answer:Initially, there are 200 balloons in the hot air balloon. After about half an hour, 1/5 of the total number of balloons have blown up. So, the number of balloons that have blown up is: 1/5 * 200 = 40 balloons After another hour, twice the number of balloons that had already blown up also blow up. So, the additional number of balloons that blow up is: 2 * 40 = 80 balloons The total number of balloons that have blown up after both events is: 40 (initially blown up) + 80 (additional blown up) = 120 balloons The number of balloons that remain intact is the original number minus the total number that have blown up: 200 (original number) - 120 (total blown up) = 80 balloons So, boxed{80} balloons in the hot air balloon remain intact.

answer:First, calculate the total number of sequences without any conditions: 2^{10} = 1024 sequences. Next, consider sequences where the first flip is heads and the next four flips meet the condition (at least three heads): 1. Calculate the number of ways to get at least three heads in four flips: - Exactly three heads: {4 choose 3} = 4 ways (HHTT, HTHT, HTTH, THTH) - Exactly four heads: {4 choose 4} = 1 way (HHHH) Total ways to get three or four heads in four flips: 4 + 1 = 5 ways 2. Multiply by the possible outcomes of the remaining five flips, which are unrestricted: 2^5 = 32 sequences. 3. Thus, there are 5 times 32 = 160 sequences that start with a head and meet the condition. Now, calculate the sequences where the first flip is tails. These flips have no further conditions, so they are just the number of unrestricted sequences of the remaining nine flips, which is 2^9 = 512. Adding up both cases, the total number of distinct sequences matching the problem's criteria is 160 + 512 = 672. Conclusion: The total number of valid sequences where the first flip determines a condition on the subsequent flips is boxed{672}.

answer:Given that the vectors {overrightarrow{a},overrightarrow{b},overrightarrow{c}} form a basis for a space, and the vectors {overrightarrow{a}+overrightarrow{b}, overrightarrow{a}-overrightarrow{b}, overrightarrow{c}} form another basis for the space. We have a vector overrightarrow{p} with coordinates left(1,-2,3right) in the basis {overrightarrow{a},overrightarrow{b},overrightarrow{c}}. First, we express overrightarrow{p} in terms of the first basis: [ overrightarrow{p} = 1overrightarrow{a} - 2overrightarrow{b} + 3overrightarrow{c} ] We need to find the coordinates of overrightarrow{p} in the new basis {overrightarrow{a}+overrightarrow{b}, overrightarrow{a}-overrightarrow{b}, overrightarrow{c}}. Let's denote these coordinates as (x, y, z). Therefore, we can express overrightarrow{p} as: [ overrightarrow{p} = x(overrightarrow{a}+overrightarrow{b}) + y(overrightarrow{a}-overrightarrow{b}) + zoverrightarrow{c} ] Expanding this, we get: [ overrightarrow{p} = (x+y)overrightarrow{a} + (x-y)overrightarrow{b} + zoverrightarrow{c} ] Since this expression for overrightarrow{p} must be equal to the expression in terms of the original basis, we have: [ 1overrightarrow{a} - 2overrightarrow{b} + 3overrightarrow{c} = (x+y)overrightarrow{a} + (x-y)overrightarrow{b} + zoverrightarrow{c} ] Equating the coefficients of overrightarrow{a}, overrightarrow{b}, and overrightarrow{c} gives us a system of equations: [ begin{array}{l} x + y = 1 x - y = -2 z = 3 end{array} ] Solving this system of equations, we find: [ begin{aligned} x + y &= 1 x - y &= -2 end{aligned} ] Adding these two equations, we get 2x = -1 Rightarrow x = -frac{1}{2}. Substituting x = -frac{1}{2} into the first equation, we get -frac{1}{2} + y = 1 Rightarrow y = 1 + frac{1}{2} = frac{3}{2}. And from the third equation, we already have z = 3. Thus, the coordinates of overrightarrow{p} in the new basis are left(-frac{1}{2}, frac{3}{2}, 3right). Therefore, the correct answer is: [ boxed{A} ]

answer:**Analysis** This problem examines the general term and summation of a sequence, and the application of the splitting method. Determining the general term of the sequence is key. From the given information, we have a_{1}+a_{2}+ldots+a_{n}=n(2n+1)=S_{n}. After finding S_{n}, we use a_{n}=S_{n}-S_{n-1} when ngeqslant 2 to find the general term a_{n}. Finally, by applying the splitting method, we can find the sum. This is a challenging problem. **Solution** Given dfrac {n}{a_{1}+a_{2}+ldots+a_{n}}= dfrac {1}{2n+1}, therefore a_{1}+a_{2}+ldots+a_{n}=n(2n+1)=S_{n} When ngeqslant 2, a_{n}=S_{n}-S_{n-1}=4n-1, and it is verified that this also holds when n=1, therefore a_{n}=4n-1, therefore b_{n}= dfrac {a_{n}+1}{4}=n, therefore dfrac {1}{b_{n}cdot b_{n+1}}= dfrac {1}{n}- dfrac {1}{n+1} therefore dfrac {1}{b_{1}b_{2}}+ dfrac {1}{b_{2}b_{3}}+ldots+ dfrac {1}{b_{10}b_{11}} =(1- dfrac {1}{2})+( dfrac {1}{2}- dfrac {1}{3})+( dfrac {1}{3}- dfrac {1}{4})+ldots+( dfrac {1}{10}- dfrac {1}{11}) =1- dfrac {1}{11}= dfrac {10}{11}． Therefore, the correct answer is boxed{B}.