answer:To solve the inequality given ( 2frac{sin A}{A} + 2frac{sin B}{B} + 2frac{sin C}{C} leq left( frac{1}{B} + frac{1}{C} right) sin A + left( frac{1}{C} + frac{1}{A} right) sin B + left( frac{1}{A} + frac{1}{B} right) sin C ), we'll follow the steps below: 1. **Assume an order on the angles:** Assume ( A leq B leq C ). This assumption will help us leverage the properties of sine function. 2. **Compare sine values using this assumption:** Since ( A leq B leq C ) and noting that the sine function is increasing in the interval ( [0, pi] ), [ sin A leq sin B leq sin C. ] 3. **Construct the necessary inequalities:** Now, let's construct the inequalities using the assumed order: - Compare values ( left( frac{1}{A} - frac{1}{B} right) (sin B - sin A) ): Since ( frac{1}{A} geq frac{1}{B} ) and ( sin B geq sin A ), the product ( left( frac{1}{A} - frac{1}{B} right) (sin B - sin A) ) is non-negative. - Compare values ( left( frac{1}{B} - frac{1}{C} right) (sin C - sin B) ): Similarly, ( frac{1}{B} geq frac{1}{C} ) and ( sin C geq sin B ) yields the product ( left( frac{1}{B} - frac{1}{C} right) (sin C - sin B) ) is non-negative. - Compare values ( left( frac{1}{C} - frac{1}{A} right) (sin A - sin C) ): Since ( frac{1}{C} leq frac{1}{A} ) and ( sin A leq sin C ), the product ( left( frac{1}{C} - frac{1}{A} right) (sin A - sin C) ) is non-negative. 4. **Sum the inequalities:** The sum of these non-negative terms is: [ left( frac{1}{A} - frac{1}{B} right) (sin B - sin A) + left( frac{1}{B} - frac{1}{C} right) (sin C - sin B) + left( frac{1}{C} - frac{1}{A} right) (sin A - sin C) geq 0. ] 5. **Break down the summation into individual terms:** Expanding and simplifying this sums into the original inequality, we get: [ 2left( frac{sin A}{A} + frac{sin B}{B} + frac{sin C}{C} right) leq left( frac{1}{B} + frac{1}{C} right) sin A + left( frac{1}{C} + frac{1}{A} right) sin B + left( frac{1}{A} + frac{1}{B} right) sin C. ] Concluding, we obtain the desired result: (boxed{text{The given inequality holds true.}})

answer:1. Let ( ABCDEF ) be the regular hexagonal base of the pyramid and ( P ) be the apex of the pyramid. 2. The side length of the base hexagon and the height of the pyramid are both given as ( a ). 3. Denote ( M ) as the center of the regular hexagon ( ABCDEF ). By the symmetry of the regular hexagon, ( M ) is equidistant from all vertices ( A, B, C, D, E, ) and ( F ), and this distance is the circumradius of the hexagon. 4. In a regular hexagon with side length ( a ), the circumradius ( R ) is given by ( R = a ). 5. Since ( M ) is the center of the hexagon and the pyramid is regular, the distance from ( M ) to the apex ( P ) (which is the height of the pyramid) is also ( a ). 6. Therefore, in the regular hexagonal pyramid (P ABCDEF), the distance from ( M ) to ( P ), which is ( PM = a ). 7. This indicates that ( M ) is equidistant from all vertices ( A, B, C, D, E, F, ) and ( P ). 8. Consequently, ( M ) is the center of the sphere that is circumscribed around the pyramid (P ABCDEF). 9. The radius of this sphere is equal to the distance from ( M ) to any of these vertices, which is ( a ). Conclusion: The radius of the sphere circumscribed around the pyramid is [ boxed{a} ]

answer:From the problem, we have the following system of equations: [ begin{cases} a^2 - b^2 = c^2 m^2 - n^2 = -c^2 c^2 = am 2n^2 = 2m^2 + c^2 end{cases} ] By the property of the ellipse and hyperbola having the same foci, we obtain: [ a^2 - b^2 = m^2 + n^2 = c^2 ] According to the information given, since c is the geometric mean of a and m, we deduce that: [ c^2 = am ] As n^2 is the arithmetic mean of 2m^2 and c^2, we can write: [ 2n^2 = 2m^2 + c^2 ] Now, we solve for the ratio between c^2 and a: [ c^2 = am implies a = frac{c^2}{m} ] Substitute a into the first equation: [ left(frac{c^2}{m}right)^2 - b^2 = c^2 ] From the relation 2n^2 = 2m^2 + c^2, we have m^2 + m^2 + frac{c^2}{2} = c^2, which can be simplified to get: [ 2 left(frac{c^2}{m}right)^2 = frac{c^2}{2} ] This implies: [ 2 left(frac{c^2}{m}right)^2 = frac{c^2}{2} implies a^2 = frac{c^2}{2} cdot 4 = 4c^2 ] Substitute a^2 = 4c^2 into the first equation to solve for c: [ 4c^2 - b^2 = c^2 implies b^2 = 3c^2 ] Now we can calculate the eccentricity e of the ellipse: [ e = frac{c}{a} = frac{c}{sqrt{4c^2}} = frac{1}{2} ] Thus, we choose the correct answer: [ boxed{D: frac{1}{2}} ]

answer:**Analysis** This question examines the spatial relationships. By evaluating each statement based on the given conditions, we can draw conclusions. **Answer** Let's examine each of the given propositions: (1) In this case, if m nparallel n, it does not satisfy alpha nparallel beta, making this statement incorrect; (2) For this statement, a line is perpendicular to a plane only if it is perpendicular to two intersecting lines within the plane, making this statement incorrect; (3) In this case, only if m and n are within the same plane can we use the theorem of parallel planes to prove this conclusion, making this statement incorrect; (4) By using the theorem of line perpendicularity to a plane, we know this conclusion is correct. Therefore, only one proposition is correct, Hence, the correct choice is boxed{text{B}}.