美国数学邀请赛2007AMC
数学amc竞赛
数学amc竞赛
数学amc竞赛
美国数学竞赛（American Mathematics Competition，简称AMC）是一个由美国数学家协会（MAA）准备的年度北美数学竞赛，它吸引了来自世界各地大多数由7 到 12 年级学生构成的参赛团队。
AMC 竞赛旨在帮助学习数学的学生和老师对数学基础有更多的了解，引发学生对自然科学和数学科学的激情，提高学生的数学兴趣，并推动数学教育发展。
AMC 竞赛分第一至六级，级别越高，答题难度越高，每一关分数要求也越高，学生需要有步步增进的学习计划和高度的技巧，非常有意义的。
参赛学生可以了解越来越多的数学基础知识，可以利用问题解决的思维模式来解决复杂的科学问题，可以培养数学思考能力和好奇心，以及完成数学实验，研究题目和思考答案的兴趣。
最后想说的是，AMC 竞赛对于促进学生的数学学习起到了重要的作用，希望更多的学生能参与到这场有趣的竞赛中，以更深入地了解数学。
2007年AMC8试题
This file was downloaded from the AoPS Math Olympiad Resources Page /
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AMC 8 2007
paper route. He will use all of his birthday money and all of the money he earns from his paper route. In how many weeks will he be able to buy the mountain bike? (A) 24 (B) 25 (C) 26 (D) 27 (E) 28
This file was downloaded from the AoPS Math Olympiad Resources Page /
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AMC 8 2007
11 Tiles I, II, III and IV are translated so one tile coincides with each of the rectangles A, B, C and D. In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle C?
1
2
23
4
(A) 1 (B) 2 (C) 3 (D) 4 (E) cannot be determined 10 For any positive integer n, define n to be the sum of the positive factors of n. For example,
2007年第36届美国数学奥林匹克 试题
Day I12:30PM–5PM EDTApril24,20071.Let n be a positive integer.Define a sequence by setting a1=n and,for each k>1,letting a k be the unique integer in the range0≤a k≤k−1for which a1+a2+···+a k is divisible by k.For instance,when n=9the obtained sequence is9,1,2,0,3,3,3,....Prove that for any n the sequence a1,a2,a3,...eventually becomes constant.2.A square grid on the Euclidean plane consists of all points(m,n),where m and n areintegers.Is it possible to cover all grid points by an infinite family of discs with non-overlapping interiors if each disc in the family has radius at least5?3.Let S be a set containing n2+n−1elements,for some positive integer n.Suppose thatthe n-element subsets of S are partitioned into two classes.Prove that there are at least n pairwise disjoint sets in the same class.Copyright c Committee on the American Mathematics Competitions,Mathematical Association of AmericaDay II12:30PM–5PM EDTApril25,20074.An animal with n cells is a connectedfigure consisting of n equal-sized square cells.1Thefigure below shows an8-cell animal.A dinosaur is an animal with at least2007cells.It is said to be primitive if its cells cannotbe partitioned into two or more dinosaurs.Find with proof the maximum number of cells in a primitive dinosaur.5.Prove that for every nonnegative integer n,the number77n+1is the product of at least2n+3(not necessarily distinct)primes.6.Let ABC be an acute triangle withω,Ω,and R being its incircle,circumcircle,and cir-cumradius,respectively.CircleωA is tangent internally toΩat A and tangent externally toω.CircleΩA is tangent internally toΩat A and tangent internally toω.Let P A and Q A denote the centers ofωA andΩA,respectively.Define points P B,Q B,P C,Q C analogously.Prove that8P A Q A·P B Q B·P C Q C≤R3,with equality if and only if triangle ABC is equilateral.Copyright c Committee on the American Mathematics Competitions,Mathematical Association of America1Animals are also called polyominoes.They can be defined inductively.Two cells are adjacent if they share a complete edge.A single cell is an animal,and given an animal with n-cells,one with n+1cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.。
