AMC美国高中数学竞赛难题精选.doc

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 Problems12. 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 Problems33. 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 Problems64. Suppose is in the interval and . Find .Answer: Solution:Omited.Resource: 2011 AIME I Problems95.For some integer , the polynomial has the three integer roots , , and . Find .Answer: Solution:Omited.Resource: 2011 AIME I Problems156. 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 57.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 18. On square , point lies on side and point lies on side , so that . Find the area of the square .Answer: 810Solution:Omited.Resource: 2011 AIME II Problems/Problem 29. The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.Answer: The first term is then Solution: The average angle in an 18-gon is . In an arithmetic sequence the average is the same as the median, so the middle two terms of the sequence average to . Thus for some positive (the sequence is increasing and thus non-constant) integer , the middle two terms are and . Since the step is the last term of the sequence is , which must be less than , since the polygon is convex. This gives , so the only suitable positive integer is 1.Resource: 2011 AIME II Problems/Problem 310. In triangle , . The angle bisector of intersects at point , and point is the midpoint of . Let be the point of the intersection of and . The ratio of to can be expressed in the form , where and are relatively prime positive integers. Find . Answer:.Solution:Omited.Resource: 2011 AIME II Problems/Problem 411. 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 termsAnswer:542.Solution:Omited.Resource: 2011 AIME II Problems/Problem 512. Let . A real number is chosen at random from the interval . The probability that is equal to , where , , , , and are positive integers. Find .Answer:850Solution:Omited.Resource: 2011 AIME II Problems/Problem1513. Maya lists all the positive divisors of. She then randomly selects two distinct divisors from this list. Let be the probability that exactly one of the selected divisors is aperfect square. The probability can be expressed in the form, where and are relatively prime positive integers. Find .Answer:107Solution:Omited.Resource: 2010 AIME I Problems/Problem114. Suppose thatand. The quantitycan be expressed as a rational number, whereandare relatively prime positive integers. Find.Answer:529Solution:Omited.Resource: 2010 AIME I Problems/Problem315. Positive integers, , , and satisfy , , and . Find the number of possible values of.Answer:501Solution:Omited.Resource: 2010 AIME I Problems/Problem516. Rectangle and a semicircle with diameter are coplanar and have nonoverlapping interiors. Let denote the region enclosed by the semicircle and the rectangle. Line meets the semicircle, segment , and segment at distinct points , , and, respectively. Linedivides region into two regions with areas in the ratio . Suppose that, and. Then can be represented as, where and are positive integers and is not divisible by the square of any prime. Find .Answer:069Solution:Omited.Resource: 2010 AIME I Problems/Problem1317. In with , and , let be a point on such that the incircles of and have equal radii. Let and be positive relatively prime integers such that . Find .Answer:045Solution:Omited.Resource: 2010 AIME I Problems/Problem1518. The terms of an arithmetic sequence add to. The first term of the sequence is increased by, the second term is increased by, the third term is increased by, and in general, theth term is increased by theth odd positive integer. The terms of the new sequence add to. Find the sum of the first, last, and middle terms of the original sequence.Answer:195Solution:Omited.Resource: 2012 AIME I Problems/Problem219. Let be the set of all perfect squares whose rightmost three digits in base are. Let be the set of all numbers of the form, where is in . In other words, is the set of numbers that result when the last three digits of each number in are truncated. Find the remainder when the tenth smallest element of is divided by .Answer:170Solution:Omited.Resource: 2012 AIME I Problems/Problem1020. Letbe the set of all binary integers that can be written using exactlyzeros andones where leading zeros are allowed. If all possible subtractions are performed in which one element ofis subtracted from another, find the number of times the answeris obtained.Answer:330Solution:Omited.Resource: 2012 AIME I Problems/Problem521. At each of the sixteen circles in the network below stands a student. A total ofcoins are distributed among the sixteen students. All at once, all students give away all their coins by passing an equal number of coins to each of their neighbors in the network. After the trade, all students have the same number of coins as they started with. Find the number of coins the student standing at the center circle had originally.Answer:280Solution:Omited.Resource: 2012 AIME I Problems/Problem722. Three concentric circles have radiiandAn equilateral triangle with one vertex on each circle has side lengthThe largest possible area of the triangle can be written aswhereandare positive integers,andare relatively prime, andis not divisible by the square of any prime. FindAnswer:041Solution:Omited.Resource: 2012 AIME I Problems/Problem1323. Ana, Bob, and Cao bike at constant rates ofmeters per second,meters per second, andmeters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a pointon the south edge of the field. Cao arrives at pointat the same time that Ana and Bob arrive atfor the first time. The ratio of the fields length to the fields width to the distance from pointto the southeast corner of the field can be represented as, where, andare positive integers withandrelatively prime. Find.Answer:061Solution:Omited.Resource: 2012 AIME II Problems/Problem424. In the accompanying figure, the outer squarehas side length. A second squareof side lengthis constructed insidewith the same center asand with sides parallel to those of. From each midpoint of a side of, segments are drawn to the two closest vertices of. The result is a four-pointed starlike figure inscribed in. The star figure is cut out and then folded to form a pyramid with base. Find the volume of this pyramid.Answer:750Solution:Omited.Resource: 2012 AIME II Problems/Problem425. Let be the complex number with and such that the distance between and is maximized, and let . Find .Answer:750Solution:Omited.Resource: 2012 AIME II Problems/Problem426. Letbe the increasing sequence of positive integers whose binary representation has exactlyones. Letbe the 1000th number in. Find the remainder whenis divided by.Answer:32Solution:Omited.Resource: 2012 AIME II Problems/Problem727. Let, and for, define. The value ofthat satisfiescan be expressed in the form, whereandare relatively prime positive integers. Find.Answer:008Solution:Omited.Resource: 2012 AIME II Problems/Problem1128. n a group of nine people each person shakes hands with exactly two of the other people from the group. Letbe the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the