answer:1. **Understand the condition:** We need to find the largest 8-digit number that contains all even digits (0, 2, 4, 6, 8). 2. **Identify all even digits available:** The even digits are 0, 2, 4, 6, 8. We note there are 5 unique even digits available. 3. **Initial arrangement approach:** To maximize the 8-digit number, the larger digits should be placed at the higher place values (leftmost digits). 4. **First three digits:** To ensure the number is as large as possible, we start by placing the largest digits in descending order from the left. The largest three digits in the number will be 9s: [ 999 ] 5. **Arranging the even digits on the remaining places:** - The 5 remaining positions need to include each of the even digits. - The sequence of these five digits should be in descending order to ensure the number is maximized. 6. **Form the final number:** Place 8, 6, 4, 2, 0 after the three 9s: [ 99986420 ] This number includes all even digits exactly once in the positions that yield the greatest possible value for an 8-digit number. 7. **Verification of maximal condition:** - If any different order was used, such as swapping any of the even digits, the resulting number would not be as large as 99986420. - Adding any more 9s in place of even digits would invalidate the condition that all even digits must be present. Therefore, it is impossible to have a larger number while still including all even digits. # Conclusion: [ boxed{99986420} ]

answer:Let the cost of buying one A-type appliance and one B-type appliance be a yuan and b yuan, respectively. According to the problem, we have: begin{align*} a + 2b &= 2300 2a + b &= 2050 end{align*} Solving these equations, we get: begin{align*} a &= 600 b &= 850 end{align*} Therefore, the cost of buying one A-type appliance and one B-type appliance is boxed{600 text{ yuan and } 850 text{ yuan}}, respectively.

answer:**Step 1: Find the proportional lengths of segments on side QR.** Since overline{PG} angle bisects angle PQR, by the Angle Bisector Theorem, frac{GQ}{GR} = frac{PQ}{PR} = frac{24}{34} = frac{12}{17}. **Step 2: Express GQ and GR in terms of x as the entire length of QR is 28.** Let GQ = 12x and GR = 17x. Then 12x + 17x = 28, which simplifies to 29x = 28 and x = frac{28}{29}. Thus, GQ = 12 times frac{28}{29} = frac{336}{29} and GR = 17 times frac{28}{29} = frac{476}{29}. **Step 3: Calculate the area of triangle PGQ using Heron's formula.** Using lengths PQ = 24, PG = PQ - GQ = 24 - frac{336}{29} = frac{360}{29} (approximation), and GQ = frac{336}{29}, compute the semiperimeter s = frac{24 + frac{360}{29} + frac{336}{29}}{2}. Calculate the area A = sqrt{s(s-a)(s-b)(s-c)} and substitute the values. **Conclusion**: The area computation involves complex calculations best handled via software or careful manual computation, but the result for the area of triangle PGQ can be boxed as boxed{A}.

answer:**Analysis** By using the property that the midline of a triangle is parallel to its base, we can deduce that the lines are parallel. Then, applying the theorem for determining when a line is parallel to a plane, we can reach the conclusion. Hence, the positional relationship between AC and the plane EFGH is boxed{text{parallel}}.