2007AMC10B答案和部分问题参考解答
AMC10B 2007 勘误一. 上次练习中第21题结论: “of the square ”没印出来.21. Right △ABC has AB =3,BC =4,and AC =5.Square XYZW is inscribed in △ABC with X and Y on AC ,W on AB , and Z on BC ( )(A) 23 (B) 3760 (C) 712 (D) 1323(E) 2二. 第24题印漏了.24. A player chooses one of the numbers 1 through 4. After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides of the dice numbered 1 through 4. If the number chosen appears on the bottom of exactly one die after it is rolled, then the player wins $1. If the number chosen appears on the bottom of both of the dice, then the player wins $2. If the number chosen does not appear on the bottom of either of the dice, the player loses $1. What is the expected return to the player, in dollars, for one roll of the dice? ( )(A) -81 (B) -161(C) 0(D) 161 (E) 81三. 第25题关键词: “gcd ”错, 应该为: “ged ”.以上问题请同学们重新练习,再参考答案和解答!AMC10B 2007 答案1(E) 2(E) 3(B) 4(D) 5(D) 6(D) 7(E) 8(D) 9(D) 10(A) 11(C) 12(D) 13(D) 14(C) 15(D) 16(C) 17(D) 18(B) 19(C) 20(C) 21(B) 22(B) 23(C) 24(E) 25(A)AMC10B 2007部分题 参 考 解 答18T.[译题]: 一个半径为1的圆被4个半径均为r 的圆环绕, ) (A)2 (B)1＋2 (C)6 (D) 3 (E) 2＋2 (分析): 设半径均为r 四个圆的圆心分别为A 、B 、C 、D .半径为1的圆的圆心为O.则正方形ABCD 中: O 为中心, AC =2r+2, AD =2r∴2r+2=2(2r) ∴ r=2+1 选(B )19T.[译题]: 一个如图所示的轮子旋转两次, 并且指针 随机地指向的数字被记录下来. 被4除的余数和第二个数字被5除的余数 分别作为列数和行数, 对应标示出棋盘的一格.求: 这对数标示黑格的机率. ( )(A)31 (B)94 (C)21 (D)95 (E)32(分析): 1与9、2与6、3与7被4整除余数依次为 的列数1、2 、3的机率相等, 均为31. 而每列白格和黑格的数目一样, 所以从列数上合格与白格被标示机率一样.1与6、2与7、3、9被4整除余数依次为1、2 、3、4, 相应机率依次为:31、31、61、61. 而第一行和第三行均为两个黑格一个白格, 第二行和第四行均为两个白格一个黑格, 黑格相应机率为32、31、32、31.综上, 最终黑格被标示机率为: (61+61)⨯32+(61+61)⨯31+61⨯32+61⨯31=21选(C )20T[译题]: 25个小正方块排成5⨯5的大正方形, 从中选出3个小方块, 且每两块均不同行不同列. 求不同顺序的选法数有多少种? ( ) (A) 100 (B) 125 (C) 600 (D) 2300 (E) 3600(分析) 给每个小方块按行列编号: 第i 行, 第j 列(i , j =1,2,3,4,5).则小方块的行号不同, 方块不同; 小方块的列号不同, 方块也不同.选出3个不同行的方法数如下:(1) 选第一块的行有5种; (2)上一块的行不能再选, 第二块的行有4种选法; (3) 上两块的行不能再选, 第三块的行有3种选法; ∴不同顺序的3个不同行的方法数为: 5⨯4⨯3=60种.而选出3个不同列的方法数与选3个不同行的方法数基本一样: 5⨯4⨯3=60种 但3个小方块不同顺序仅由行或列之一确定, 在选行时已定顺序, 则列无需再有顺序. 不同列的方法数5⨯4⨯3=60中, 每六种实际为为同一种. 例如: 532, 623, 325, 352, 253, 235 实际为为同一种 .∴不同行的方法数为: 60/6=10种 . 综上, 不同顺序的选法数有60⨯10=100种. 选(C )21T[译题]: 直角△ABC 的边AB =3, BC =4, 且AC =5. 正方形XYAW 内接于△ABC , 且点X 、Y 在线段 AC 上, 点W 在线段BC 上. (分析) 设正方形XYZW 边长为a .易知RT ⊿BWZ ∽RT ⊿XA W ∽RT ⊿BAC ∴W Z BW =AC BA =53, W X W A =BC AC =45∴ BW =53a , AW =45a又BW +AW =AB =3 ∴ 3=53a +45a ∴a =376022T[译题]: 一个底面为正方形的锥体被一个与底面平行且距离为2的平面分成两个部分, 其中顶部 被分割出的小锥的表面积为原来的大锥体的表面积的一半. 求原锥体的高为多少?( )(A) 2 (B) 2＋2 (C) 12 (分析) 锥PO 1与锥PO 相似.∴PO PO 1=SS 1=21 ∴PO O O 1=PO PO PO 1-=212- ∴ PO =1221-OO =1222-=22()12+=4+22 选(E )23T[译题]: 定义n 为满足如下条件的最小正整数: 能被4和9整除, 且十进制形式之下至少含一个数字4和9. 求n 的后四位上数字. ( )(A) 4444 (B) 4494 (C) 4944 (D) 9444 (E) 9944(分析) ∵9|n ∴n 的各位数字之和能被9整除, 又其至少含一个4, 则共至少有9个4. (含4的个数为9的倍数). ∵4|n ∴n 的后两位数字对应两位数能被4整除. ∴只能为44. 以上条件之下n 的最小值为4444444944 ∴4944为所求. 选(D ).24T[译题]: 一个玩家从1至4中选定一个数字, 选定后, 摇下两个四面上分别标有1至4个点的正四面体骰子(如下图)两个骰子. 摇完后, 若其中恰有一个骰子底面的数字为选定数,则玩家赢得1元, 若两个骰子的底面数字都为选定数字, 则玩家赢得2元. 若两个骰子的底面数字都不是选定数字, 则玩家输1元. 撒完一次骰子,玩家的赢钱期望值为多少?