answer：Okay, so I have this problem where a sequence starts with 219, and from the 4th digit onwards, each digit is the units digit of the sum of the previous three digits. I need to figure out how many of the four-digit numbers 1113, 2226, 2125, and 2215 do not appear in this sequence. The options are A. 1, B. 2, C. 3, D. 4.First, let me understand the sequence. It starts with 2, 1, 9. Then, each subsequent digit is the units digit of the sum of the previous three. So, let me try to generate the sequence step by step.Starting with 2, 1, 9.To find the 4th digit: 2 + 1 + 9 = 12. The units digit is 2. So, the sequence is now 2, 1, 9, 2.Next, the 5th digit: 1 + 9 + 2 = 12. Units digit is 2. So, 2, 1, 9, 2, 2.6th digit: 9 + 2 + 2 = 13. Units digit is 3. So, 2, 1, 9, 2, 2, 3.7th digit: 2 + 2 + 3 = 7. Units digit is 7. So, 2, 1, 9, 2, 2, 3, 7.8th digit: 2 + 3 + 7 = 12. Units digit is 2. So, 2, 1, 9, 2, 2, 3, 7, 2.9th digit: 3 + 7 + 2 = 12. Units digit is 2. So, 2, 1, 9, 2, 2, 3, 7, 2, 2.10th digit: 7 + 2 + 2 = 11. Units digit is 1. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1.11th digit: 2 + 2 + 1 = 5. Units digit is 5. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5.12th digit: 2 + 1 + 5 = 8. Units digit is 8. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8.13th digit: 1 + 5 + 8 = 14. Units digit is 4. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4.14th digit: 5 + 8 + 4 = 17. Units digit is 7. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7.15th digit: 8 + 4 + 7 = 19. Units digit is 9. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9.16th digit: 4 + 7 + 9 = 20. Units digit is 0. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0.17th digit: 7 + 9 + 0 = 16. Units digit is 6. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0, 6.18th digit: 9 + 0 + 6 = 15. Units digit is 5. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0, 6, 5.19th digit: 0 + 6 + 5 = 11. Units digit is 1. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0, 6, 5, 1.20th digit: 6 + 5 + 1 = 12. Units digit is 2. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0, 6, 5, 1, 2.21st digit: 5 + 1 + 2 = 8. Units digit is 8. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0, 6, 5, 1, 2, 8.22nd digit: 1 + 2 + 8 = 11. Units digit is 1. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0, 6, 5, 1, 2, 8, 1.23rd digit: 2 + 8 + 1 = 11. Units digit is 1. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0, 6, 5, 1, 2, 8, 1, 1.24th digit: 8 + 1 + 1 = 10. Units digit is 0. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0, 6, 5, 1, 2, 8, 1, 1, 0.25th digit: 1 + 1 + 0 = 2. Units digit is 2. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0, 6, 5, 1, 2, 8, 1, 1, 0, 2.26th digit: 1 + 0 + 2 = 3. Units digit is 3. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0, 6, 5, 1, 2, 8, 1, 1, 0, 2, 3.27th digit: 0 + 2 + 3 = 5. Units digit is 5. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0, 6, 5, 1, 2, 8, 1, 1, 0, 2, 3, 5.28th digit: 2 + 3 + 5 = 10. Units digit is 0. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0, 6, 5, 1, 2, 8, 1, 1, 0, 2, 3, 5, 0.29th digit: 3 + 5 + 0 = 8. Units digit is 8. