120057 Three circles of radii \(a, b, c(a\lt b\lt c)\) touch each other externally. If they have \(\mathrm{X}\)-axis as a common tangent, then

120058 For the four circles \(M, N, O\) and \(P\), following four equations are given
Circle \(M: x^2+y^2=1\)
Circle \(N: x^2+y^2-2 x=0\)
Circle \(O: x^2+y^2-2 x-2 y+1=0\)
Circle \(P: x^2+y^2-2 y=0\)
If the centre of circle \(M\) is joined with centre of the circle \(N\), further centre of circle \(N\) is joined with centre of circle \(O\), centre of circle \(O\) is joined with the centre of circle \(P\) and lastly, centre of circle \(M\), then these lines from the sides of a

120059 \(\text { Let } A=\{(x, y) \in \mathbf{R} \times \mathbf{R} \mid\)
\(\left.2 x^2+2 y^2-2 x-2 y=1\right\} \text {, }\)
\(\mathbf{B}=\{\mathbf{x}, \mathbf{y}) \in \mathbf{R} \times \mathbf{R} \mid\)
\(\left.4 x^2+4 y^2-16 y+7=0\right\}\)
\(\mathbf{C}=\{(\mathbf{x}, \mathbf{y}) \in \mathbf{R} \times \mathbf{R}\)
\(\left.x^2+y^2-4 x-2 y+5 \leq r^2\right\}\) Then the minimum value of \(|r|\) such that \(A \cup B \subseteq C\) is equal to

120060 Let \(Z\) be the set of all integers,
\(A=\left\{(x, y) \in Z \times Z:(x-2)^2+y^2 \leq 4\right\}\)
\(B=\left\{(x, y) \in Z \times Z: x^2+y^2 \leq 4\right\} \text { and }\)
\(C=\left\{(x, y) \in Z \times Z:(x-2)^2+(y-2)^2 \leq 4\right\}\)
If the total number of relation from \(A \cap B\) to \(A\) \(\cap C\) is \(2^p\), then the value of \(p\) is

120057 Three circles of radii \(a, b, c(a\lt b\lt c)\) touch each other externally. If they have \(\mathrm{X}\)-axis as a common tangent, then

120058 For the four circles \(M, N, O\) and \(P\), following four equations are given
Circle \(M: x^2+y^2=1\)
Circle \(N: x^2+y^2-2 x=0\)
Circle \(O: x^2+y^2-2 x-2 y+1=0\)
Circle \(P: x^2+y^2-2 y=0\)
If the centre of circle \(M\) is joined with centre of the circle \(N\), further centre of circle \(N\) is joined with centre of circle \(O\), centre of circle \(O\) is joined with the centre of circle \(P\) and lastly, centre of circle \(M\), then these lines from the sides of a

120059 \(\text { Let } A=\{(x, y) \in \mathbf{R} \times \mathbf{R} \mid\)
\(\left.2 x^2+2 y^2-2 x-2 y=1\right\} \text {, }\)
\(\mathbf{B}=\{\mathbf{x}, \mathbf{y}) \in \mathbf{R} \times \mathbf{R} \mid\)
\(\left.4 x^2+4 y^2-16 y+7=0\right\}\)
\(\mathbf{C}=\{(\mathbf{x}, \mathbf{y}) \in \mathbf{R} \times \mathbf{R}\)
\(\left.x^2+y^2-4 x-2 y+5 \leq r^2\right\}\) Then the minimum value of \(|r|\) such that \(A \cup B \subseteq C\) is equal to

120060 Let \(Z\) be the set of all integers,
\(A=\left\{(x, y) \in Z \times Z:(x-2)^2+y^2 \leq 4\right\}\)
\(B=\left\{(x, y) \in Z \times Z: x^2+y^2 \leq 4\right\} \text { and }\)
\(C=\left\{(x, y) \in Z \times Z:(x-2)^2+(y-2)^2 \leq 4\right\}\)
If the total number of relation from \(A \cap B\) to \(A\) \(\cap C\) is \(2^p\), then the value of \(p\) is

