《数学奥林匹克报》 Mathematical Olympiad Express 26th Annual American Invitational Mathematics Examination 2008 第 26 届美国数学邀请赛(AIME1 2008 年 3 月 18 日注意: 1,直到监考老师给出信号方可答题. 2,本卷共 15 道题,答题时间 3 小时.每题的答案都是 000～999 之间的整数.如果某题你的答案是 7, 请在相应位置涂黑 007;如果某题你的答案是 43,请在相应位置涂黑 043.你答对的题数就是你的得分,每题没有部分得分,做错也不倒扣分. 3,可以使用草稿纸,方格纸,直尺,圆规,量角器,橡皮.特别地,不许使用计算器和计算机. 4,AIME 与 AMC10 或AMC12 的总分将用来决定美国数学奥林匹克(USAMO的参赛资格. 2008 第 37 届USAMO 于 4 月 29 日～30 日举行. 5,请把试题答案和相关信息填涂于 AIME 答题卡. 6,答题卡请用 2B 铅笔在相应的圆圈内涂黑,方框处可用签字笔书写. 7,答题卡不许折叠,不许使用涂改液,涂改带.若需修改请用橡皮擦拭干净后再修改. 8,请不要忘记答题卡背面的签名(先英文再中文"Yes"处也请涂黑. , 9,Last Name 姓,First Name名,Gender 性别,Female 女,Male 男简体中文版试题 1,在参加学校聚会的学生中,60%的学生是女生,40%的学生喜欢跳舞.随后又多了 20 名都会跳舞的男生,现在聚会的人中有 58%是女生.那么现在聚会的人中有多少人喜欢跳舞? 2,正方形 AIME 的边长为 10,等腰△ GEM 的底是 EM ,且△ GEM 与正方形 AIME 的公共部分的面积为 80.求△ GEM 的底 EM 上的高. 3,艾德和苏骑自行车的速度相等且恒定,他们慢跑的速度也是相等且恒定,他们游泳的速度也是相等且恒定.艾德在骑车 2 小时,慢跑 3 小时,游泳 4 小时后共行 74 千米,苏在慢跑 2 小时,游泳 3 小时, 骑车 4 小时后共行 91 千米.他们骑车,慢跑,游泳的速度都是以每小时整千米数行进.求艾德骑车, 慢跑,游泳的速度的平方和. 普及数学知识,传播奥林文化,快递竞赛信息.《数学奥林匹克报》 Mathematical Olympiad Express 2 2 4,存在唯一的正整数 x , y 满足方程 x + 84 x + 2008 = y .求 x + y . 5,圆锥的底面圆半径为r ,高为 h .把此圆锥的侧面平放于桌面,它沿着桌面无滑动地滚动.当圆锥再次回到起始位置时正好滚动 17 周,这时圆锥的底座边沿在桌面上形成一个以圆锥顶点为圆心的圆弧. 比值 h 能写成 m n 的形式,其中 m , n 是正整数且 n 不能被任何质数的平方整除.求 m + n . r 6,三角形阵列中的第一行数字按照奇数1,3,5,……,99 的递增顺序书写,在第一行下面的每一行比上一行少一个数字,最底部的一行只有一个数字.每行的相邻两个数字之和写在这两个数字中间的下一行的位置.这个阵列中有多少个数是67 的倍数? 1 4 3 8 5 ……… 12 …… …… … 7,设 Si 是满足条件100i ≤ n < 100 ( i + 1 的所有整数 n 的集合.例如, S 4 = {400, 401, 402, 在集合S0 , S1 , S 2 ,……, S999 中有多少个不包括完全平方数? 97 196 99 , 499} . 8,求出满足条件arctan 1 1 1 1 π + arctan + arctan + arctan = 的正整数 n . 3 4 5 n 4 9,同样规格的木箱的长宽高分别是 3 英尺,4 英尺,6 英尺.第一个木箱放置在地面上,其余九个木箱依次被放置于前一个木箱的上面,且每个木箱的朝向是被随机地选择来放置.设高度正好达 41 英尺的概率,这里 m , n 是互质的正整数.求 m . m 是使得堆放的木箱的 n 10,等腰梯形 ABCD 中, AD ‖ BC ,下底 AD 上的底角是π 3 .对角线长 10 21 . EA = 10 7 , ED = 30 7 .CF ⊥ AD 于 F .线段 EF 的长度可以表示为 m n 的形式,其中 m ,n 是正整数且 n 不能被任何质数的平方整除.求 m + n . 11,考察完全由字母 A 和 B 组成的具有如下性质的序列:每个连写的 A 有偶数长度,每个连写的 B 有奇普及数学知识,传播奥林文化,快递竞赛信息.《数学奥林匹克报》 Mathematical Olympiad Express 数长度.例如 AA , B , AABAA 都是这样的序列,而 BBAB 就不是这样的序列. 在长度为 14 的序列中有多少个具有这种性质? 12,在一段单车道单行线的长直高速公路上,汽车均以相同的速度且都遵循安全规则:每个 15 千米/小时的速度, 前一辆车的车尾到下一辆车的车头的距离是一辆车的长度 (不足 15 千米的按 15 千米/千米算, 因此,当前一辆车每小时行 52 千米时,前一辆车的车尾与后一辆车的车头将相距 4 个车长 .安放在路边的电子眼摄像头用来记录一小时内通过的车辆数目,假定每辆车是 4 米长,且汽车可以以任何速度穿过.设 M 是一小时内电子眼摄像头能够记录的通过的最大整数.求 M 被 10 除所得的商. 13,设 p ( x, y = a0 + a1 x + a2 y + a3 x + a4 xy + a5 y + a6 x + a7 x y + a8 xy + a9 y , 2 2 3 2 2 3 且 p ( 0, 0 = p (1, 0 = p ( 1, 0 = p ( 0,1 = p ( 0, 1 = p (1,1 = p (1, 1 = p ( 2, 2 =0. 对所有这样的多项式存在点 a b a b , 使得 p , =0, c c c c 这里 a , b , c 是正整数,且 a , c 互质, c >1.求 a + b + c . 14,⊙ ω 的直径为 AB ,延长 AB 到 C ,过 C 作 CT 切⊙ ω 于 T , AP ⊥ CT 于 P .设 AB =18, m 表示线段 BP 的最大值.求 m . 