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0, 6, 5, 1, 2, 8, 1, 1, 0, 2, 3, 5, 0, 8.30th digit: 5 + 0 + 8 = 13. Units digit is 3. So, 2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0, 6, 5, 1, 2, 8, 1, 1, 0, 2, 3, 5, 0, 8, 3.Hmm, this is getting quite long. Maybe I should look for a repeating pattern or cycle in the sequence. Let me see if the sequence starts repeating after some point.Looking back at the sequence:2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0, 6, 5, 1, 2, 8, 1, 1, 0, 2, 3, 5, 0, 8, 3...I don't immediately see a clear cycle, but maybe if I continue a bit more.31st digit: 0 + 8 + 3 = 11. Units digit is 1. So, ..., 3, 1.32nd digit: 8 + 3 + 1 = 12. Units digit is 2. So, ..., 1, 2.33rd digit: 3 + 1 + 2 = 6. Units digit is 6. So, ..., 2, 6.34th digit: 1 + 2 + 6 = 9. Units digit is 9. So, ..., 6, 9.35th digit: 2 + 6 + 9 = 17. Units digit is 7. So, ..., 9, 7.36th digit: 6 + 9 + 7 = 22. Units digit is 2. So, ..., 7, 2.37th digit: 9 + 7 + 2 = 18. Units digit is 8. So, ..., 2, 8.38th digit: 7 + 2 + 8 = 17. Units digit is 7. So, ..., 8, 7.39th digit: 2 + 8 + 7 = 17. Units digit is 7. So, ..., 7, 7.40th digit: 8 + 7 + 7 = 22. Units digit is 2. So, ..., 7, 2.41st digit: 7 + 7 + 2 = 16. Units digit is 6. So, ..., 2, 6.42nd digit: 7 + 2 + 6 = 15. Units digit is 5. So, ..., 6, 5.43rd digit: 2 + 6 + 5 = 13. Units digit is 3. So, ..., 5, 3.44th digit: 6 + 5 + 3 = 14. Units digit is 4. So, ..., 3, 4.45th digit: 5 + 3 + 4 = 12. Units digit is 2. So, ..., 4, 2.46th digit: 3 + 4 + 2 = 9. Units digit is 9. So, ..., 2, 9.47th digit: 4 + 2 + 9 = 15. Units digit is 5. So, ..., 9, 5.48th digit: 2 + 9 + 5 = 16. Units digit is 6. So, ..., 5, 6.49th digit: 9 + 5 + 6 = 20. Units digit is 0. So, ..., 6, 0.50th digit: 5 + 6 + 0 = 11. Units digit is 1. So, ..., 0, 1.51st digit: 6 + 0 + 1 = 7. Units digit is 7. So, ..., 1, 7.52nd digit: 0 + 1 + 7 = 8. Units digit is 8. So, ..., 7, 8.53rd digit: 1 + 7 + 8 = 16. Units digit is 6. So, ..., 8, 6.54th digit: 7 + 8 + 6 = 21. Units digit is 1. So, ..., 6, 1.55th digit: 8 + 6 + 1 = 15. Units digit is 5. So, ..., 1, 5.56th digit: 6 + 1 + 5 = 12. Units digit is 2. So, ..., 5, 2.57th digit: 1 + 5 + 2 = 8. Units digit is 8. So, ..., 2, 8.58th digit: 5 + 2 + 8 = 15. Units digit is 5. So, ..., 8, 5.59th digit: 2 + 8 + 5 = 15. Units digit is 5. So, ..., 5, 5.60th digit: 8 + 5 + 5 = 18. Units digit is 8. So, ..., 5, 8.61st digit: 5 + 5 + 8 = 18. Units digit is 8. So, ..., 8, 8.62nd digit: 5 + 8 + 8 = 21. Units digit is 1. So, ..., 8, 1.63rd digit: 8 + 8 + 1 = 17. Units digit is 7. So, ..., 1, 7.64th digit: 8 + 1 + 7 = 16. Units digit is 6. So, ..., 7, 6.65th digit: 1 + 7 + 6 = 14. Units digit is 4. So, ..., 6, 4.66th digit: 7 + 6 + 4 = 17. Units digit is 7. So, ..., 4, 7.67th digit: 6 + 4 + 7 = 17. Units digit is 7. So, ..., 7, 7.68th digit: 4 + 7 + 7 = 18. Units digit is 8. So, ..., 7, 8.69th digit: 7 + 7 + 8 = 22. Units digit is 2. So, ..., 8, 2.70th digit: 7 + 8 + 2 = 17. Units digit is 7. So, ..., 2, 7.71st digit: 8 + 2 + 7 = 17. Units digit is 7. So, ..., 7, 7.72nd digit: 2 + 7 + 7 = 16. Units digit is 6. So, ..., 7, 6.73rd digit: 7 + 7 + 6 = 20. Units digit is 0. So, ..., 6, 0.74th digit: 7 + 6 + 0 = 13. Units digit is 3. So, ..., 0, 3.75th digit: 6 + 0 + 3 = 9. Units digit is 9. So, ..., 3, 9.76th