Three circles of radius $r$ touch each other externally. What is the radius of the circle that touches all three circles internally?

Let $S$ be a subset of the plane defined by $S = \{(x, y) : |x| + 2|y| = 1\}$. Then,the radius of the smallest circle with centre at the origin and having non-empty intersection with $S$ is

Consider a semicircle of radius $1$ unit constructed on the diameter $AB$ and let $O$ be its centre. Let $C$ be a point on $AO$ such that $AC:CO = 2:1$. Draw $CD$ perpendicular to $AO$ with $D$ on the semicircle. Draw $OE$ perpendicular to $AD$ with $E$ on $AD$. Let $OE$ and $CD$ intersect at $H$. Then,$DH$ equals

If the chord $y = mx + 1$ of the circle ${x^2} + {y^2} = 1$ subtends an angle of measure ${45^\circ}$ at the major segment of the circle,then the value of $m$ is:

The number of common tangents that can be drawn to the circles $x^2+y^2=1$ and $x^2+y^2-2x-6y+6=0$ is

The number of integers in the range of $m$ for which the point $(m, 1)$ lies in the smaller region enclosed by the circle $x^2 + y^2 - 3x + 1 = 0$ and the line $2x - y = 2$ is