answer:Since 361 is a prime number and cannot be divided by 473, and 172 = 2 times 2 times 43, 473 = 43 times 11, 360 = 4 times 90, the greatest common divisor of the products 360 times 473 and 172 times 361 is 4 times 43 = 172. Therefore, the correct answer is boxed{C}.

answer:1. **Identify the roots of ( P(x) ):** Let ( P(x) ) be an odd degree polynomial with real coefficients. Since ( P(x) ) is of odd degree, it has at least one real root. Let ( a_1, a_2, ldots, a_k ) be the distinct real roots of ( P(x) ). This means: [ P(a_i) = 0 quad text{for} quad i = 1, 2, ldots, k ] 2. **Consider the equation ( P(P(x)) = 0 ):** We need to show that the equation ( P(P(x)) = 0 ) has at least as many distinct real roots as the equation ( P(x) = 0 ). 3. **Analyze the inner polynomial ( P(x) ):** Since ( P(x) ) is an odd degree polynomial, its range is ( mathbb{R} ). This implies that for each ( a_i ) (where ( P(a_i) = 0 )), the equation ( P(x) = a_i ) has at least one real solution. Let ( b_i ) be a real number such that: [ P(b_i) = a_i quad text{for} quad i = 1, 2, ldots, k ] 4. **Verify the roots of ( P(P(x)) = 0 ):** For each ( b_i ) found in the previous step, we have: [ P(P(b_i)) = P(a_i) = 0 ] This means that each ( b_i ) is a root of the equation ( P(P(x)) = 0 ). 5. **Check the distinctness of the roots ( b_i ):** Suppose ( b_i = b_j ) for some ( i neq j ). Then: [ P(b_i) = P(b_j) implies a_i = a_j implies i = j ] This is a contradiction since ( a_i ) are distinct. Therefore, ( b_i ) must be distinct for ( i = 1, 2, ldots, k ). 6. **Conclusion:** We have shown that there are at least ( k ) distinct real roots ( b_1, b_2, ldots, b_k ) of the equation ( P(P(x)) = 0 ), where ( k ) is the number of distinct real roots of ( P(x) = 0 ). (blacksquare)

answer:We can find the analytical expression of x in [frac{1}{3},3] based on the function f(x) satisfying f(x)=2f(frac{1}{x}). It is known that in the interval [frac{1}{3},3], the function g(x)=f(x)-ax has three distinct zeros. By differentiating g(x) and studying its monotonicity, we can find the range of a. This problem fully utilizes the idea of case analysis and is a comprehensive problem with higher difficulty. We need to exclude the case when a < 0, and note that solving the equation involves a large amount of calculation, so it is important to learn how to analyze cases. Solution steps: 1. When a > 0, if x in [1,3], then f(x)=ln x, so g(x)=ln x-ax, (x > 0). 2. Differentiating g(x) gives g'(x)=frac{1}{x}-a=frac{1-ax}{x}. 3. If g'(x) < 0, then x > frac{1}{a}, and g(x) is a decreasing function. If g'(x) > 0, then x < frac{1}{a}, and g(x) is an increasing function. 4. For g(x) to have two intersections in [1,3], we must have g(frac{1}{a}) > 0, g(3) leqslant 0, and g(1) leqslant 0. Solving this system of inequalities gives frac{ln 3}{3} leqslant a < frac{1}{e} tag{1}. 5. If frac{1}{3} < x < 1, then 1 < frac{1}{x} < 3, and f(x)=2f(frac{1}{x})=2ln frac{1}{x}. Thus, g(x)=-2ln x-ax. 6. Differentiating g(x) gives g'(x)=-frac{2+ax}{x}. If g'(x) > 0, then x < -frac{2}{a} < 0, and g(x) is an increasing function. If g'(x) < 0, then x > -frac{2}{a}, and g(x) is a decreasing function. 7. For g(x) to have one intersection in [frac{1}{3},1], we must have g(frac{1}{3}) geqslant 0 and g(1) leqslant 0. Solving this system of inequalities gives 0 < a leqslant 6ln 3 tag{2}. 8. Combining (1) and (2), we get frac{ln 3}{3} leqslant a < frac{1}{e}. 9. If a < 0, then for x in [1,3], g(x)=ln x-ax > 0, which contradicts the requirement that g(x) has three distinct zeros in [frac{1}{3},3]. 10. If a=0, there is only one solution, which is discarded. In conclusion, boxed{frac{ln 3}{3} leqslant a < frac{1}{e}}.

answer:1. **Calculate the volume ratio between the actual observatory tower and the miniature model**: The actual observatory holds 200,000 liters, and Carson's miniature holds 0.2 liters. The ratio of the volumes is: [ frac{200000 text{ liters}}{0.2 text{ liters}} = 1000000 ] 2. **Relate the volume ratio to the scale of the model**: Since the volume of a cylinder is given by: [ V = pi r^2 h ] and utilizing the volume ratio, the ratio of linear dimensions (height and radius of the cylinder) is the cube root of the volume ratio: [ sqrt[3]{1000000} = 100 ] 3. **Calculate the height of the miniature tower**: The actual observatory tower is 60 meters high. Since the miniature should be 100 times smaller, the height of the miniature tower should be: [ frac{60 text{ meters}}{100} = 0.6 text{ meters} ] Conclusion: Carson should make his miniature observatory tower 0.6 meters high. [ 0.6 ] The final answer is boxed{0.6}