digit: 0 + 3 + 9 = 12. Units digit is 2. So, ..., 9, 2.77th digit: 3 + 9 + 2 = 14. Units digit is 4. So, ..., 2, 4.78th digit: 9 + 2 + 4 = 15. Units digit is 5. So, ..., 4, 5.79th digit: 2 + 4 + 5 = 11. Units digit is 1. So, ..., 5, 1.80th digit: 4 + 5 + 1 = 10. Units digit is 0. So, ..., 1, 0.81st digit: 5 + 1 + 0 = 6. Units digit is 6. So, ..., 0, 6.82nd digit: 1 + 0 + 6 = 7. Units digit is 7. So, ..., 6, 7.83rd digit: 0 + 6 + 7 = 13. Units digit is 3. So, ..., 7, 3.84th digit: 6 + 7 + 3 = 16. Units digit is 6. So, ..., 3, 6.85th digit: 7 + 3 + 6 = 16. Units digit is 6. So, ..., 6, 6.86th digit: 3 + 6 + 6 = 15. Units digit is 5. So, ..., 6, 5.87th digit: 6 + 6 + 5 = 17. Units digit is 7. So, ..., 5, 7.88th digit: 6 + 5 + 7 = 18. Units digit is 8. So, ..., 7, 8.89th digit: 5 + 7 + 8 = 20. Units digit is 0. So, ..., 8, 0.90th digit: 7 + 8 + 0 = 15. Units digit is 5. So, ..., 0, 5.91st digit: 8 + 0 + 5 = 13. Units digit is 3. So, ..., 5, 3.92nd digit: 0 + 5 + 3 = 8. Units digit is 8. So, ..., 3, 8.93rd digit: 5 + 3 + 8 = 16. Units digit is 6. So, ..., 8, 6.94th digit: 3 + 8 + 6 = 17. Units digit is 7. So, ..., 6, 7.95th digit: 8 + 6 + 7 = 21. Units digit is 1. So, ..., 7, 1.96th digit: 6 + 7 + 1 = 14. Units digit is 4. So, ..., 1, 4.97th digit: 7 + 1 + 4 = 12. Units digit is 2. So, ..., 4, 2.98th digit: 1 + 4 + 2 = 7. Units digit is 7. So, ..., 2, 7.99th digit: 4 + 2 + 7 = 13. Units digit is 3. So, ..., 7, 3.100th digit: 2 + 7 + 3 = 12. Units digit is 2. So, ..., 3, 2.Okay, this is getting really long, and I don't see an obvious cycle yet. Maybe I should check if any of the four-digit numbers appear in the sequence so far.The four-digit numbers to check are 1113, 2226, 2125, 2215.Let me look through the sequence I've generated so far:2, 1, 9, 2, 2, 3, 7, 2, 2, 1, 5, 8, 4, 7, 9, 0, 6, 5, 1, 2, 8, 1, 1, 0, 2, 3, 5, 0, 8, 3, 1, 2, 6, 9, 7, 2, 8, 7, 7, 2, 6, 5, 3, 4, 2, 9, 5, 6, 0, 1, 7, 8, 6, 1, 5, 2, 8, 5, 5, 8, 8, 1, 7, 6, 4, 7, 7, 8, 2, 7, 7, 6, 0, 3, 9, 2, 4, 5, 1, 0, 6, 7, 3, 6, 6, 5, 7, 8, 0, 5, 3, 8, 6, 7, 1, 4, 2, 7, 3, 2.Looking for 1113: I don't see four consecutive 1s anywhere. The closest is at positions 21-24: 8,1,1,0. So, 1113 is not present.Looking for 2226: I see multiple 2s, but not three consecutive 2s followed by a 6. The closest is at the beginning: 2,1,9,2,2,3. So, 2226 is not present.Looking for 2125: Let's see if 2,1,2,5 appears. Looking through the sequence, at positions 1-4: 2,1,9,2. Not 2,1,2,5. Then later, positions 9-12: 2,2,1,5. That's 2,2,1,5, which is close but not 2,1,2,5. Hmm, maybe later? Let me check further. At positions 19-22: 1,5,8,4. Not. Positions 27-30: 5,0,8,3. Not. Maybe not present.Looking for 2215: Let's see if 2,2,1,5 appears. Yes, at positions 9-12: 2,2,1,5. So, 2215 is present.Wait, but earlier I thought 2125 might not be present. Let me double-check. Is there a 2,1,2,5 in the sequence? Let me scan through:Positions 1-4: 2,1,9,2.Positions 5-8: 2,3,7,2.Positions 9-12: 2,2,1,5.Positions 13-16: 4,7,9,0.Positions 17-20: 6,5,1,2.Positions 21-24: 8,1,1,0.Positions 25-28: 2,3,5,0.Positions 29-32: 8,3,1,2.Positions 33-36: 6,9,7,2.Positions 37-40: 8,7,7,2.Positions 41-44: 6,5,3,4.Positions 45-48: 2,9,5,6.Positions 49-52: 0,1,7,8.Positions 