answer:Let us denote the sides of the pentagon as a, b, c, d, e, along with r and s representing the legs of the triangular region that is removed, respectively. For simplification, consider designating prevailing sides as follows using a diagram (not provided here): 1. Use the Pythagorean theorem: r^2 + s^2 = e^2. 2. Set up possible leg length conditions: r = b - d, s = c - a. 3. Look for possible Pythagorean triples from the side lengths 17, 23, 26, 28, 34. Determine feasible configurations of r, s, e where 'e' can be hypothesized from the Pythagorean relation. Possible legs from lengths provided: - Pairwise differences from provided sides (17, 23, 26, 28, 34): 6, 3, 9, 7, 11, 8, 4, 2, 6. - Potential Pythagorean triples including e as one of the lengths are examined: - None of {17, 23, 26, 28, 34} specifically forms a standard small-sized Pythagorean triple, let’s try assuming e=26 and attempting to match it with r^2 + s^2 = 26^2, equaling 676. - Check if feasible values exist for r, s = (15, 15), (10, 24), etc., are derivable from differences. We find r = b - d = 10, s = c - a = 24 works: - Let's assign b = 34, d = 24, c = 28, a = 4, e = 26 (check careful arrangement). Then, the area of the rectangle is bc: 28 cdot 34 = 952. The area of the triangle is frac{1}{2}rs = frac12 cdot 24 cdot 10 = 120. Finally, the area of the pentagon is 952 - 120 = boxed{832}.

answer:Since the inequality x_1f(x_1)+x_2f(x_2) > x_1f(x_2)+x_2f(x_1) holds for any given distinct real numbers x_1, x_2, it is equivalent to (x_1-x_2)[f(x_1)-f(x_2)] > 0, which means function f(x) is an increasing function defined on mathbb{R}. 1. For y=-x^{3}+x+1, y'=-3x^{2}+1. The function is not monotonic on its domain. 2. For y=3x-2(sin x-cos x), y'=3-2(cos x+ sin x)=3-2 sqrt {2}sin (x+ frac {π}{4}) > 0. The function is strictly increasing, satisfying the condition. 3. For y=e^{x}+1, the function is increasing, satisfying the condition. 4. For f(x)= begin{cases} ln |x|,xneq 0 0,x=0 end{cases}, the function is strictly increasing when x > 0 and strictly decreasing when x < 0, not satisfying the condition. Therefore, functions (2) and (3) are "H functions". The answer is: boxed{text{(2)(3)}}. The inequality x_1f(x_1)+x_2f(x_2) > x_1f(x_2)+x_2f(x_1) is equivalent to (x_1-x_2)[f(x_1)-f(x_2)] > 0, meaning that the functions satisfying the condition are strictly increasing functions. Examining the monotonicity of the functions leads to the conclusion. This problem mainly tests the application of function monotonicity. Transforming the conditions into the form of function monotonicity is the key to solving this problem.

answer:Check each point by substituting x-value into y = frac{x-1}{x+2} and comparing the output to the provided y-coordinate: 1. **Point (a) (1,0):** [ y = frac{1-1}{1+2} = frac{0}{3} = 0 ] Matches (1,0). 2. **Point (b) (-1, -1):** [ y = frac{-1-1}{-1+2} = frac{-2}{1} = -2 ] **Does not match (-1, -1)**, as the calculated y-coordinate is -2, not -1. 3. **Point (c) (3, frac{1}{2}):** [ y = frac{3-1}{3+2} = frac{2}{5} ] **Does not match (3, frac{1}{2})**, as frac{2}{5} simplifies to frac{2}{5}, not frac{1}{2}. 4. **Point (d) (0, -frac{1}{2}):** [ y = frac{0-1}{0+2} = frac{-1}{2} = -frac{1}{2} ] Matches (0, -frac{1}{2}). 5. **Point (e) (-2, 3):** [ y = frac{-2-1}{-2+2} = frac{-3}{0} ] **Undefined** as denominator becomes zero. Conclusion: Point (b) and point (c) are incorrect, while point (e) is undefined. But, in terms of belonging to the graph, points (b) and (c) are directly calculably incorrect. The correct answer is the point (-1, -1) or textbf{(b)}. The final answer is boxed{B) (-1, -1)}

answer:Let's denote the total profit as P, which is given as 14999.999999999995 (which we can round to 15000 for simplicity). Let's denote the portion of the profit that is divided equally for their efforts as E. The remaining profit, which is P - E, is divided in the ratio of their investments. Mary invested 550, and Mike invested 450. The total investment is 550 + 450 = 1000. The ratio of Mary's investment to the total investment is 550 / 1000 = 11/20, and the ratio of Mike's investment to the total investment is 450 / 1000 = 9/20. So, the remaining profit is divided such that Mary gets (11/20) * (P - E) and Mike gets (9/20) * (P - E). Mary received 1000 more than Mike did, so we can set up the following equation: Mary's share = Mike's share + 1000 (E/2) + (11/20) * (P - E) = (E/2) + (9/20) * (P - E) + 1000 Now, let's solve for E: (E/2) + (11/20) * (P - E) = (E/2) + (9/20) * (P - E) + 1000 (E/2) + (11/20) * P - (11/20) * E = (E/2) + (9/20) * P - (9/20) * E + 1000 (E/2) - (11/20) * E = (E/2) - (9/20) * E - (11/20) * P + (9/20) * P + 1000 (E/2) - (11/20) * E - (E/2) + (9/20) * E = - (11/20) * P + (9/20) * P + 1000 (-2/20) * E = - (2/20) * P + 1000 (-1/10) * E = - (1/10) * P + 1000 Now, multiply both sides by -10 to get rid of the negative and the fraction: E = P - 10000 We know that P is 15000, so: E = 15000 - 10000 E = 5000 Therefore, the portion of the profit that was divided equally for their efforts is boxed{5000} .