answer:We start by analyzing the given equation: [ x(x-y) + y(y-z) + z(z-x) = 1. ] We can rewrite this equation by expanding and rearranging terms: [ x^2 - xy + y^2 - yz + z^2 - zx = 1. ] Grouping and factoring the terms differently, we get: [ (x^2 - xy + y^2) + (y^2 - yz + z^2) + (z^2 - zx + x^2) - (x^2 + y^2 + z^2) = 1. ] This simplifies to: [ x^2 + y^2 + z^2 - xy - yz - zx = 1. ] We can rewrite this equation as: [ (x-y)^2 + (y-z)^2 + (z-x)^2 = 2. ] This equation tells us that the sum of the squares of the differences between each pair of variables is 2. Since (x-y)^2, (y-z)^2, and (z-x)^2 are all non-negative integers, the possible values for these squares are 0, 1, 2, etc. The only way their sum can be 2 is if two of these squares are 1 and one is 0, or if one of them is 2 and the others are 0. Now, we check each of the given conditions to see which one satisfies this requirement: - **Option A: x>y and y=z** - Here, (x-y)^2 > 0 and (y-z)^2 = 0, (z-x)^2 > 0. We need to check if (x-y)^2 + (z-x)^2 = 2. This is not necessarily true. - **Option B: x=y-1 and y=z-1** - Here, (x-y)^2 = 1, (y-z)^2 = 1, (z-x)^2 = 4. This does not satisfy the equation. - **Option C: x=z+1 and y=x+1** - Here, (x-y)^2 = 1, (y-z)^2 = 4, (z-x)^2 = 1. This does not satisfy the equation. - **Option D: x=z and y-1=x** - Here, (x-y)^2 = 1, (y-z)^2 = 1, (z-x)^2 = 0. This satisfies the equation as 1 + 1 + 0 = 2. - **Option E: x+y+z=1** - This condition does not directly relate to the squares of differences and does not guarantee the required sum of squares. Thus, the only option that guarantees the equation x(x-y) + y(y-z) + z(z-x) = 1 is satisfied is: [ boxed{textbf{(D)} : x=z text{ and } y-1=x}. ]

answer:The incorrect mean was calculated with the wrong observation of 23 instead of the correct observation of 48. To find the corrected mean, we need to first calculate the total sum of the observations with the incorrect value and then adjust it for the correct value. The incorrect total sum of the observations (S_incorrect) can be calculated as: S_incorrect = mean * number of observations S_incorrect = 41 * 50 S_incorrect = 2050 Now, we need to remove the incorrect observation and add the correct observation to the total sum: S_correct = S_incorrect - incorrect observation + correct observation S_correct = 2050 - 23 + 48 S_correct = 2050 - 23 + 48 S_correct = 2027 + 48 S_correct = 2075 Now we can calculate the corrected mean (mean_correct) by dividing the corrected total sum by the number of observations: mean_correct = S_correct / number of observations mean_correct = 2075 / 50 mean_correct = 41.5 The corrected new mean of the observations is boxed{41.5} .

answer:Let's denote the ending number as N. Since we are looking for numbers divisible by 4, we start with the first number after 6 that is divisible by 4, which is 8. The sequence of numbers divisible by 4 between 6 and N is: 8, 12, 16, ..., N (where N is also divisible by 4) The average of this sequence is given as 22. To find the average of an arithmetic sequence, we use the formula: Average = (First term + Last term) / 2 In this case, the first term is 8, and the last term is N. So we have: 22 = (8 + N) / 2 Multiplying both sides by 2 to get rid of the denominator, we have: 44 = 8 + N Now, subtract 8 from both sides to solve for N: N = 44 - 8 N = 36 Therefore, the ending number is boxed{36} .

answer:To determine the period of the function y = tan frac{x}{2}, we need to understand how the variable x affects the tangent function. The standard period of the tangent function, tan x, is pi, but since we have tan frac{x}{2}, the input to the tangent function is scaled by a factor of frac{1}{2}. This scaling affects the period of the function. 1. The tangent function, tan x, has a period of pi. This means that tan x repeats its values every pi units. 2. For the function y = tan frac{x}{2}, the input x is scaled by frac{1}{2}. This scaling extends the period of the function because the input to tan is now frac{x}{2} instead of x. 3. To find the new period, we look at the range of frac{x}{2} that completes one full cycle of the tangent function. The tangent function completes one cycle as its input ranges from -frac{pi}{2} to frac{pi}{2}. 4. Therefore, for y = tan frac{x}{2}, the input x must range from -pi to pi to cover the same range for frac{x}{2} from -frac{pi}{2} to frac{pi}{2}. 5. The period of y = tan frac{x}{2} is the difference between the end and start of this range for x: pi - (-pi) = pi + pi = 2pi. Thus, the period of y = tan frac{x}{2} is boxed{2pi}.