53-56: 6,1,5,2.Positions 57-60: 8,5,5,8.Positions 61-64: 8,1,7,6.Positions 65-68: 4,7,7,8.Positions 69-72: 2,7,7,6.Positions 73-76: 0,3,9,2.Positions 77-80: 4,5,1,0.Positions 81-84: 6,7,3,6.Positions 85-88: 6,5,7,8.Positions 89-92: 0,5,3,8.Positions 93-96: 6,7,1,4.Positions 97-100: 2,7,3,2.I don't see 2,1,2,5 anywhere. So, 2125 is not present.Wait, but earlier I thought 2215 is present. Let me confirm: 2,2,1,5 is at positions 9-12. So, yes, 2215 is present.So, among the four numbers:1113: Not present2226: Not present2125: Not present2215: PresentSo, three numbers do not appear in the sequence. But wait, the options are A.1, B.2, C.3, D.4.But the question is asking how many do not appear. So, 1113, 2226, 2125 do not appear, and 2215 does. So, three numbers do not appear. So, the answer should be C.3.But wait, in my earlier thought process, I thought 2125 might be present, but upon checking, it's not. So, three numbers do not appear.But let me double-check if 2125 is really not present. Let me look again:Looking for 2,1,2,5.Looking through the sequence:Positions 1-4: 2,1,9,2Positions 5-8: 2,3,7,2Positions 9-12: 2,2,1,5Positions 13-16: 4,7,9,0Positions 17-20: 6,5,1,2Positions 21-24: 8,1,1,0Positions 25-28: 2,3,5,0Positions 29-32: 8,3,1,2Positions 33-36: 6,9,7,2Positions 37-40: 8,7,7,2Positions 41-44: 6,5,3,4Positions 45-48: 2,9,5,6Positions 49-52: 0,1,7,8Positions 53-56: 6,1,5,2Positions 57-60: 8,5,5,8Positions 61-64: 8,1,7,6Positions 65-68: 4,7,7,8Positions 69-72: 2,7,7,6Positions 73-76: 0,3,9,2Positions 77-80: 4,5,1,0Positions 81-84: 6,7,3,6Positions 85-88: 6,5,7,8Positions 89-92: 0,5,3,8Positions 93-96: 6,7,1,4Positions 97-100: 2,7,3,2I don't see 2,1,2,5 anywhere. So, 2125 is not present.Therefore, three numbers do not appear: 1113, 2226, 2125. Only 2215 appears.So, the answer should be C.3.But wait, the initial problem statement says "how many do not appear in this sequence." So, if three do not appear, the answer is C.3.But in the initial assistant answer, it concluded that only 1113 does not appear, which seems incorrect based on my analysis.Alternatively, maybe I missed something. Let me check again.Looking for 2125: 2,1,2,5.Is there a place where 2,1,2,5 occurs? Let me check positions 1-4: 2,1,9,2. Not.Positions 5-8: 2,3,7,2. Not.Positions 9-12: 2,2,1,5. Close, but it's 2,2,1,5, not 2,1,2,5.Positions 17-20: 6,5,1,2. Not.Positions 21-24: 8,1,1,0. Not.Positions 25-28: 2,3,5,0. Not.Positions 29-32: 8,3,1,2. Not.Positions 33-36: 6,9,7,2. Not.Positions 37-40: 8,7,7,2. Not.Positions 41-44: 6,5,3,4. Not.Positions 45-48: 2,9,5,6. Not.Positions 49-52: 0,1,7,8. Not.Positions 53-56: 6,1,5,2. Not.Positions 57-60: 8,5,5,8. Not.Positions 61-64: 8,1,7,6. Not.Positions 65-68: 4,7,7,8. Not.Positions 69-72: 2,7,7,6. Not.Positions 73-76: 0,3,9,2. Not.Positions 77-80: 4,5,1,0. Not.Positions 81-84: 6,7,3,6. Not.Positions 85-88: 6,5,7,8. Not.Positions 89-92: 0,5,3,8. Not.Positions 93-96: 6,7,1,4. Not.Positions 97-100: 2,7,3,2. Not.So, 2125 is definitely not present.Therefore, three numbers do not appear: 1113, 2226, 2125. Only 2215 appears.So, the answer should be C.3.But the initial assistant answer concluded only 1113 does not appear, which is incorrect. So, I think the correct answer is C.3.

