answer:We can represent the sum as [sum_{n=1}^{1991} n(1992 - n).] This sum is equal to [sum_{n=1}^{1991} (1992n - n^2) = 1992 sum_{n=1}^{1991} n - sum_{n=1}^{1991} n^2.] Using the formula for the sum of the first (n) natural numbers, [sum_{n=1}^{1991} n = frac{1991 times 1992}{2},] and the formula for the sum of the squares of the first (n) natural numbers, [sum_{n=1}^{1991} n^2 = frac{1991 times 1992 times 3983}{6},] we substitute these into our equation: [sum_{n=1}^{1991} (1992n - n^2) = 1992 cdot frac{1991 times 1992}{2} - frac{1991 times 1992 times 3983}{6}.] Simplifying further, [frac{1991 times 1992}{6} (3 times 1992 - 3983) = frac{1991 times 1992}{6} cdot 1993.] This simplifies to [frac{1991 times 1992 times 1993}{6} = 1991 times 996 times 664.] Thus, (x = boxed{664}.)

answer:To determine which equation has been factored, we need to understand the definition of factoring in the context of polynomials. Factoring, in general, refers to the process of transforming a polynomial into the product of two or more polynomials. Let's analyze each option: **Option A:** (2x-3)^{2}=4x^{2}-12x+9 This option represents an expanded form of a binomial squared, not a factored form. Therefore, it is not factored. **Option B:** x^{2}-5x+6=left(x-2right)left(x-3right) Here, the polynomial x^{2}-5x+6 has been transformed into the product of two binomials, left(x-2right)left(x-3right). This is a clear example of factoring. **Option C:** a^{2}-5=left(a+2right)left(a-2right)-1 In this case, the expression left(a+2right)left(a-2right)-1 is close to being factored but is not purely in the form of the product of polynomials due to the subtraction of 1. Therefore, it is not considered factored. **Option D:** (3+x)left(3-xright)=left(3-xright)left(3+xright) This option shows a product of two polynomials, but there is no transformation or simplification process involved. It is merely a rearrangement of the factors, not a process of factoring from a polynomial to its factors. Based on the definition and analysis, the equation that has been factored is: boxed{text{B}}

answer:1. Let ( y_i = frac{1}{1 + x_i} ) for ( i = 1, 2, ldots, n+1 ). Then, [ sum_{i=1}^{n+1} y_i = 1. ] 2. Define: [ s_i = sum_{j=1, j neq i}^{n+1} y_j ] and [ p_i = prod_{j=1, j neq i}^{n+1} y_j. ] 3. Using the arithmetic-mean geometric-mean inequality (AM-GM inequality): [ frac{s_i}{y_i} = frac{1 - y_i}{y_i} geq frac{n sqrt[n]{p_i}}{y_i}. ] 4. By multiplying both sides of the inequality by ( y_i ) and then for each i: [ frac{1 - y_i}{y_i} geq frac{n sqrt[n]{p_i}}{y_i}. ] 5. Notice that in the product ( prod_{i=1}^{n+1} sqrt[n]{p_i} ), each ( sqrt[n]{y_i} ) appears ( n ) times. Therefore, [ prod_{i=1}^{n+1} sqrt[n]{p_i} = prod_{i=1}^{n cdot 1} y_i. ] 6. Hence, it follows that [ x_1 x_2 cdots x_{n+1} = prod_{i=1}^{n+1} left( frac{1 - y_i}{y_i} right) geq prod_{i=1}^{n+1} frac{n sqrt[n]{p_i}}{y_i} = n^{n+1} frac{prod_{i=1}^{n+1} y_i}{prod_{i=1}^{n+1} y_i} = n^{n+1}. ] Therefore, we have demonstrated that: [ x_1 x_2 cdots x_{n+1} geq n^{n+1}. ] 7. Let's express the condition involving ( x_1, x_2, ldots, x_{n+1} ) under our initial transformation: [ frac{1}{1 + x_{n+1}} leq 1 - sum_{i=1}^n frac{1}{1 + x_i}. ] Consequently, if we assume: [ x_{n+1} geq frac{n^{n+1}}{prod_{i=1}^n x_i}, ] then we have: [ sum_{i=1}^n frac{1}{1 + x_i} geq 1 - frac{1}{1 + frac{n^{n+1}}{prod_{i=1}^n x_i}} = frac{n^{n+1}}{n^{n+1} + x_1 x_2 cdots x_n}. ] 8. Considering the results from steps above and by correctly embodying our assumption, we've validated the fundamental inequality: [ sum_{i=1}^n frac{1}{1 + x_i} geq frac{n^{n+1}}{n^{n+1} + x_1 x_2 cdots x_n}. ] 9. Proof concludes by using the given summation in the problem. Thus, we confirm ( x_1 x_2 cdots x_{n+1} geq n^{n+1} ). (blacksquare)

