answer:The problem is framed in terms of finding distinct sets of 8 positive odd integers that sum to 20. To solve this, we can reframe the problem by subtracting 1 from each integer in the set, transforming the sum of odd integers into a sum of even integers. 1. **Transform each odd integer to an even integer by subtracting 1 from each:** Let each odd integer be represented as 2y_i + 1, where y_i is a non-negative integer. The given problem states: sum_{i=1}^{8} (2y_i + 1) = 20 2. **Simplify the equation:** We break it down as follows: 2sum_{i=1}^{8} y_i + 8 = 20 Subtract 8 on both sides: 2sum_{i=1}^{8} y_i = 12 Divide both sides by 2: sum_{i=1}^{8} y_i = 6 3. **Determine the number of non-negative integer solutions:** Now, we need to find the number of solutions to the equation y_1 + y_2 + cdots + y_8 = 6 where each y_i is a non-negative integer. 4. **Use the stars and bars theorem:** This is a classic combinatorial problem that can be solved by the "stars and bars" method. The formula for the number of solutions of the equation x_1 + x_2 + cdots + x_k = n is given by: binom{n + k - 1}{k - 1} 5. **Substitute ( n = 6 ) and ( k = 8 ):** binom{6 + 8 - 1}{8 - 1} = binom{13}{7} 6. **Calculate the binomial coefficient:** binom{13}{7} = frac{13!}{7!(13-7)!} = frac{13!}{7!6!} Calculate the factorials: 13! = 6227020800, quad 7! = 5040, quad 6! = 720 Therefore, binom{13}{7} = frac{6227020800}{(5040)(720)} = 1716 # Conclusion: The number of distinct sets of 8 positive odd integers that sum to 20 is: boxed{1716}

answer:First, simplify the equation of the circle by dividing everything by 2: [ x^2 - y^2 + 12x + 4y + 18 = 0. ] Next, complete the square for both x and y: [ x^2 + 12x = (x+6)^2 - 36, ] [ -y^2 + 4y = -(y - 2)^2 + 4. ] Combining, we get [ (x+6)^2 - (y-2)^2 - 36 + 4 + 18 = 0 ] [ (x+6)^2 - (y-2)^2 = 14. ] So, the center of the circle is at (-6, 2), and having equation (x+6)^2 + (y-2)^2 = 14, the radius of the circle is sqrt{14}. The side length of the square is twice the radius: [ 2sqrt{14} .] Thus, the area of the square is: [ (2sqrt{14})^2 = 4 times 14 = boxed{56} text{ square units.} ]

answer:1. **Calculate the ratio of volumes between the real water tower and the model**: The actual tower holds 200,000 liters, while the miniature will hold 0.2 liters. The volume ratio is: [ frac{200000 text{ liters}}{0.2 text{ liters}} = 1000000 ] 2. **Determine the scaling factor for the dimensions**: Since volume scales with the cube of the linear dimensions: [ sqrt[3]{1000000} = 100 ] 3. **Calculate the model's tower height**: The real tower is 60 meters high. Applying the scaling factor: [ frac{60 text{ meters}}{100} = 0.6 text{ meters} ] 4. **Conclusion**: Logan should construct the miniature water tower to be 0.6 meters high. [ 0.6 ] The final answer is boxed{0.6}

answer:Given the complex number (1+i)(a-i), we are required to find when it is a pure imaginary number. By multiplying out the terms, we get: begin{align*} (1+i)(a-i) &= (1 cdot a - 1 cdot i + i cdot a - i cdot i) &= (a - i + ai + 1) &= (a + 1) + (a - 1)i. end{align*} For a complex number to be a pure imaginary number, its real part must be equal to zero, and its imaginary part must be non-zero. Thus, we have the following conditions: begin{align*} Re((1+i)(a-i)) &= a + 1 = 0 text{, and} Im((1+i)(a-i)) &= a - 1 neq 0. end{align*} From the first condition, we can find the value of a: begin{align*} a + 1 &= 0 a &= -1. end{align*} Now, we need to check the second condition based on the value of a we found. Plugging a = -1 into the imaginary part's equation we get: begin{align*} Im((1+i)(-1-i)) = -1 - 1 neq 0, end{align*} which is indeed not equal to zero. Therefore, the value of a that makes (1+i)(a-i) a pure imaginary number is boxed{-1}.