answer：First, I need to determine the percentage of cultivated land that is taxed in Mr. Willam's village. The total farm tax collected from the village is 3840, and Mr. Willam paid 480 of this amount. Additionally, Mr. Willam's land constitutes 25% of the total taxable land in the village.To find the percentage of cultivated land that is taxed, I'll start by calculating the total taxable land in the village. Since Mr. Willam's 480 represents 25% of the total tax collected, I can set up the equation:480 = 0.25 * Total Tax CollectedSolving for the Total Tax Collected gives me:Total Tax Collected = 480 / 0.25 = 1920Now, knowing that the total tax collected is 1920 and the total taxable land is 3840, I can determine the percentage of cultivated land that is taxed by dividing the total tax collected by the total taxable land and multiplying by 100:Percentage Taxed = (1920 / 3840) * 100 = 50%Therefore, 50% of the cultivated land is taxed.

answer：Alright, so I've got this problem about a parallelogram ABCD with diagonals intersecting at point O. In triangles OAB, OBC, and OCD, they've drawn medians OM, OM', OM'' and angle bisectors OL, OL', OL''. I need to prove that angles MM'M'' and LL'L'' are equal. Hmm, okay, let's break this down.First, I know that in a parallelogram, the diagonals bisect each other. So, point O is the midpoint of both diagonals AC and BD. That means AO = OC and BO = OD. That might come in handy later.Now, let's think about the triangles mentioned: OAB, OBC, and OCD. Each of these triangles has a median and an angle bisector drawn from point O. For triangle OAB, the median is OM, and the angle bisector is OL. Similarly, for OBC, it's OM' and OL', and for OCD, it's OM'' and OL''.I need to find angles MM'M'' and LL'L''. So, these are angles formed by the medians and angle bisectors respectively. I guess I need to analyze the properties of these medians and angle bisectors in each triangle and see how they relate to each other.Let me start by recalling what a median does in a triangle. A median connects a vertex to the midpoint of the opposite side. So, in triangle OAB, OM is the median from O to AB, meaning M is the midpoint of AB. Similarly, M' is the midpoint of BC, and M'' is the midpoint of CD.On the other hand, an angle bisector divides the angle at a vertex into two equal angles. So, OL is the angle bisector of angle OAB, OL' is the angle bisector of angle OBC, and OL'' is the angle bisector of angle OCD.I think I should try to find some relationship between these medians and angle bisectors. Maybe they are parallel or something? Or perhaps they form similar triangles?Wait, since ABCD is a parallelogram, opposite sides are equal and parallel. So, AB is equal and parallel to CD, and AD is equal and parallel to BC. Also, the diagonals bisect each other, so AO = OC and BO = OD.Maybe I can use coordinate geometry to model this. Let me assign coordinates to the points. Let's place point O at the origin (0,0). Since it's a parallelogram, if I let point A be at (a, b), then point C would be at (-a, -b). Similarly, if point B is at (c, d), then point D would be at (-c, -d).So, points:- O: (0,0)- A: (a, b)- B: (c, d)- C: (-a, -b)- D: (-c, -d)Now, let's find the midpoints M, M', M''. For triangle OAB, the median OM goes to the midpoint of AB. The midpoint M of AB is ((a + c)/2, (b + d)/2).Similarly, for triangle OBC, the median OM' goes to the midpoint of BC. The midpoint M' of BC is ((c - a)/2, (d - b)/2).For triangle OCD, the median OM'' goes to the midpoint of CD. The midpoint M'' of CD is ((-a - c)/2, (-b - d)/2).Okay, so now I have coordinates for M, M', M''. Let me write them down:- M: ((a + c)/2, (b + d)/2)- M': ((c - a)/2, (d - b)/2)- M'': ((-a - c)/2, (-b - d)/2)Now, let's think about the angle bisectors OL, OL', OL''. These are the bisectors of angles at O in triangles OAB, OBC, and OCD respectively.I remember that the angle bisector theorem