(A) -81 (B) -161(C) 0(D)161 (E) 81(分析) 玩家赢1元的概率P(ξ=1)=2⨯41⨯43=83玩家赢1元的概率P(ξ=2)=41⨯41=161, 玩家输1元的概率P(ξ=-1)=43⨯43=169. ∴玩家赢钱的期望为: E ξ=1⨯83+2⨯161+(-1)⨯ 169= -161选 (B )25T. [译题]: 正整数对(a ,b )满足如下条件: 它们最大公因数为1, 且b a +ab914为整数. 这样的数对有多少对?( )(A) 4 (B) 6 (C) 9 (D) 12 (E) 无穷多对(分析) 设b a +ab 914= k (k ∈N *), 则易整理得9a 2-9kab +14b 2=0 . 视其为关于a 的二次方程,有正整数解. ∴∆=9b 2(9k 2-56)为完全平方数. 设9k 2-56=m 2 (m ∈N *) , 变形有 (3k -m)(3k +m)=56 ∴(3k -m)(3k +m)=1⨯56=2⨯28=4⨯14=8⨯7 . 由6| (3k -m)+(3k +m)知:3k -m=2且3k +m=28; 或3k -m=4且3k +m=14 .∴(a ,b )=(1,3)或(2,3)、(7,3)、(14,3) 验证这四组解均合题意. 选 ( A )。
精选美国amc中文版试题及答案
2000到2012年A M C10美国数学竞赛P 0 A 0B 0 C0 D 0 全美中学数学分级能力测验(AMC 10)2000年 第01届 美国AMC10 (2000年2月 日 时间75分钟)1. 国际数学奥林匹亚将于2001年在美国举办，假设I 、M 、O 分别表示不同的正整数，且满足I ?M ?O =2001，则试问I ?M ?O 之最大值为 。
(A) 23 (B) 55 (C) 99 (D) 111 (E) 6712. 2000(20002000)为 。
(A) 20002001 (B) 40002000 (C) 20004000 (D) 40000002000 (E) 200040000003. Jenny 每天早上都会吃掉她所剩下的聪明豆的20%，今知在第二天结束时，有32颗剩下，试问一开始聪明豆有 颗。
(A) 40 (B) 50 (C) 55 (D) 60 (E) 754. Candra 每月要付给网络公司固定的月租费及上网的拨接费，已知她12月的账单为12.48元，而她1月的账单为17.54元，若她1月的上网时间是12月的两倍，试问月租费是 元。
(A) 2.53 (B) 5.06 (C) 6.24 (D) 7.42 (E) 8.775. 如图M ，N 分别为PA 与PB 之中点，试问当P 在一条平行AB 的直在线移动时，下列各数值有 项会变动。
(a) MN 长 (b) △PAB 之周长 (c) △PAB 之面积 (d) ABNM 之面积 (A) 0项 (B) 1项 (C) 2项 (D) 3项 (E) 4项6. 费氏数列是以两个1开始，接下来各项均为前两项之和，试问在费氏数列各项的个位数字中， 最后出现的阿拉伯数字为 。
(A) 0 (B) 4 (C) 6 (D) 7 (E) 97. 如图，矩形ABCD 中，AD =1，P 在AB 上，且DP 与DB 三等分 ?ADC ，试问△BDP 之周长为 。
2006年2007 美国AMC10
2006年第7届美国AMC10 (2006年2月日时间75分钟) 2006年第7届AMC 10 考试须知1. 未经监考人员宣布打开测验卷之前，不可先行打开试卷作答。
2. 本测验为选择题共有25题，每一题各有A、B、C、D、E五种选项，其中祇有一种选项是正确的答案。
3. 请将正确答案用2B铅笔在「答案欄」上适当的圆圈内涂黑，请检查所圈选的答案是否正确，并将错误及模糊不清部分擦拭干净。
请注意，祇有将答案圈选清楚在答案卡上才得以计分。
4. 计分方式：每一题答对可得6分，不作答得2.5分，答错没有分數。
5. 除了考试所准许使用的尺、圆规、量角器、橡皮擦、计算器、方格纸及计算纸外，请勿携带任何东西进入考场，考卷上所有的题目均不需使用计算器便可作答。
6. 考试之前，监考人员会指示你填写一些基本资料于答案卡上，待监考人员给予指示后开始作答，你有75分钟的时间来回答所有的题目。
7. 当你完成作答后，请在答案卡的签名空格内签名。
8. AMC 10考生考120分以上，或者成绩名列前1%者，将会受邀參加2006年3月26日星期日所举行的第24届American Invitational Mathematics Examination (AIME)考试。
1. 快餐店每个三明治卖美金3元、每杯汽水卖美金2元。
买5个三明治及8杯汽水总共要多少美元？(A) 31 (B) 32 (C) 33 (D) 34 (E) 35 。
2. 若定义x⊗y=x3-y，则h⊗(h⊗h)可化简为下列哪一个选项？(A) -h (B) 0 (C) h (D) 2h (E) h3。
3. 玛莉的岁数与爱丽斯的岁数之比为3：5。
若爱丽斯是30岁，则玛莉是几岁？(A) 15 (B) 18 (C) 20 (D) 24 (E) 50 。
4. 某数字手表会显示上午(AM)与下午(PM)的小时数与分钟数。
手表上所有显示数的每一位数字之总和可能的最大值是多少？(A) 17 (B) 19 (C) 21 (D) 22 (E) 23。
amc数学竞赛成绩划分
amc数学竞赛成绩划分摘要：一、AMC数学竞赛简介二、AMC数学竞赛成绩划分及意义1.评分标准2.奖项设置3.全球排名三、如何提高AMC数学竞赛成绩1.学习策略2.解题技巧3.模拟练习四、总结正文：AMC数学竞赛是全球范围内最具影响力的数学竞赛之一，每年吸引着众多数学爱好者参加。
在我国，AMC竞赛同样具有很高的认可度和关注度。
本文将为大家介绍AMC数学竞赛的成绩划分及如何提高竞赛成绩。
一、AMC数学竞赛简介AMC数学竞赛分为两个阶段：AMC 8、AMC 10/12、AIME（美国数学邀请赛）。
参赛者需在规定时间内完成一定数量的数学题目。
题目难度逐渐递增，涵盖了从基础数学知识到高级数学技巧的各个方面。
二、AMC数学竞赛成绩划分及意义1.评分标准AMC数学竞赛采用积分制，正确答案得1分，部分正确或错误的答案不得分。
题目数量和分数决定了参赛者的总成绩。
2.奖项设置AMC竞赛设置有不同的奖项，如全球奖项、全国奖项、学校奖项等。
奖项的划分依据参赛者的总成绩在全球范围内的排名。
具体奖项设置如下：- 全球奖项：- 满分奖（Perfect Score）：全球排名前1%- 荣誉奖（Honor Roll）：全球排名前5%- 入围奖（Quick Start Award）：全球排名前10%- 全国奖项：- 一等奖（First Place）：全国排名前10名- 二等奖（Second Place）：全国排名前20名- 三等奖（Third Place）：全国排名前30名3.全球排名参赛者可以根据自己的成绩在全球排名中查找自己的位置。
全球排名有助于了解自己在国际范围内的数学水平，为今后的学术发展提供参考。
三、如何提高AMC数学竞赛成绩1.学习策略- 扎实掌握基础知识：AMC竞赛涉及的知识点较多，参赛者需要扎实掌握基础知识，才能在竞赛中游刃有余。
- 学习高级数学技巧：竞赛题目难度逐渐递增，掌握高级数学技巧对于解决难题至关重要。
- 总结经验：每次竞赛后，参赛者应总结自己的经验教训，找出不足之处，有针对性地进行提高。
2007青少年数学国际城市邀请赛
两个队之间比赛两次 ,每个队每天比赛一场. 图 9 (a) 中给出了比赛的最后记分牌的一部 分 ,其余部分裂成了图 9 ( b) 中的四块 ,这些 碎块只在一面写有得分情况. 一个黑圈表示 胜一局 ,白圈表示负一局. 问 :冠军是哪个参 赛队 ?
