Let's analyze the two problems:

Problem 1

We are given two circles $C_1(O_1, r_1)$ and $C_2(O_2, r_2)$ with the following data:

• $r_1 = 3 \, \text{cm}$, $r_2 = 2 \, \text{cm}$,
• $O_1O_2 = x - 2 \, \text{cm}$.

We need to determine the natural values of $x$ such that the circles are:

1. Concentric: Circles have the same center $O_1 = O_2$, so $O_1O_2 = 0$.
   Thus, $x - 2 = 0 \implies x = 2$.

2. Tangent internally: The distance between the centers $O_1O_2$ is equal to the difference of their radii $|r_1 - r_2| = 1$:
   $x - 2 = 1 \implies x = 3$.

3. Secant: The circles overlap but are not tangent. This occurs when the distance between centers satisfies:
   $|r_1 - r_2| < O_1O_2 < r_1 + r_2$, i.e., $1 < x - 2 < 5$.
   Solving: $3 < x < 7$.
   Thus, $x \in \{4, 5, 6\}$.

4. Tangent externally: The distance between the centers equals the sum of their radii $r_1 + r_2 = 5$:
   $x - 2 = 5 \implies x = 7$.

Problem 2

We are given a circle with $\angle BOC = 30^\circ$ and $DO \perp CO$.
We need to calculate the measures of the following arcs:

1. $\overset{\frown}{ADC}$,
2. $\overset{\frown}{DAB}$,
3. $\overset{\frown}{DCB}$,
4. $\overset{\frown}{ABC}$.

The solution involves using the fact that:

• A full circle measures $360^\circ$,
• The central angle determines the measure of the corresponding arc.

Using the given $\angle BOC = 30^\circ$:

• $\overset{\frown}{BC} = 30^\circ$,
• The remaining portion of the circle $\overset{\frown}{ADC} = 360^\circ - 30^\circ = 330^\circ$.

For the perpendicular relationship $DO \perp CO$:

• The diameter divides the circle into two semicircles, each measuring $180^\circ$.

From this, we calculate:

1. $\overset{\frown}{ADC} = 330^\circ$,
2. $\overset{\frown}{DAB} = 180^\circ + 30^\circ = 210^\circ$,
3. $\overset{\frown}{DCB} = 180^\circ - 30^\circ = 150^\circ$,
4. $\overset{\frown}{ABC} = 180^\circ$.

Would you like detailed explanations of any step or concept?

Further Exploration

1. What are the steps to verify tangent conditions for circles algebraically?
2. How can the perpendicular diameter divide the arcs geometrically?
3. Can we generalize this method for overlapping circles with unequal radii?
4. Why is $O_1O_2 = |r_1 - r_2|$ important in tangency calculations?
5. How does central angle calculation extend to sectors of a circle?

Tip: Remember to always analyze geometric relationships (e.g., tangency, distance, diameter) through both algebra and visualization!