states that the angle bisector divides the opposite side in the ratio of the adjacent sides. So, in triangle OAB, OL bisects angle OAB, so it divides side OB into segments proportional to OA and AB.Wait, actually, in triangle OAB, the angle bisector from O would divide side AB into segments proportional to OA and OB. Wait, no, the angle bisector theorem says that the bisector of an angle in a triangle divides the opposite side into segments proportional to the adjacent sides.So, in triangle OAB, angle bisector OL from O would divide side AB into segments AL and LB such that AL / LB = OA / OB.Similarly, for triangle OBC, angle bisector OL' from O would divide side BC into segments BL' and L'C such that BL' / L'C = BO / OC.And for triangle OCD, angle bisector OL'' from O would divide side CD into segments CL'' and L''D such that CL'' / L''D = CO / OD.But since ABCD is a parallelogram, OA = OC and OB = OD. So, OA = OC = d (let's say), and OB = OD = c (let's say). Wait, actually, OA and OC are equal because diagonals bisect each other, so OA = OC, and similarly OB = OD.Therefore, in triangle OAB, AL / LB = OA / OB = d / c.Similarly, in triangle OBC, BL' / L'C = BO / OC = c / d.And in triangle OCD, CL'' / L''D = CO / OD = d / c.Hmm, interesting. So, the ratios are either d/c or c/d depending on the triangle.Now, let's try to find the coordinates of points L, L', L''.Starting with point L in triangle OAB. Since OL is the angle bisector, it divides AB into AL and LB such that AL / LB = OA / OB = d / c.Given that AB is from point A (a, b) to point B (c, d), the coordinates of L can be found using the section formula. If AL / LB = d / c, then L divides AB in the ratio d:c.So, coordinates of L are:L_x = (d * c + c * a) / (d + c) = (dc + ac) / (c + d)L_y = (d * d + c * b) / (c + d) = (d² + bc) / (c + d)Wait, no, the section formula is (mx2 + nx1)/(m + n), where m:n is the ratio.So, since AL / LB = d / c, the ratio is d:c, so m = d, n = c.Therefore, L_x = (d * c + c * a) / (d + c) = c(d + a) / (c + d)Similarly, L_y = (d * d + c * b) / (c + d) = (d² + bc) / (c + d)Wait, that seems a bit messy. Maybe I can write it as:L = ( (c(d + a))/(c + d), (d² + bc)/(c + d) )Similarly, for point L' in triangle OBC. The angle bisector OL' divides BC into BL' / L'C = BO / OC = c / d.Since BO = c and OC = d, the ratio is c/d.Point B is at (c, d), point C is at (-a, -b). So, L' divides BC in the ratio c:d.Using the section formula again, coordinates of L' are:L'_x = (c*(-a) + d*c) / (c + d) = (-ac + dc) / (c + d) = c(d - a) / (c + d)L'_y = (c*(-b) + d*d) / (c + d) = (-bc + d²) / (c + d)So, L' = ( c(d - a)/(c + d), (d² - bc)/(c + d) )Similarly, for point L'' in triangle OCD. The angle bisector OL'' divides CD into CL'' / L''D = CO / OD = d / c.Point C is at (-a, -b), point D is at (-c, -d). So, L'' divides CD in the ratio d:c.Using the section formula:L''_x = (d*(-c) + c*(-a)) / (d + c) = (-dc - ac) / (c + d) = -c(d + a)/(c + d)L''_y = (d*(-d) + c*(-b)) / (c + d) = (-d² - bc)/(c + d) = -(d² + bc)/(c + d)So, L'' = ( -c(d + a)/(c + d), -(d² + bc)/(c + d) )Okay, so now I have coordinates for L, L', L''. Similarly, I have coordinates for M, M', M''.Now, I need to find angles MM'M'' and LL'L''.Wait, angles MM'M'' and LL'L'' are formed by the points M, M', M'' and L, L', L'' respectively.So, angle MM'M'' is the angle at point M' between points M and M''. Similarly, angle LL'L'' is the angle at point L' between points L and L''.To find these angles, I can use vector analysis or coordinate geometry to find the slopes of the lines and then compute the angles between them.Alternatively, maybe I can show that the triangles MM'M'' and LL'L'' are similar, which would imply that their corresponding angles are equal.Let me try to compute the vectors involved.First, let's compute vectors for angle MM'M''.At point M', the vectors are M'M and M'M''.Coordinates:M: ((a + c)/2, (b + d)/2)M': ((c - a)/2, (d - b)/2)M'': ((-a - c)/2, (-b - d)/2)So, vector M'M is M - M' = [ (a + c)/2 - (c - a)/2, (b + d)/2 - (d - b)/2 ]Simplify:x-coordinate: (a + c - c + a)/2 = (2a)/2 = ay-coordinate: (b + d - d + b)/2 = (2b)/2 = bSo, vector M'M is (a, b)Similarly, vector M'M'' is M'' - M' = [ (-a - c)/2 - (c - a)/2, (-b - d)/2 - (d - b)/2 ]Simplify:x-coordinate: (-a - c - c + a)/2 = (-2c)/2 = -cy-coordinate: (-b - d - d + b)/2 = (-2d)/2 = -dSo, vector M'M'' is (-c, -d)Therefore, at point M', the vectors are (a, b) and (-c, -d). So, the angle between these two vectors is angle MM'M''.Similarly, let's compute vectors for angle LL'L''.At point L', the vectors are L'L and L'L''.Coordinates:L: ( c(d + a)/(c + d), (d² + bc)/(c + d) )L': ( c(d - a)/(c + d), (d² - bc)/(c + d) )L'': ( -c(d + a)/(c + d), -(d² + bc)/(c + d) )So, vector L'L is L - L' = [ c(d + a)/(c + d) - c(d - a)/(c + d), (d² + bc)/(c + d) - (d² - bc)/(c + d) ]Simplify:x-coordinate: [ c(d + a - d + a) ] / (c + d) = [ c(2a) ] / (c + d) = 2ac / (c + d)y-coordinate: [ (d² + bc - d² + bc) ] / (c + d) = [ 2bc ] / (c + d)So, vector L'L is (2ac/(c + d), 2bc/(c + d)) = (2c(a, b)/(c + d))Similarly, vector L'L'' is L'' - L' = [ -c(d + a)/(c + d) - c(d - a)/(c + d), -(d² + bc)/(c + d) - (d² - bc)/(c + d) ]Simplify:x-coordinate: [ -c(d + a) - c(d - a) ] / (c + d) = [ -c(d + a + d - a) ] / (c + d) = [ -c(2d) ] / (c + d) = -2cd / (c + d)y-coordinate: [ -(d² + bc) - (d² - bc) ] / (c + d) = [ -d² - bc - d² + bc ] / (c + d) = [ -2d² ] / (c + d)So, vector L'L'' is (-2cd/(c + d), -2d²/(c + d)) = (-2d(c, d)/(c + d))Wait a second, so vector L'L is (2c(a, b)/(c + d)) and vector L'L'' is (-2d(c, d)/(c + d)).Now, let's compare these vectors to the vectors at M'.At M', vectors are (a, b) and (-c, -d). So, if I factor out 2c/(c + d) from vector L'L, I get 2c/(c + d)*(a, b). Similarly, if I factor out -2d/(c + d) from vector L'L'', I get -2d/(c + d)*(c, d).So, essentially, vectors L'L and L'L'' are scalar multiples of vectors (a, b) and (-c, -d), which are the vectors M'M and M'M'' respectively.Therefore, the vectors at L' are scaled versions of the vectors at M'. This suggests that the angles formed by these vectors should be equal because scaling doesn't change the angle between vectors.In other words, since vector L'L is a scalar multiple of vector M'M, and vector L'L'' is a scalar multiple of vector M'M'', the angle between L'L and L'L'' is the same as the angle between M'M and M'M''.Hence, angle MM'M'' is equal to angle LL'L''.Wait, let me make sure I didn't make any mistakes in the calculations. Let me double-check the coordinates and the vectors.Starting with point L:L divides AB in the ratio d:c, so using the section formula, it's ( (d*c + c*a)/(d + c), (d*d + c*b)/(d + c) ). That seems correct.Similarly, point L' divides BC in the ratio c:d, so coordinates are ( (c*(-a) + d*c)/(c + d), (c*(-b) + d*d)/(c + d) ). That also looks correct.Point L'' divides CD in the ratio d:c, so coordinates are ( (d*(-c) + c*(-a))/(d + c), (d*(-d) + c*(-b))/(d + c) ). Correct.Then, vectors M'M and M'M'' were calculated as (a, b) and (-c, -d). That seems right because M is the midpoint of AB, so subtracting M' from M gives the vector from M' to M.Similarly, vectors L'L and L'L'' were found to be scalar multiples of (a, b) and (-c, -d). So, since they are scalar multiples, the angles between them should be equal.Therefore, angles MM'M'' and LL'L'' are equal.I think that makes sense. The key was realizing that the vectors involved in the angles are scalar multiples of each other, which preserves the angle between them. So, despite the different positions of the points, the angles remain equal because the direction of the vectors is the same, just scaled differently.Yeah, I think that's the proof. It was a bit involved with all the coordinates and vectors, but breaking it down step by step helped me see the relationship between the medians and angle bisectors in the parallelogram.