市 ,再取一个与去掉的路不相连的城市 ,称这 三个城市为村庄. 则这三个村庄和三个城市 有路相连. 由已知条件 ,必有一个交叉路口 , 矛盾.
3. 一个绝对质数如果同时含有数字 1 、 3 、7 、9 ,则在这个质数的十进制表示中 ,不可 能含有数字 0 、2 、4 、5 、6 、8 ,否则 ,通过适当排 列后 ,这个数能被 2 或 5 整除.
设 N 是一个同时含有数字 1 、3 、7 、9 的 绝对质数.
因为 k0 = 7 931 , k1 = 1 793 , k2 = 9 137 ,
k3 = 7 913 , k4 = 7 193 , k5 = 1 937 , k6 = 7 139 被 7 除所得的余数分别是 0 、1 、2 、3 、4 、5 、6 ,所 以 ,如下 7 个正整数
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AMC Intermediate 2007
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(C) x + y = 13
(D) x + y = 207
(E) x − y =colates at $1.35 each. How much change should he get from $5?
(A) 60 (B) 70 (C) 80 (D) 90 (E) 100
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A u s t r a l i a n M a t h e ma t i c s C o m p e t i t i o n f o r t h e W e s t p a c A wa r d s
an activity of the australian mathematics trust
美国AMC10中文版试题及答案
2000到20XX年AMC10美国数学竞赛0 0P 0 A 0 B 0 C 0D 0 全美中学数学分级能力测验(AMC 10)2000年 第01届 美国AMC10 (2000年2月 日 时间75分钟)1. 国际数学奥林匹亚将于 在美国举办，假设I 、M 、O 分别表示不同的正整数，且满足I ⨯M ⨯O =2001，则试问I +M +O 之最大值为 。
(A) 23 (B) 55 (C) 99 (D) 111 (E) 6712. 2000(20002000)为 。
(A) 20002001 (B) 40002000 (C) 20004000 (D) 40000002000 (E) 200040000003. Jenny 每天早上都会吃掉她所剩下的聪明豆的20%，今知在第二天结束时，有32颗剩下，试问一开始聪明豆有 颗。
(A) 40 (B) 50 (C) 55 (D) 60 (E) 754. Candra 每月要付给网络公司固定的月租费及上网的拨接费，已知她12月的账单为12.48元，而她1月的账单为17.54元，若她1月的上网时间是12月的两倍，试问月租费是 元。
(A) 2.53 (B) 5.06 (C) 6.24 (D) 7.42 (E) 8.775. 如图M ，N 分别为PA 与PB 之中点，试问当P 在一条平行AB 的直 在线移动时，下列各数值有 项会变动。
(a) MN 长 (b) △P AB 之周长 (c) △P AB 之面积 (d) ABNM 之面积(A) 0项 (B) 1项 (C) 2项 (D) 3项 (E) 4项 6. 费氏数列是以两个1开始，接下来各项均为前两项之和，试问在费氏数列各项的个位数字中， 最后出现的阿拉伯数字为 。
(A) 0 (B) 4 (C) 6 (D) 7 (E) 97. 如图，矩形ABCD 中，AD =1，P 在AB 上，且DP 与DB 三等分∠ADC ，试问△BDP 之周长为 。
2007 AMC 10B Problems
2007 AMC 10B ProblemsProblem 1Isabella's house has3bedrooms. Each bedroom is12feet long,10feet wide, and8feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy60square feet in each bedroom. How many square feet of walls must be painted?（A）678 （B）768 （C）786 （D）867 （E）876Problem 2Define the operation by What is（A）-16 （B）-8 （C）0 （D）8 （E）16Problem 3A college student drove his compact car 120miles home for the weekend and averaged 30miles per gallon. On the return trip the student drove his parents' SUV and averaged only 20miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?（A）22 （B）24 （C）25 （D）26 （E）28Problem 4The point O is the center of the circle circumscribed about △ABC with∠BOC=120ºand ∠AOB=140º. What is the degree measure of ∠ABC?（A）35 （B）40 （C）45 （D）50 （E）60Problem 5In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?（A）All Dramps are Brafs and are Crups （B）All Brafs are Crups and are Dramps （C）All Arogs are Crups and are Dramps （D）All Crups are Arogs and are Brafs （E）All Arogs are Dramps and some Arogs may not be CrumpsProblem 6The 2007 AMC10 will be scored by awarding 6 points for each correct response, 0 points for each incorrect response, and 1.5 points for each problem left unanswered. After looking over the 25 problems, Sarah has decided to attempt the first 22 and leave only the last 3 unanswered. How many of the first 22 problems must she solve correctly in order to score at least 100 points?（A）13 （B）14 （C）15 （D）16 （E）17Problem 7All sides of the convex pentagon ABCDE are of equal length, and ∠A=∠B=90º What is the degree measure of ∠E?（A）90 （B）108 （C）120 （D）144 （E）150On the trip home from the meeting where this AMC10 was constructed, the Contest Chair noted that his airport parking receipt had digits of the form bbcac , where 90≤<<≤c b a and was the average of and How many different five-digitnumbers satisfy all these properties?（A ）12 （B ）16 （C ）18 （D ）20 （E ）25Problem 9A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is 1 place to its right in the alphabet (asumming that the letter A is one place to the right of the letter Z ). The second time this same letter appears in the given message, it is replaced by the letter that is 1+2 places to the right, the third time it is replaced by the letter that is 1+2+3 places to the right, and so on. For example, with this code the word "banana" becomes "cbodqg". What letter will replace the last letter in the message “ Lee ’s sis is a Mississippi miss, Chriss!”（A ）g （B ）h （C ）o （D ）s （E ）tProblem 10Two points B and C are in a plane. Let S be the set of all points A in the plane for which △ABC has area 1. Which of the following describes S?（A ）two parallel lines （B ）a parabola （C ）a circle （D ）a line segment （E ）two pointsProblem 11A circle passes through the three vertices of an isosceles triangle that has two sides of length 3 and a base of length 2. What is the area of this circle?（A ）π2 （B ）π25（C ）π3281 （D ）π3 （E ）π27 Problem 12Tom's age is T years, which is also the sum of the ages of his three children. His age N years ago was twice the sum of their ages then. What is T/N ?