15, 正方形纸片的边长为 100, 在其每个角上按如下方式剪下一个楔形: 距离正方形顶点 17 处入刀下剪, 使得两刀口在正方形的对角线相交得 60°角(如图 ,接着把这个顶点处的两条切口折起并胶粘在一起,这样这张纸就做成一个纸盘,这个纸盘的侧面和底面所成的角不是直角.这个纸盘的高度(即纸盘最高处到纸盘底面的垂直距离可以写成 n m 的形式,这里 m ,n 是正整数,其中 m <1000,且 m 不被任何质数的 n 次方整除.求 m + n . 折痕折痕 30° 30°切口 2 切口 17 17 普及数学知识,传播奥林文化,快递竞赛信息.《数学奥林匹克报》 Mathematical Olympiad Express 2008 第26 届美国数学邀请赛(AIME1 American Invitational Mathematics Examination Tuesday,March 18,2008 1,Of the students attending a school party,60% of the students are girls,and 40% of the students like to dance. After these students are joined by 20 more boy students,all of whom like to dance,the party is now 58% girls. How many students now at the party like to dance? 2,Square AIME has sides of length 10 units. Isosceles triangle GEM has base EM , and the area common to triangle GEM and square AIME is 80 square units. Find the length of the altitude to EM in GEM . 3, and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim Ed at equal and constant rates. Ed covers 74 kilometers after biking for 2 hours, jogging for 3 hours, and swimming for 4 hours, while Sue covers 91 kilometers after jogging for 2 hours, swimming for 3 hours, and biking for 4 hours. Their biking, jogging, and swimming rates are all whole numbers of kilometers per hour. Find the sum of the squares of Ed's biking,jogging,and swimming rates. 4,There exist unique positive integers x and y that satisfy the equation x + 84 x + 2008 = y . 2 2 Find x + y . 5, right circular cone has base radius r and height h . The cone lies on its side on a flat table. As the cone A rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to its original position on the table after making 17 complete rotations. The value of h can be written in the form r m n , where m and n are positive integers and n is not divisible by the square of any prime. Find m + n . 6, triangular array of numbers has a first row consisting of the odd integers 1, 5, A 3, ……, in increasing order. 99 Each row below the first has one fewer entry than the row above it, and the bottom row has asingle entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it. How many entries in the array are multiples of 67? 普及数学知识,传播奥林文化,快递竞赛信息.《数学奥林匹克报》Mathematical Olympiad Express 1 4 3 8 5 ………12 …… …… … 97 99 196 7,Let Si be the set of all integers n such that 100i ≤ n < 100 ( i + 1 . For example, S 4 is the set {400, 401, 402, , 499} . How many of the sets S0 ,S1 ,S2 , ……,S999 do not contain a perfect square? 8,Find the positive integer n such that arctan 1 1 1 1 π + arctan + arctan + arctan = . 3 4 5 n 4 9, identical crates each of dimensions 3 ft×4 ft×6 ft. The first crate is placed flat on the floor. Each of the Ten remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let m be the probability that the stack of crates is exactly 41 ft tall, where m and n n are relatively prime positive integers.Find m . 