answer:Let's tackle the problem step by step. 1. **Identify the base of the solid:** The base of the solid is a square with side length ( s ). 2. **Determine all dimensions of the solid:** - The height from the base to the top face, which is parallel to the base, is ( s ). - The length of the top parallel face is ( 2s ). - The remaining edges of the solid are ( s ). 3. **Given value:** The side length of the base square ( s = 6sqrt{2} ). 4. **Construct vertical planes:** To aid in calculating the volume, we analyze two key vertical planes: - Plane ( BCM ) is perpendicular to the base and intersects ( EF ) at ( M ). - Plane ( ADN ) is perpendicular to the base and intersects ( EF ) at ( N ). 5. **Calculate intersection properties:** - Since ( BE = AF = AB = s ) and ( EF = 2s ), we find that point ( M ) is halfway, i.e. ( ME = frac{1}{2}s ). - Now, calculate the height ( BM ) within the triangle ( BCM ): [ BM = sqrt{ s^2 - left(frac{1}{2}sright)^2 } = sqrt{ s^2 - frac{1}{4}s^2 } = sqrt{ frac{3}{4}s^2 } = frac{sqrt{3}}{2}s ] - Similarly, the length of ( CM ): [ CM = frac{sqrt{3}}{2}s ] 6. **Calculate the area of triangle ( triangle BCM ):** The height from ( C ) to ( BM ) can be calculated: [ text{Height of } triangle BCM = sqrt{left(frac{sqrt{3}}{2}sright)^2 - left(frac{1}{2}sright)^2} = sqrt{frac{3}{4}s^2 - frac{1}{4}s^2} = sqrt{frac{2}{4}s^2} = frac{sqrt{2}}{2}s ] The area of ( triangle BCM ): [ text{Area} = frac{1}{2} cdot BC cdot text{Height} = frac{1}{2} cdot s cdot frac{sqrt{2}}{2}s = frac{sqrt{2}}{4}s^2 ] 7. **Calculate the volume of ( V_{text{E-BCM}} ):** The volume of the pyramidal section ( V_{text{E-BCM}} ): [ V_{text{E-BCM}} = frac{1}{3} cdot text{Base Area} cdot text{Height} = frac{1}{3} cdot frac{sqrt{2}}{4}s^2 cdot frac{1}{2}s = frac{sqrt{2}s^3}{24} ] 8. **Similarly, the volume for ( V_{text{F-ADN}} ):** [ V_{text{F-ADN}} = frac{sqrt{2}s^3}{24} ] 9. **Calculate the volume of ( BCMA-DN ):** The volume of the parallelepiped ( BCMA-DN ): [ V_{BCMA-DN} = text{Base Area} cdot text{Height} = frac{sqrt{2}}{4}s^2 cdot s = frac{sqrt{2}s^3}{4} ] 10. **Combine all volumes:** Finally, the total volume of the solid is: [ V = V_{text{E-BCM}} + V_{text{F-ADN}} + V_{BCMA-DN} = frac{sqrt{2}s^3}{24} + frac{sqrt{2}s^3}{24} + frac{sqrt{2}s^3}{4} ] Simplifying, we get: [ V = frac{sqrt{2}s^3}{24} + frac{sqrt{2}s^3}{24} + 6 frac{sqrt{2}s^3}{24} = frac{sqrt{2}s^3}{3} ] 11. **Insert the value of ( s = 6sqrt{2} ):** [ V = frac{sqrt{2}(6sqrt{2})^3}{3} = frac{sqrt{2}(6^3 cdot (sqrt{2})^3)}{3} = frac{sqrt{2}(216 cdot 2 sqrt{2})}{3} = frac{432sqrt{2}}{3} = 288 ] Conclusion: [ boxed{288} ]