（A ）2 （B ）3 （C ）4 （D ）5 （E ）6Problem 13Two circles of radius 2 are centered at (2, 0) and at (0, 2). What is the area of the intersection of the interiors of the two circles?（A ）2-π （B ）2π （C ）33π （D ）)2(2-π （E ）π Problem 14Some boys and girls are having a car wash to raise money for a class trip to China. Initially 40% of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then 30% of the group are girls. How many girls were initially in the group?（A ）4 （B ）6 （C ）8 （D ）10 （E ）12The angles of quadrilateral ABCD satisfy ∠A =2∠B =3∠C =4∠D. What is the degree measure of ∠A rounded to the nearest whole number?（A ）125 （B ）144 （C ）153 （D ）173 （E ）180Problem 16A teacher gave a test to a class in which 10% of the students are juniors and 90% are seniors. The average score on the test was 84 The juniors all received the same score, and the average score of the seniors was 83 What score did each of the juniors receive on the test?（A ）85 （B ）88 （C ）93 （D ）94 （E ）98Problem 17Point P is inside equilateral △ABC . Points Q,R, and S are the feet of the perpendiculars from P to AB ,BC , and CA , respectively. Given that PQ =1, PR =2, and PS =3 , what is AB ?（A ）4 （B ）33 （C ）6 （D ）34 （E ）9Problem 18A circle of radius is surrounded by circles of radius as shown. What is ?（A ）2 （B ）21+ （C ）6 （D ）3 （E ）22+ Problem 19The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by 4 and the second number is divided by 5. The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. What is the probability that the pair of numbers designates a shaded square?（A ）31（B ）94 （C ）21 （D ）95 （E ）32 Problem 20A set of 25 square blocks is arranged into a 5*5 square. How many different combinations of 3 blocks can be selected from that set so that no two are in the same row or column?（A ）100 （B ）125 （C ）600 （D ）2300 （E ）3600 Problem 21Right △ABC has AB =3, BC =4 and AC=5 Square XYZW is inscribed in △ABC with X and Y on AC , W on AB , and Z on BC . What is the side length of the square?（A ）23 （B ）3760 （C ）712 （D ）1323 （E ）2 Problem 22A player chooses one of the numbers 1 through 4. After the choice has been made, two regular four-sided (tetrahedral) dice are rolled, with the sides of the dice numbered 1 through 4. If the number chosen appears on the bottom of exactly one die after it has been rolled, then the player wins 1 dollar. If the number chosen appears on the bottom of both of the dice, then the player wins 2 dollars. If the number chosen does not appear on the bottom of either of the dice, the player loses1 dollar. What is the expected return to the player, in dollars, for one roll of the dice?（A ）81- （B ）161- （C ）0 （D ）161 （E ）81 Problem 23A pyramid with a square base is cut by a plane that is parallel to its base and 2 units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid?（A ）2 （B ）22+ （C ）221+ （D ）4 （E ）224+ Problem 24Let n denote the smallest positive integer that is divisible by both 4 and 9, and whose base-10 representation consists of only 4's and 9's, with at least one of each. What are the last four digits of n ?（A ）4444 （B ）4494 （C ）4944 （D ）9444 （E ）9944Problem 25How many pairs of positive integers ),(b a are there such that and have no commonfactors greater than 1 and ab b a 914+is an integer? （A ）4 （B ）6 （C ）9 （D ）12 （E ）infinitely many2007 AMC 10B SolutionsProblem 1There are four walls in each bedroom, since she can't paint floors or ceilings. So we calculate the number of square feet of wall there is in one bedroom : (12*8)+(12*8)+(10*8)+(10*8)-60=160+192-60=292. We have three bedrooms, so shesquare feet of wall. Problem 2Problem 3The trip was 240 miles long and took 10642012030120=+=+ gallons. Therefore, theaverage mileage was 240/10=24 Problem 4Because all the central angles of a circle add up to 360º∠BOC +∠AOB +∠AOC =360 120+140+∠AOC =360 so ∠AOC =100Therefore, the measure of is also 100º. Since the measure of an inscribed angle isequal to half the measure of the arc it intercepts, ∠ABC=50 Problem 5It may be easier to visualize this by drawing some sort of diagram. From the first statement, you can draw an Arog circle inside of the Braf circle, since all Arogs are Brafs, but no all Brafs are Arogs. Ignore the second statement for now, and draw a Dramp circle in the Arog circle and a Crup circle in the Dramp circle. You can see the second statement is already true because all Crups are Arogs. As you can see, the only statement that istrue is Problem 6Sarah is leaving 3 questions unanswered, guaranteeing her 3*1.5=4.5 points. She will either get 6 points or 0 points for the rest of the questions. Let be the number ofquestions Sarah answers correctly. 