as the first coordinate of the points of intersection between the elements of

\mathcal{C}

and the

x

-axis.

Problem Statement