answer:# Solution 1: Step-by-Step 1. Identify the amount of gold taken as tax and the total tax paid in coins: - The customs take 2 pounds of gold as tax. - The person is returned 5000 coins after paying the tax. 2. Calculate the effective amount of gold after tax is taken: - The tax rate is one-tenth, so for 12 pounds, the tax would be 12 times frac{1}{10} = 1.2 pounds. - However, 2 pounds of gold are taken, so we adjust the calculation to account for the actual gold taken versus the expected tax in gold. This gives us 2 - 12 times frac{1}{10} = 2 - 1.2 = 0.8 pounds. 3. Determine the value of one pound of gold: - The value of the gold taken as tax is equivalent to 5000 coins. - Therefore, the value of 0.8 pounds of gold is 5000 coins. - To find the value of one pound of gold, we calculate 5000 div 0.8 = 6250 coins. 4. Conclude the value of one pound of gold: - One pound of gold is worth 6250 coins. boxed{text{D: 6250 coins}} # Solution 2: Step-by-Step 1. Let the value of one pound of gold be x coins. 2. Formulate the equation based on the problem: - The customs take 2 pounds of gold as tax, and the tax rate is one-tenth of the total gold carried, which is 12 pounds. - The equation for the tax paid in gold and its equivalent in coins is (2 - 12 times frac{1}{10})x = 5000. 3. Simplify and solve the equation: - Simplify the expression inside the parentheses: 2 - 12 times frac{1}{10} = 2 - 1.2 = 0.8. - Substitute back into the equation: 0.8x = 5000. - Solve for x: x = 5000 div 0.8 = 6250 coins. 4. Conclude the value of one pound of gold: - One pound of gold is worth 6250 coins. boxed{text{D: 6250 coins}}

answer:1. Since the ellipse C: frac{x^{2}}{a^{2}} + frac{y^{2}}{b^{2}} = 1 (a > b > 0), has the right focus F(sqrt{3}, 0) and eccentricity e = frac{sqrt{3}}{2}, From the given information, we know begin{cases} c = sqrt{3} e = frac{c}{a} = frac{sqrt{3}}{2} a^2 = b^2 + c^2 end{cases}. Solving this system of equations, we get a = 2 and b = 1. Therefore, the equation of the ellipse C is frac{x^{2}}{4} + y^{2} = 1. 2. Let the equation of the line MN be y = x - sqrt{3}. Solving the system of equations begin{cases} y = x - sqrt{3} frac{x^{2}}{4} + y^{2} = 1 end{cases}, we get 5x^2 - 8sqrt{3}x + 8 = 0. Let M(x_1, y_1) and N(x_2, y_2), then begin{cases} x_1 + x_2 = frac{8sqrt{3}}{5} x_1x_2 = frac{8}{5} end{cases}. Thus, |MN| = sqrt{1 + 1^2} cdot sqrt{(x_1 + x_2)^2 - 4x_1x_2} = sqrt{2} cdot sqrt{(frac{8sqrt{3}}{5})^2 - 4 cdot frac{8}{5}x_1x_2} = frac{8}{5}. The distance d from point O to line MN is d = frac{|0 - 0 - sqrt{3}|}{sqrt{2}} = frac{sqrt{6}}{2}. Therefore, the area of triangle OMN is S_{triangle OMN} = frac{1}{2} |MN| cdot d = frac{1}{2} cdot frac{8}{5} cdot frac{sqrt{6}}{2} = boxed{frac{2sqrt{6}}{5}}.

answer:Given the problem, we start by analyzing the given greatest common divisors (GCDs) of pairs of numbers (a, b), (b, c), and (c, d). We have: 1. gcd(a, b) = 24 2. gcd(b, c) = 36 3. gcd(c, d) = 54 4. 70 < gcd(d, a) < 100 From these GCDs, we can infer the prime factorizations of the GCDs as follows: - 24 = 2^3 cdot 3^1 - 36 = 2^2 cdot 3^2 - 54 = 2^1 cdot 3^3 Given these prime factorizations, we can express a, b, c, and d in terms of their prime factors and some positive integers w, x, y, z respectively: - a = 2^3 cdot 3^1 cdot w - b = 2^3 cdot 3^2 cdot x - c = 2^2 cdot 3^3 cdot y - d = 2^1 cdot 3^3 cdot z Next, we consider the restrictions imposed by the GCDs. If 3 divides w, then gcd(a, b) would be at least 2^3 cdot 3^2, which contradicts the given gcd(a, b) = 24. Therefore, 3 does not divide w. Similarly, if 2 divides z, then gcd(c, d) would be at least 2^2 cdot 3^3, which contradicts the given gcd(c, d) = 54. Hence, 2 does not divide z. Thus, we have gcd(a, d) = 2 cdot 3 cdot gcd(w, z), where neither 2 nor 3 divides gcd(w, z). This means gcd(w, z) must be composed of primes greater than or equal to 5. Given the constraint 70 < gcd(d, a) < 100, and considering our expression for gcd(a, d), the only prime factorization that fits this criterion and the restrictions is 78 = 2 cdot 3 cdot 13. Therefore, the divisor of a that must be present, given the conditions, is boxed{13}.

answer:From (1-2i)i=a+bi, we get: 2+i=a+bi, Therefore, a=2, b=1, so ab=2. Hence, the answer is boxed{2}.