answer:To solve the problem, let's break down the payment calculation into two parts based on the quantity purchased, x, where x > 20: 1. For the first 20 units, the price per unit is the original price, which is 80. So, the total cost for these 20 units is 80 times 20. 2. For any units purchased beyond the first 20, there is a 30% discount on the original price. The original price per unit is 80, so the discounted price per unit is 80 times (1 - 0.3) = 80 times 0.7. If x is the total number of units purchased, then (x - 20) units are purchased at the discounted price. Therefore, the total cost for the units purchased beyond the first 20 is 0.7 times 80 times (x - 20). Adding these two parts together gives us the total payment amount, y: [y = 0.7 times 80 times (x - 20) + 80 times 20] Therefore, the correct expression that represents the relationship between y and x is: [y = 0.7 times 80 times (x - 20) + 80 times 20] This matches option A, so the correct answer is boxed{A}.

answer:First, we find (n^2 = (2^{20}3^{25})^2 = 2^{40}3^{50}). The number of factors of (n^2) is given by ((40 + 1)(50 + 1) = 41 times 51 = 2091). Now, excluding (n) itself from this count and recognizing that each factor pairs uniquely with another factor to give (n^2), half of these factors (except (n) itself) are less than (n). This gives us (frac{2091 - 1}{2} = 1045) factors of (n^2) less than (n). The number of factors of (n) is ((20 + 1)(25 + 1) = 21 times 26 = 546). Using complementary counting, the number of factors of (n^2) that are less than (n) and do not divide (n) are (1045 - 546 = boxed{499}).

answer:To find the values of the given distances ( M O ), ( M C ), ( M H ), and ( M P ), and their arrangement in ascending order, follow these steps: 1. **Determine ( M O ):** - The point ( O ) is the midpoint of the segment ( A B ). - By applying the midpoint formula in geometry, we have: [ M O= frac{M A + M B}{2} = frac{a + b}{2} ] 2. **Determine ( M C ):** - Using the theorem about the tangent and secant from ( M ), we have the following relation: [ M C = sqrt{M A cdot M B} = sqrt{a b} ] 3. **Determine ( M H ):** - The length of ( M H ) can be derived as: [ M H = frac{M C^2}{M O} ] Now substituting the values: [ M H = frac{a b}{frac{a+b}{2}} = frac{2 a b}{a + b} ] 4. **Determine ( M P ):** - Using the Pythagorean theorem in the right triangle ( M O P ), where ( O P ) is perpendicular to ( A B ): [ M P = sqrt{M O^2 + O P^2} ] Since ( O P ) is equal to ( O A ): [ M P = sqrt{M O^2 + O A^2} ] We know ( O A = frac{|a - b|}{2} ), so: [ M P = sqrt{left( frac{a + b}{2} right)^2 + left( frac{a - b}{2} right)^2} ] Simplify the expression: [ M P = sqrt{frac{(a + b)^2}{4} + frac{(a - b)^2}{4}} ] [ M P = sqrt{frac{(a^2 + 2ab + b^2) + (a^2 - 2ab + b^2)}{4}} ] [ M P = sqrt{frac{2a^2 + 2b^2}{4}} ] [ M P = sqrt{frac{a^2 + b^2}{2}} ] # Conclusion: We have determined: [ begin{aligned} M O &= frac{a + b}{2} M C &= sqrt{a b} M H &= frac{2 a b}{a + b} M P &= sqrt{frac{a^2 + b^2}{2}} end{aligned} ] To compare: - Notice that ( M H < M C ) because: [ frac{2 a b}{a+b} < sqrt{a b} ] After squaring both sides and simplifying, you will find that the inequality holds for ( a > b ). - Similarly, ( M C < M O ) because: [ sqrt{a b} < frac{a + b}{2} ] After squaring both sides and simplifying, you will find that the inequality holds. - And, ( M O < M P ) because: [ frac{a + b}{2} < sqrt{frac{a^2 + b^2}{2}} ] After squaring both sides and simplifying, you will find that the inequality holds. Thus, the final arrangement in increasing order is: [ boxed{M H < M C < M O < M P} ]

answer:This problem tests the relationship between a line and a circle, the application of function thinking, computational ability, and the ability to transform thoughts. By analyzing the equation of the curve, we can determine the characteristics of the curve. According to overrightarrow{AP}+overrightarrow{AQ}=overrightarrow{0}, point A is the midpoint of PQ. By combining the range of x, we can find the range of m. The curve C: x=-sqrt{4-y^{2}}, is a circle with the origin as its center and a radius of 2. The range of x_P is [-2,0]. For point A(m,0), there exists a point P on curve C and a point Q on line l such that overrightarrow{AP}+overrightarrow{AQ}=overrightarrow{0}. This indicates that A is the midpoint of PQ. The x-coordinate of Q is 6. Therefore, m=frac{6+x_P}{2} in [2,3]. Hence, the answer is boxed{[2,3]}.