92.151005.46≥⇒≥+x x .The number ofquestions she answers correctly has to be a whole number, so round up to get 16. Problem 7AB=EC because they are opposite sides of a square. Also, ED=DC=AB because all sides of the convex pentagon are of equal length. Since ABCE is a square and △CED is an equilateral triangle, ∠AEC =90 and ∠CED =60 . Use angle addition1506090=+=∠+∠=∠CED AEC E Problem 8For the average,b , to be an integer, and have to either both be odd or both be even.There are 101245=⨯⨯ ways to choose a set of two even numbers and 101245=⨯⨯ ways to choose a set of two odd numbers. Therefore, the number of five-digit numbers that satisfythese properties is 10+10=20. Problem 9 Since the letter that will replace the last does not depend on any letter except the other's, you can disregard anything else. There are 12 's, so the last will be replaced theletter 1+2+3+…+12 places to the right of .7821211212...321=+⨯=++++ Every 26 places, you will end up with the same letter so you can just take the remainder of 78 when you divide by 26, which is 0. Therefore, the letter that will is replace thelast is sProblem 10Let h be the length of the altitude of △ABC . Since segment BC is the base of the triangle and cannot change, the area of the triangle is 1))((21=h BC andBC h 2= . Thus consists of two lines parallel to BC and is BC2 units away from it. Problem 11Solution 1Let have vertex and center , with foot of altitude from at .Then by Pythagorean Theorem (with radius , height OD=h ) on △OBD, ABD ,91)(,1222=++=+r h r h . Substituting and solving gives 249=r . Then the area of thecircle is πππ3281)249(22==r . Solution 2By R abc Bh A 421== (or we could use and Heron's formula ),249)22)(2(22332=∙∙==Bh abc Rand the answer is ππ32812=RAlternatively, by the Extended Law of Sines , 2493sin 2322=⇒=∠=R ABC AC R Answer follows as above. Solution 3Extend segment AD to R on Circle O .By the Pythagorean Theorem 2213222=⇒-=AD AD△ADC is similar to △ACR , so r 23322=which gives us 4292292==r therefore829=r . The area of the circle is therefore πππ3281)829(22==r Problem 12Tom's age N years ago was T-N . The ages of his three children N years ago was T-3N , since there are three people. If his age N years ago was twice the sum of thechildren's ages then, 5/5)3(2=⇒=⇒-=-N T N T N T N T Problem 13 You can find the area of half the intersection by subtracting the isosceles triangle in the sector from the whole sector. This sector is a quarter of the circle with radius 2 and the isosceles triangle is a right triangle. Therefore, the area of half the intersectionisc 2)2)(2(21441-=-ππThat means the area of the whole intersection is Problem 14If we let p be the number of people initially in the group, the 0.4p is the number of girls. If two girls leave and two boys arrive, the number of people in the group is still but thenumber of girls is 24.0-p . Since only 30% of the group are girls, 2010324.0=⇒=-p p pThe number of girls is 84.0=p Alternate SolutionThere are the same number of total people before and after, but the number of girls hasdropped by two and 10%. 2/0.1=20, and 40%·Problem 15The sum of the interior angles of any quadrilateral is 360º1738.1721225413121360≈=∠⇒∠=∠+∠+∠+∠=∠+∠+∠+∠=A A A A A A D C B AProblem 16We can assume there are 10 people in the class. Then there will be 1 junior and 9 seniors. The sum of everyone's scores is 8408410=∙ Since the average score of the seniors was 83, the sum of all the senior's scores is 747839=∙. The only score that has not beenadded to that is the junior's score, which is 840-747=93. Problem 17Drawing PA ,PB , and PC , △ABC is split into three smaller triangles. The altitudes of these triangles are given in the problem as PQ , PR , and PS . Summing the areas of each of these triangles and equating it to the area of the entiretriangle, we get:432)3(2)2(2)1(2s s s s =++where is the length of a side 34=s Problem 18Solution 1You can express the line connecting the centers of an outer circle and the inner circle in two different ways. You can add the radius of both circles to get 1+r . You can also add the radius of two outer circles and use a 45—45—90 triangle to get 222r r = . Sinceboth representations are for the same thing, you can set them equal to each other.1212121+=-=⇒=+r r r Solution 2You can solve this problem by setting up a simple equation with the Pythagorean Theorem. The hypotenuse would be a segment that includes the radius of two circles on opposite corners and the diameter of the middle circle. This results in 22+r . The two legs are each the length between two large, adjacent circles, thus r 2. Using the Pythagorean Theorem:Problem 19Solution 1When dividing