answer:**Analysis** This question examines the properties of quadratic functions, linear functions, and inverse proportion functions. By applying the properties of linear functions, inverse proportion functions, and quadratic functions, we can make a judgment. **Solution** According to the properties of functions, the function that decreases as x increases is: y=-x+1. Therefore, the correct answer is boxed{text{B}}.

answer:Let's denote the distance walked each day as {a_n}. Given that {a_n} forms a geometric sequence with a common ratio of frac{1}{2}, From S_6=378, we have S_6= frac{a_1(1- frac{1}{2^6})}{1- frac{1}{2}}=378, Solving this, we find a_1=192, Therefore, a_5=192× frac{1}{2^4}=12 li. Hence, the correct choice is boxed{C}. From the problem, we know that the distances walked each day form a geometric sequence with a common ratio of frac{1}{2}. By calculating the sum of the first 6 terms, S_6=378, we can find the first term, and then use the formula for the nth term of a geometric sequence to find the distance walked on the fifth day. This problem tests the application of the formula for the nth term of a geometric sequence. It is a basic problem, and when solving it, it is important to carefully read the problem and correctly apply the properties of geometric sequences.

answer:Since it is known that tan x = 3, we have frac {1}{sin^{2}x-2cos^{2}x} = frac {sin^{2}x+cos^{2}x}{sin^{2}x-2cos^{2}x} = frac {tan^{2}x+1}{tan^{2}x-2} = frac {9+1}{9-2} = frac {10}{7}, Therefore, the answer is: boxed{frac {10}{7}}. This solution utilizes the fundamental relationships between trigonometric functions of the same angle to find the value of frac {1}{sin^{2}x-2cos^{2}x}. This question primarily examines the application of the basic relationships between trigonometric functions of the same angle and is considered a basic problem.

answer:We start with the function given in the problem: f(x) = x^2 + 3x + 2. To solve the problem, we need to compute the following value: prod_{n=1}^{2019} left(1 - left|frac{2}{f(n)}right|right). Let's break down each step to compute (f(n)) and simplify the expression inside the product. 1. **Compute (f(n)):** [ f(n) = n^2 + 3n + 2. ] 2. **Rewrite the fraction:** [ left|frac{2}{f(n)}right| = left|frac{2}{n^2 + 3n + 2}right|. ] 3. **Simplify the expression:** Note that the polynomial (f(n)) can be factored: [ f(n) = n^2 + 3n + 2 = (n + 1)(n + 2). ] Therefore, the fraction simplifies to: [ left|frac{2}{f(n)}right| = left|frac{2}{(n + 1)(n + 2)}right|. ] 4. **Form the product term:** [ 1 - left|frac{2}{f(n)}right| = 1 - frac{2}{(n + 1)(n + 2)}. ] 5. **Combine into a single product:** [ prod_{n=1}^{2019} left(1 - frac{2}{(n + 1)(n + 2)}right). ] 6. **Simplify the inner fraction:** [ 1 - frac{2}{(n + 1)(n + 2)} = frac{(n + 1)(n + 2) - 2}{(n + 1)(n + 2)} = frac{n^2 + 3n + 2 - 2}{(n + 1)(n + 2)} = frac{n^2 + 3n}{(n + 1)(n + 2)} = frac{n(n + 3)}{(n + 1)(n + 2)}. ] 7. **Rewrite as a product of fractions:** [ prod_{n=1}^{2019} frac{n(n + 3)}{(n + 1)(n + 2)}. ] 8. **Write out the first few terms and last few terms to notice a pattern:** [ left( frac{1 cdot 4}{2 cdot 3} right) cdot left( frac{2 cdot 5}{3 cdot 4} right) cdot ldots cdot left( frac{2018 cdot 2021}{2019 cdot 2020} right) cdot left( frac{2019 cdot 2022}{2020 cdot 2021} right). ] Many terms in the numerator and denominator cancel out, leading to: [ frac{1 cdot 4}{2020 cdot 3}. ] 9. **Solve the final fraction:** [ frac{4}{2020 cdot 3} = frac{4}{6060} = frac{2}{3030} = frac{1}{1515}. ] Thus, the product simplifies to: [ boxed{frac{337}{1010}} ]