120057 Three circles of radii \(a, b, c(a\lt b\lt c)\) touch each other externally. If they have \(\mathrm{X}\)-axis as a common tangent, then

120058 For the four circles \(M, N, O\) and \(P\), following four equations are given
Circle \(M: x^2+y^2=1\)
Circle \(N: x^2+y^2-2 x=0\)
Circle \(O: x^2+y^2-2 x-2 y+1=0\)
Circle \(P: x^2+y^2-2 y=0\)
If the centre of circle \(M\) is joined with centre of the circle \(N\), further centre of circle \(N\) is joined with centre of circle \(O\), centre of circle \(O\) is joined with the centre of circle \(P\) and lastly, centre of circle \(M\), then these lines from the sides of a

120059 \(\text { Let } A=\{(x, y) \in \mathbf{R} \times \mathbf{R} \mid\)
\(\left.2 x^2+2 y^2-2 x-2 y=1\right\} \text {, }\)
\(\mathbf{B}=\{\mathbf{x}, \mathbf{y}) \in \mathbf{R} \times \mathbf{R} \mid\)
\(\left.4 x^2+4 y^2-16 y+7=0\right\}\)
\(\mathbf{C}=\{(\mathbf{x}, \mathbf{y}) \in \mathbf{R} \times \mathbf{R}\)
\(\left.x^2+y^2-4 x-2 y+5 \leq r^2\right\}\) Then the minimum value of \(|r|\) such that \(A \cup B \subseteq C\) is equal to

120060 Let \(Z\) be the set of all integers,
\(A=\left\{(x, y) \in Z \times Z:(x-2)^2+y^2 \leq 4\right\}\)
\(B=\left\{(x, y) \in Z \times Z: x^2+y^2 \leq 4\right\} \text { and }\)
\(C=\left\{(x, y) \in Z \times Z:(x-2)^2+(y-2)^2 \leq 4\right\}\)
If the total number of relation from \(A \cap B\) to \(A\) \(\cap C\) is \(2^p\), then the value of \(p\) is

120057 Three circles of radii \(a, b, c(a\lt b\lt c)\) touch each other externally. If they have \(\mathrm{X}\)-axis as a common tangent, then

120058 For the four circles \(M, N, O\) and \(P\), following four equations are given
Circle \(M: x^2+y^2=1\)
Circle \(N: x^2+y^2-2 x=0\)
Circle \(O: x^2+y^2-2 x-2 y+1=0\)
Circle \(P: x^2+y^2-2 y=0\)
If the centre of circle \(M\) is joined with centre of the circle \(N\), further centre of circle \(N\) is joined with centre of circle \(O\), centre of circle \(O\) is joined with the centre of circle \(P\) and lastly, centre of circle \(M\), then these lines from the sides of a

120059 \(\text { Let } A=\{(x, y) \in \mathbf{R} \times \mathbf{R} \mid\)
\(\left.2 x^2+2 y^2-2 x-2 y=1\right\} \text {, }\)
\(\mathbf{B}=\{\mathbf{x}, \mathbf{y}) \in \mathbf{R} \times \mathbf{R} \mid\)
\(\left.4 x^2+4 y^2-16 y+7=0\right\}\)
\(\mathbf{C}=\{(\mathbf{x}, \mathbf{y}) \in \mathbf{R} \times \mathbf{R}\)
\(\left.x^2+y^2-4 x-2 y+5 \leq r^2\right\}\) Then the minimum value of \(|r|\) such that \(A \cup B \subseteq C\) is equal to

120060 Let \(Z\) be the set of all integers,
\(A=\left\{(x, y) \in Z \times Z:(x-2)^2+y^2 \leq 4\right\}\)
\(B=\left\{(x, y) \in Z \times Z: x^2+y^2 \leq 4\right\} \text { and }\)
\(C=\left\{(x, y) \in Z \times Z:(x-2)^2+(y-2)^2 \leq 4\right\}\)
If the total number of relation from \(A \cap B\) to \(A\) \(\cap C\) is \(2^p\), then the value of \(p\) is