each number on the wheel by 4, the remainders are 1,1,2,2,3, and 3 Each column on the checkerboard is equally likely to be chosen.When dividing each number on the wheel by 5, the remainders are 1,1,2,2,3, and 4The probability that a shaded square in the 1st or 3rd row of the 1st or 3rd column is 316332=⨯ The probability that a shaded square in the 2nd or 4th row of the 2nd column is 616331=⨯Add those two together 216131=+ Solution 2Alternatively, we may analyze this problem a little further.First, we isolate the case where the rows are numbered 1 or 2. Notice that as listed before,the probability for picking a shaded square here is 1/2 because the column/row probabilities are the same, with the same number of shaded and non-shaded squares Next we isolate the rows numbered 3 or 4. Note that the probability of picking the rows is same, because of our list up above. The columns, of course, still have the same probability. Because the number of shaded and non-shaded squares are equal, we have1/2 Combining these we have a general probability of Problem 20There are 25 ways to choose the first square. The four remaining squares in its column and row, and the square you chose exclude nine squares from being chosen next time. There are 16 remaining blocks to be chosen for the second square. The three remaining spaces in its row and column and the square you chose must be excluded from being chosen next time.Finally, the last square has 9 remaining choices.The number of ways to choose 3 squares is 25·16·9, but you must divide by 3! sincesome sets are the same. 600382512391625=∙∙=∙∙∙∙ Problem 21Solution 1There are many similar triangles in the diagram, but we will only use △WBZ ∽△ABC . If h is the altitude from B to AC and is the side length of the square, then h-s is thealtitude from B to WZ . By similar triangles,555+=⇒=-h h s h s s h Find the length of the altitude of △ABC. Since it is a right triangle, the area of △ABC is 6)4)(3(21= The area can also be expressed as ),)(5(21h so 4.2625=⇒=h hSubstitute back into37604.71255==+=h h s Solution 2 Let l be the side length of the inscribed square. Note that △ZYC ∽△WBZ ∽△ABC . Then we can setup the following ratios:l CZ l CZ 3535=⇒= l ZB l CB 5454=⇒= But then 37604153745435=⇒=⇒==+=+l l CB ZB CZ l l Problem 22There are 2·3·1=6 ways for your number to show up once, 1·1=1 way for your number to show up twice, and 3·3=9 ways for your number to not show up at all. Think of this as aset of 16 numbers with six 1s , one 2, and nine -1s . The average of this set is theexpected return to the player. 1611692616)1(9)2(1)1(6-=-+=-+ Problem 23Since the two pyramids are similar, the ratio of the altitudes is the square root of the ratio of the surface areas. If is the altitude of the larger pyramid, then 2-a is the altitude of the smaller pyramid.2241222122+=-=⇒=-a a a Problem 24For a number to be divisible by 4, the last two digits have to be divisible by 4. That means the last two digits of this integer must be 4.For a number to be divisible by 9, the sum of all the digits must be divisible by 9. The only way to make this happen is with 9 4's. However, we also need one 9.The smallest integer that meets all these conditions is 4444444944. The last four digitsare 4944. Problem 25Let b a x =. We can then write the given expression as k xx =+914 where k is an integer. We can rewrite this as a quadratic,014992=+-kx x . By the Quadratic Formula,656921850481922-±=-±=k k k k x . We know that must be rational, so 5692-k must be aperfect square. Let 22569n k =-. Then, )3)(3(95622n k n k n k +-=-=. The factorspairs of 56 are 1 and 56, 2 and 28, 4 and 14, and 7 and 8. Only 2 and 28 and 4 and 14 give integer solutions, 5=k and 13=n and and 5=n , respectively. Pluggingpossibilities for , namely ,32,314,31 and 37.2007 AMC 10B Answer Key1. E2. E3.B4. D5. D6.D7.E8.D 8. D 10.A11.C 12. D 13.D 14.C 15. D 16.C 17. D 18.B 19.C 20.C21.B 22.B 23.E 24.C 25.A。
美国留学：美国数学竞赛AMC
美国留学：美国数学竞赛AMC近年来，随着标准化考试越来越普遍，仅仅靠SAT、托福等成绩越来越难和同龄人区分开来，因此，有一门数学竞赛成绩就成了申请环节中加分的利器。
今天就来为大家科普一下什么是AMC。
1. AMC基本信息AMC全称为American Mathematics Competition，是“美国数学竞赛”的简称，1950年美国数学协会Mathematics Association of America （简称MAA），开始举办美国高中数学考试（AHSME）。
在1985年时，MAA又增加了初中数学的考试（AJHSME），2000年以后这些考试统一被称为AMC，AMC总部现设在美国加州內布拉斯加大学林肯校区。
AMC的目标是发掘、培养学生的数学潜能，为从事数学及其他学科的学习和研究打下坚实基础。
2. AMC比赛类型与年级AMC8由8年级和8年级以下学生参加。
注：许多小学生参加AMC8考试，特别是来自中国的学生，不少四至六年级的中国学生参加AMC8考试并获奖。
AMC10由10年级和10年级以下的学生参加。
AMC12由12年级和12年级以下的学生参加。
AIME由AMC10和AMC12测试中成绩达到要求的学生参加。
USJMO由AMC10的约前230名美国学生参加。
USAMO由AMC10的约前270名美国学生参加。
3. AMC举办时间AMC系列赛的举办时间AMC8在每年的11月中举行。
AMC10和AMC12的比赛时间相同，分A、B两次在每年的2月初和2月中举行。
AIME在每年的3月底举行。
USJMO和USAMO在每年4月的最后一周举行。
4. AMC主试委员会AMC测试的试题研发、命制到统一阅卷等工作由AMC主试委员会全权负责。
