20102014AMC12数学竞赛试题及详解.pdf

2010 AMC 12-A Problem 1 What is ? Solution . Problem 2 A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day? Solution It is easy to see that the ferry boat takes trips total. The total number of people taken to the island is Problem 3 Rectangle , pictured below, shares of its area with square . Square shares of its area with rectangle . What is ? Solution If we shift to coincide with , and add new horizontal lines to divide into five equal parts: This helps us to see that and , where . Hence . Problem 4 If , then which of the following must be positive? Solution is negative, so we can just place a negative value into each expression and find the one that is positive. Suppose we use . Obviously only is positive. Problem 5 Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelseas next shots are bullseyes she will be guara