answer:**Analysis** This problem mainly examines the graph and properties of the function y=Asin (omega x+varphi). Mastering these properties is key to solving such problems. **Solution** Given that the smallest positive period is frac{2pi}{omega}=2, we solve to get omega=pi. Since the graph of the function f(x)=2sin (pi x+varphi)+2 (omega > 0, |varphi| < frac{pi}{2}) passes through the point (0,3), therefore f(0)=2sin (0+varphi)+2=3, Solving this, we get varphi= frac{pi}{6}, Therefore, f(x)=2sin (pi x+ frac{pi}{6} )+2, When pi x+ frac{pi}{6}=-2pi, the symmetry center is left(- frac{13}{6},2right), The minimum value of |a+b| is left|- frac{13}{6}+2right|= frac{1}{6}, Thus, the answer is boxed{frac{1}{6}}.

answer:1. **Consider the given equation:** [ sqrt[3]{1-x} + sqrt[3]{1+x} = p ] where ( p ) is an arbitrary real number. 2. **Cube both sides to eliminate the cube roots:** [ (sqrt[3]{1-x} + sqrt[3]{1+x})^3 = p^3 ] Expanding the left side using the binomial theorem for cube of a sum: [ (sqrt[3]{1-x} + sqrt[3]{1+x})^3 = (sqrt[3]{1-x})^3 + (sqrt[3]{1+x})^3 + 3(sqrt[3]{1-x})(sqrt[3]{1+x})(sqrt[3]{1-x} + sqrt[3]{1+x}) ] Since ((sqrt[3]{1-x})^3 = 1-x) and ((sqrt[3]{1+x})^3 = 1+x), this simplifies to: [ 1-x + 1+x + 3(sqrt[3]{1-x})(sqrt[3]{1+x})(sqrt[3]{1-x} + sqrt[3]{1+x}) = p^3 ] Simplifying further: [ 2 + 3 (sqrt[3]{(1-x)(1+x)}(p)) = p^3 ] Notice that ((1-x)(1+x) = 1-x^2): [ 2 + 3p sqrt[3]{1-x^2} = p^3 ] 3. **Rearrange this equation to solve for (sqrt[3]{1-x^2}):** [ 3p sqrt[3]{1-x^2} = p^3 - 2 ] [ sqrt[3]{1-x^2} = frac{p^3 - 2}{3p} ] 4. **Cube both sides again to solve for (1 - x^2):** [ 1 - x^2 = left(frac{p^3 - 2}{3p}right)^3 ] 5. **Solve for (x^2):** [ x^2 = 1 - left(frac{p^3 - 2}{3p}right)^3 ] 6. **Therefore, we have:** [ x = pm sqrt{1 - left(frac{p^3 - 2}{3p}right)^3} ] This solution is meaningful only when the expression inside the square root is non-negative, i.e., when: [ 1 - left(frac{p^3 - 2}{3p}right)^3 geq 0 ] 7. **Analyze the domain of validity of the solution:** [ p = -1 quad text{and} quad 0 < p leq 2 ] Note that these transformations are not equivalent and might introduce extraneous solutions. Validate by substitution into the original equation to ensure no additional roots are considered. 8. **Final step:** Consider specific cases ( p = -1 ): [ sqrt[3]{1-x} + sqrt[3]{1+x} = -1 Rightarrow sqrt[3]{1-x} = sqrt[3]{1+x} = -1 Rightarrow 1 - x = -1 Rightarrow x = 0 ] Thus, for ( p = -1 ), the equation holds true. In summary, we conclude that ( x = 0 ) and ( p = -1 ) satisfy our transformed equations. Therefore, the solution is: ( boxed{x = 0 text{ for } p = -1} )

answer:1. **Calculate the total number of line segments connecting any two vertices:** The total number of line segments connecting any two vertices in a polyhedron with (26) vertices is given by the binomial coefficient: [ binom{26}{2} = frac{26 times 25}{2} = 325 ] 2. **Subtract the number of edges:** We are given that the polyhedron has (60) edges. These edges are line segments connecting adjacent vertices, so we need to subtract these from the total number of line segments: [ 325 - 60 = 265 ] 3. **Subtract the diagonals of the quadrilateral faces:** Each quadrilateral face has (4) vertices and (2) diagonals. Since there are (12) quadrilateral faces, the total number of diagonals in these faces is: [ 2 times 12 = 24 ] We need to subtract these (24) diagonals because they connect vertices that belong to the same face. 4. **Calculate the number of space diagonals:** Subtract the number of diagonals of the quadrilateral faces from the previous result: [ 265 - 24 = 241 ] Thus, the number of space diagonals in the polyhedron is: [ boxed{241} ]

answer:Given the equation |z^2 + 9| = |z(z + 3i)|, this can be rewritten using the identity z^2 + 9 = (z + 3i)(z - 3i). Thus: [ |z + 3i||z - 3i| = |z||z + 3i|.] If |z + 3i| = 0, then z = -3i. In this case: [ |z + 2i| = |-i| = 1. ] Otherwise, if |z + 3i| neq 0, dividing both sides by |z + 3i| yields: [ |z - 3i| = |z|. ] This implies z is equidistant from the origin and the point 3i on the imaginary axis. The locus of such points z is the perpendicular bisector of 0 and 3i, which is the horizontal line where the imaginary part equals 1.5 (halfway between 0 and 3i). This configuration can be represented as z = x + 1.5i, substituting into the equation to find: [ |z + 2i| = |x + 3.5i| = sqrt{x^2 + 3.5^2}. ] Thus, to minimize: [ |z + 2i| = sqrt{x^2 + 3.5^2} geq 3.5, ] achieving equality when x = 0. Therefore, the smallest possible value of |z + 2i| is boxed{3.5}, which occurs when z = 1.5i.