10, Let ABCD be an isosceles trapezoid with AD ‖ BC whose angle at the longer base AD is π 3 . The diagonals have length 10 21 ,and point E is at dist ances10 7 and 30 7 from vertices A and D , respectively. Let F be the foot of the altitude fromC to AD . The distance EF can be expressed in the form m n , where m and n are positive integers and n is not divisible by the square of any prime. Find m + n . 11, Consider sequences that consist entirely of A's and B's and that have the property that every run of consecutive A's has even length,and every run of consecutive B's has oddlength.Examples of such sequences are AA, B, and AABAA,while BBAB is not such a sequence.How many such sequences have length 14? 12,On a long straight stretch ofone-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling 52 kilometers per hour will be four car lengths behind the back of the car in front of it. A photoelectric eye by the side of the road counts the number of cars that pass in one hour. Assuming that each car is 4 meters long and that the cars can travel at anyspeed, let M be the maximum whole number of cars that can pass the 普及数学知识,传播奥林文化,快递竞赛信息.《数学奥林匹克报》 Mathematical Olympiad Express photoelectric eye in one hour. Find the quotient when M is divided by 10. 13,Let p ( x, y = a0 + a1 x + a2 y + a3 x + a4 xy + a5 y + a6 x + a7 x y + a8 xy + a9 y , 2 2 3 2 2 3 Suppose that p ( 0, 0 = p (1, 0 = p ( 1, 0 = p ( 0,1 = p ( 0, 1 = p (1,1 = p (1, 1 = p ( 2, 2 =0 There is a point a b a b , for which p , =0 for all such polynomials,where a , b ,and c are c c c c positive integers, a and c are relatively prime,and c >1. Find a + b + c . 14, Let AB be a diameter of circle ω . Extend AB through A to C . Point T lies on ω so that line CT is tangent to ω . Point P is the foot of the perpendicular from A to line CT . Suppose AB = 18, and let m denote the maximum possible length of segment BP . Find m . 2 15, square piece of paper has sides of length 100. From each corner a wedge is cut in the following manner: at A each corner, the two cuts for the wedge each start at distance 17 from the corner, and they meet on the diagonal at an angle of 60° (see the figure below. The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of the tray, that is, the perpendicular distance between the plane of the base and the plane formed by the upper edges, can be written in the form n m , where m and n are positive integers,m <1000 , and m is not divisible by the n-th power of any prime. Find m + n . fold cut 30° 30° cut fold 17 17 普及数学知识,传播奥林文化,快递竞赛信息.。