answer：Okay, so I have this problem about a function y = f(x) that's increasing everywhere on the real numbers. It goes through the point A, which is (0, -1), and another point B that I need to find. The key information is that the solution set for the inequality |f(x + 1)| < 1 is exactly the interval from -1 to 2. Hmm, let me try to unpack this step by step.First, since f is an increasing function, that means if x1 < x2, then f(x1) < f(x2). That should help me figure out the behavior of the function. Also, it passes through (0, -1), so when x is 0, f(0) is -1. That's a starting point.Now, the inequality |f(x + 1)| < 1. Let me rewrite that without the absolute value. That would mean -1 < f(x + 1) < 1. So, the function f evaluated at (x + 1) is between -1 and 1. And the solution set for this inequality is x between -1 and 2. So, when x is between -1 and 2, f(x + 1) is between -1 and 1.Let me think about what this means for the function f. Since f is increasing, the values of f(x + 1) will increase as x increases. So, when x is -1, f(-1 + 1) = f(0) = -1, which is the lower bound. And when x is 2, f(2 + 1) = f(3) = 1, which is the upper bound. So, f(0) = -1 and f(3) = 1.Wait, that makes sense because f is increasing, so as x increases from 0 to 3, f(x) increases from -1 to 1. So, the function f goes from -1 at x = 0 to 1 at x = 3. Therefore, the graph of f passes through (0, -1) and (3, 1). So, point B must be (3, 1).Let me double-check. If f is increasing, then f(x + 1) is also increasing because it's just a horizontal shift. So, the inequality |f(x + 1)| < 1 translates to f(x + 1) being between -1 and 1. Since f is increasing, the x-values where f(x + 1) is between -1 and 1 correspond to the x-values where x + 1 is between 0 and 3. So, x + 1 is between 0 and 3, which means x is between -1 and 2. That matches the given solution set.Therefore, f(0) = -1 and f(3) = 1. Since the function passes through (0, -1) and (3, 1), point B is (3, 1).I think that's solid. I don't see any mistakes in my reasoning. The key was understanding how the inequality relates to the function's behavior and using the fact that it's increasing to determine the corresponding x-values.**Final Answer**The coordinates of point B are boxed{(3, 1)}.

answer：First, I need to convert the train's speed from kilometers per hour to meters per second because the train's length and the time to cross the bridge are given in meters and seconds, respectively.The conversion formula is:[text{Speed in m/s} = text{Speed in km/hr} times left(frac{1000 text{ m}}{1 text{ km}}right) times left(frac{1 text{ hr}}{3600 text{ s}}right)]Plugging in the given speed:[text{Speed in m/s} = 45 times left(frac{1000}{3600}right) = 45 times left(frac{5}{18}right) = 12.5 text{ m/s}]Next, I calculate the total distance the train travels in 30 seconds using the formula:[text{Distance} = text{Speed} times text{Time}][text{Distance} = 12.5 text{ m/s} times 30 text{ s} = 375 text{ m}]This distance represents the combined length of the train and the bridge. To find the bridge's length, I subtract the train's length from the total distance:[text{Length of the bridge} = text{Distance} - text{Length of the train}][text{Length of the bridge} = 375 text{ m} - 160 text{ m} = 215 text{ m}]

answer：First, I need to determine the ages of George, Christopher, and Ford based on the given relationships and the total sum of their ages.I'll start by letting Christopher's age be represented by ( C ).According to the problem:- George is 8 years older than Christopher, so George's age is ( C + 8 ).- Ford is 2 years younger than Christopher, so Ford's age is ( C - 2 ).The sum of their ages is 60, which gives me the equation:[C + (C + 8) + (C - 2) = 60]Next, I'll simplify and solve for ( C ):[3C + 6 = 60][3C = 54][C = 18]Therefore, Christopher is 18 years old.