answer:**Analysis** This problem examines the application of function graphs and the use of basic inequalities to find the maximum and minimum values. First, we draw the graph of the function (f(x)) to determine the relationship and range of (m) and (n), then derive a function in terms of (m) and use derivatives to find the maximum and minimum values. **Solution** Solve the equation (x^{2} - 2x - 1 = 0) to get (x = 1 pm sqrt{2}). Therefore, when (1 - sqrt{2} < x < 1 + sqrt{2}), (x^{2} - 2x - 1 < 0), and when (x < 1 - sqrt{2} text{ or } x > 1 + sqrt{2}), (x^{2} - 2x - 1 > 0), so (f(x) = begin{cases} x^{2} - 2x - 1, & x < 1 - sqrt{2} text{ or } x > 1 + sqrt{2} -x^{2} + 2x + 1, & 1 - sqrt{2} < x < 1 + sqrt{2} end{cases}). Since (m > n > 1) and (f(m) = f(n)), it follows that (1 < n < 1 + sqrt{2}, m > 1 + sqrt{2}), thus ((m-1)(n-1) > 0), (f(n) = -n^{2} + 2n + 1), (f(m) = m^{2} - 2m - 1), Since (f(m) = f(n)), we have (m^{2} - 2m - 1 + n^{2} - 2n - 1 = 0), which means ((m-1)^{2} + (n-1)^{2} = 4), and ((m-1)^{2} + (n-1)^{2} > 2(m-1)(n-1)), thus ((m-1)(n-1) < dfrac{1}{2} left[(m-1)^{2} + (n-1)^{2}right] = 2), Therefore, the range of ((m-1)(n-1)) is (0 < (m-1)(n-1) < 2). Hence, the answer is boxed{(0,2)}.

answer:First, we are given the function f(x) = 3x + 1. The problem asks us to find a relationship between a and b such that when |x - 1| < b, the inequality |f(x) - 4| < a is satisfied. We start by expressing the condition in terms of f(x): begin{align*} |f(x) - 4| &< a |3x + 1 - 4| &< a |3x - 3| &< a 3|x - 1| &< a end{align*} Since |x - 1| < b, we can replace |x - 1| in the inequality with b to get: 3b < a. However, to satisfy the original inequality |f(x) - 4| < a for all values of x within the range |x - 1| < b, the inequality 3b < a might not be a strict inequality (i.e., we may have 3b = a). Hence, the correct relationship between a and b is: 3b leq a. Rearranging the terms gives us the final relationship: begin{equation} a - 3b geq 0. end{equation} boxed{a - 3b geq 0}

answer:To solve this problem, let's follow the folding and unfolding process with the new conditions: 1. **First Fold (Bottom to Top)**: Folding the paper in half by bringing the bottom edge up to the top edge reduces the paper's height by half. 2. **Second Fold (Left to Right)**: Folding the paper again by bringing the left edge to the right edge reduces the width to half, resulting in the paper being one quarter of its original size. 3. **Third Fold (Top to Bottom)**: The final fold brings the top edge down to the bottom edge, further reducing the size of the paper. The paper is now one-eighth of its original size. 4. **Punching the Hole**: A hole is punched in the top left corner of the folded paper. Now, let's reverse the folding process: 1. **Unfolding the Third Fold (Top to Bottom)**: Unfolding this layer, the hole at the top left corner of the folded section will mirror to the bottom left, resulting in holes in both the top left and bottom left of this section. 2. **Unfolding the Second Fold (Left to Right)**: This step mirrors the previous section (containing two holes) to the right, creating holes in all four corners of this half-sized paper. 3. **Unfolding the First Fold (Bottom to Top)**: Finally, unfolding the paper vertically, the pattern from the half-sized paper (holes in all four corners) is mirrored to the bottom half of the original paper. Upon complete unfolding, there will be holes in all four corners of the paper. Each corner of the original sized paper now contains one hole due to the sequential folding and punching, followed by the mirroring effects of unfolding. Conclusion: The paper, when unfolded, will have four holes, one in each corner. The final answer is text{Four holes, one in each corner}. The final answer is boxed{C}

answer:1. **Understanding the relationship**: Given that 5y varies directly as the square of x, we can express this relationship mathematically as: [ 5y = kx^2 ] where k is a constant. 2. **Finding the constant k**: We know that when y = 8 and x = 2, the equation holds. Plugging these values into the equation gives: [ 5 times 8 = k times 2^2 implies 40 = 4k implies k = 10 ] Thus, the equation becomes: [ 5y = 10x^2 ] 3. **Solving for y when x = 4**: Substitute x = 4 into the equation: [ 5y = 10 times 4^2 = 10 times 16 = 160 ] To find y, solve the equation: [ 5y = 160 implies y = frac{160}{5} = 32 ] 4. **Conclusion**: When x = 4, the value of y is 32. Therefore, the correct answer is: [ 32 ] The final answer is boxed{textbf{(B)} 32}