(AMC美国高中数学竞赛难题精选.docVIP

F组题 1.Jar A contains four liters of a solution that is 45% acid. Jar B contains five liters of a solution that is 48% acid. Jar C contains one liter of a solution that is acid. From jar C, liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the end both jar A and jar B contain solutions that are 50% acid. Given that and are relatively prime positive integers, find . Answer: Solution:Omited. Resource: 2011 AIME I Problems1 2. Let be the line with slope that contains the point , and let be the line perpendicular to line that contains the point . The original coordinate axes are erased, and line is made the -axis and line the -axis. In the new coordinate system, point is on the positive -axis, and point is on the positive -axis. The point with coordinates in the original system has coordinates in the new coordinate system. Find . Answer: Solution:Omited. Resource: 2011 AIME I Problems3 3. Suppose that a parabola has vertex and equation , where and is an integer. The minimum possible value of can be written in the form , where and are relatively prime positive integers. Find . Answer: Solution:Omited. Resource: 2011 AIME I Problems6 4. Suppose is in the interval and . Find . Answer: Solution:Omited. Resource: 2011 AIME I Problems9 5.For some integer , the polynomial has the three integer roots , , and . Find . Answer: Solution:Omited. Resource: 2011 AIME I Problems15 6. The sum of the first 2011 terms of a geometric sequence is 200. The sum of the first 4022 terms is 380. Find the sum of the first 6033 terms. Answer: the first terms is . Solution:Omited. Resource: 2011 AIME II Problems/Problem 5 7.Gary purchased a large beverage, but only drank of it, where and are relatively prime positive integers. If he had purchased half as much and drunk twice as much, he would have wasted only as much beverage. Find . Answer: Solution:Omited. Resource: 2011 AIME II Problems/Problem 1 8. On square , point lie