answer:Since YZ is the diameter of the circle, angle YWZ is a right angle. This means triangle XYW sim triangle YWZ sim triangle XYZ due to AA similarity. Using the similarity ratio frac{YW}{XW} = frac{ZW}{YW}, we have: [ frac{9}{3} = frac{ZW}{9} ] [ 3 = frac{ZW}{9} ] [ ZW = 3 times 9 = 27 ] Thus, the length of segment ZW is boxed{27}.

answer:Let's denote the total number of diplomats by T, the number of diplomats who speak French by F, the number of diplomats who speak Russian by R, and the number of diplomats who speak both French and Russian by B. We are given that T = 70. We are also given that 20% of the diplomats speak neither French nor Russian, which means that 0.20 * T diplomats speak neither language. So, 0.20 * 70 = 14 diplomats speak neither French nor Russian. We are given that 10% of the diplomats speak both languages, which means that B = 0.10 * T. So, B = 0.10 * 70 = 7 diplomats speak both French and Russian. We are given that 32 diplomats do not speak Russian. Since there are 70 diplomats in total, the number of diplomats who speak Russian is T - 32 = 70 - 32 = 38. Now, let's use the principle of inclusion-exclusion to find the number of diplomats who speak French (F). The principle of inclusion-exclusion states that for any two sets, the size of their union is the size of the first set plus the size of the second set minus the size of their intersection. So, the number of diplomats who speak French or Russian (or both) is F + R - B. But we also know that the number of diplomats who speak neither language is 14, so the number of diplomats who speak French or Russian (or both) is T - 14 = 70 - 14 = 56. We can now set up the equation: F + R - B = 56 We know R = 38 and B = 7, so we can substitute these values into the equation: F + 38 - 7 = 56 F + 31 = 56 F = 56 - 31 F = 25 Therefore, boxed{25} diplomats speak French.

answer:First, we look for an integer which leaves remainder of 3 when divided by 6, and a remainder of 5 when divided by 8. Checking the remainders of 5, 13, 21, 29, ldots when divided by 6, we find that 29 is the least positive integer satisfying this condition. By the Chinese Remainder Theorem, the positive integers which leave a remainder of 3 when divided by 6 and a remainder of 5 when divided by 8 are those that differ from 29 by a multiple of 6 cdot 8 = 48. Checking the remainders of 29, 77, 125, ldots when divided by 9, we find that 29 leaves a remainder of 2. We adjust to find the next multiple of 48 that fits, which is 77, leaving a remainder of 5 when divided by 9. Again using the Chinese Remainder Theorem, the integers which satisfy all three conditions are those that differ from 77 by a multiple of 6 cdot 8 cdot 9 = 432. Among the integers 77, 509, ldots, only boxed{269} is between 200 and 300.

answer:From the problem, we can derive that sin (2 times dfrac {3π}{8}+φ)=0, hence 2 times dfrac {3π}{8}+φ=kπ. Solving for φ, we get φ=kπ- dfrac {3π}{4}, where k∈Z. Given that 0<φ<dfrac {π}{2}, we find φ=dfrac {π}{4}. Therefore, f(x)=sin (2x+dfrac {π}{4}). Now, when 2kπ+ dfrac {π}{2} leq 2x+ dfrac {π}{4} leq 2kπ+ dfrac {3π}{2}, we get kπ+ dfrac {π}{8} leq x leq kπ+ dfrac {5π}{8}. Thus, the interval where the function f(x) is monotonically decreasing is boxed{[kπ+ dfrac {π}{8}, kπ+ dfrac {5π}{8}]}, where k∈Z. The answer is D. By understanding the symmetry of the function, we determine its analytical expression, which in turn helps us find the interval where it is monotonically decreasing. This problem tests our knowledge of trigonometric functions and their properties, particularly their monotonicity, making it a fundamental question.