该委员会由以数理闻名的内布拉斯加大学林肯分校数学系的专家和来自其它全美一流学府，如麻省理工学院（MIT）、哈佛大学（Harvard）、普林斯顿大学（Princeton）等名校的专家组成。
2007年美国中学生数学建模竞赛竞赛试题(中文)
2007年美国中学生数学建模竞赛竞赛试题（中文）
问题：烟雾警报器
火灾是导致意外死亡的一个主要原因。
这是很重要的，大家必须采取一切预防措施，做可能的防范，以随时准备应付火灾紧急状态。
所有致命火灾中的一半以上发生在下午10时到上午6时之间，这个时间段内通常大家都在家里睡着了。
烟雾警报器可以在你睡眠的时候提醒你有火情。
烟雾警报器能留出足够的时间，以便于安全疏散吗？
建立一个数学模型，确定烟雾警报器安装的数量和位置，以提供最大的疏散时间。
还包括一个模型，以确定居家消防灭火器的安装数目和位置。
建立一个整个家庭从一到两层的房子中撤离的数学模型。
为您当地的消防部门准备一个向社区的通知，其中包括你的数学模型的主要成果。
一层的房子
两层房子的底层两层房子的顶层
B题问题：租车
有些人当他们将要做一个长途旅行的时候，他们会租一辆汽车。
通常他们会选择节省金钱。
即使他们不吝惜钱，他们也会认为"如果车在旅行中坏了，问题是出租公司的" ，这一点使得租金是值得的。
分析这种情况，并确定在什么条件下，租用一辆汽车是一个比较适当的选择。
对自己拥有的汽车，确定里程限制，并确定对于司机和她的家人来说可接受的损益平衡点。
2007年AMC8试题解析
20071Theresa’s parents have agreed to buy her tickets to see her favorite band if she spends an average of 10hours per week helping around the house for 6weeks.For the first 5weeks,she helps around the house for 8,11,7,12and 10hours.How many hours must she work during the final week to earn the tickets?(A)9(B)10(C)11(D)12(E)132Six-hundred fifty students were surveyed about their pasta preferences.The choices were lasagna,manicotti,ravioli and spaghetti.The results of the survey are displayed in the bar graph.What is the ratio of the number of students who preferred spaghetti to the number of students who preferredmanicotti?50100150200250L a s a g n a M a n i c o t t i R a v i o l iS p a g h e t t i N u m b e r o f P e o p l e (A)25(B)12(C)54(D)53(E)523What is the sum of the two smallest prime factors of 250?(A)2(B)5(C)7(D)10(E)124A haunted house has six windows.In how many ways can Georgie the Ghost enter the house by one window and leave by a different window?(A)12(B)15(C)18(D)30(E)365Chandler wants to buy a $500dollar mountain bike.For his birthday,his grandparents send him $50,his aunt sends him $35and his cousin gives him $15.He earns $16per week for his This file was downloaded from the AoPS Math Olympiad Resources PagePage 12007paper route.He will use all of his birthday money and all of the money he earns from his paper route.In how many weeks will he be able to buy the mountain bike?(A)24(B)25(C)26(D)27(E)286The average cost of a long-distance call in the USA in 1985was 41cents per minute,and the average cost of a long-distance call in the USA in 2005was 7cents per minute.Find the approximate percent decrease in the cost per minute of a long-distance call.(A)7(B)17(C)34(D)41(E)807The average age of 5people in a room is 30years.An 18-year-old person leaves the room.What is the average age of the four remaining people?(A)25(B)26(C)29(D)33(E)368In trapezoid ABCD ,AD is perpendicular to DC ,AD =AB =3,and DC =6.In addition,E is on DC ,and BE is parallel to AD .Find the area of ∆BEC .ABCD E336(A)3(B)4.5(C)6(D)9(E)189To complete the grid below,each of the digits 1through 4must occur once in each row and once in each column.What number will occupy the lower right-hand square?12234(A)1(B)2(C)3(D)4(E)cannot be determined10For any positive integer n ,define n to be the sum of the positive factors of n .For example,6=1+2+3+6=12.Find 11.(A)13(B)20(C)24(D)28(E)30200711Tiles I,II,III and IV are translated so one tile coincides with each of the rectangles A,B,Cand D .In the final arrangement,the two numbers on any side common to two adjacent tiles must be the same.Which of the tiles is translated to Rectangle C ?8672III IIIIV 341972069351A BC D(A)I (B)II (C)III (D)IV (E)cannot be determined 12A unit hexagon is composed of a regular haxagon of side length 1and its equilateral triangularextensions,as shown in the diagram.What is the ratio of the area of the extensions to the area of the originalhexagon?2007(A)1:1(B)6:5(C)3:2(D)2:1(E)3:113Sets A and B,shown in the venn diagram,have the same number of elements.Thier union has2007elements and their intersection has1001elements.Find the number of elements in A.A B1001(A)503(B)1006(C)1504(D)1507(E)151014The base of isosceles ABC is24and its area is60.What is the length of one of the congruent sides?(A)5(B)8(C)13(D)14(E)1815Let a,b and c be numbers with0<a<b<c.Which of the following is impossible?=a(A)a+c<b(B)a·b<c(C)a+b<c(D)a·c<b(E)bc16Amanda Reckonwith drawsfive circles with radii1,2,3,4and5.Then for each circle she plots the point(C;A),where C is its circumference and A is its area.Which of the following could be her graph?(A)AC2007 (B)AC (C)AC (D)AC (E)AC200717A mixture of30liters of paint is25%red tint,30%yellow tint,and45%water.Five liters of yellow tint are added to the original mixture.What is the percent of yellow tint that is the mixture?(A)25(B)35(C)40(D)45(E)5018The product of the two99-digit numbers303,030,303,...,030,303and505,050,505,...,050,505has thousands digit A and units digit B.What is the sum of A and B?(A)3(B)5(C)6(D)8(E)1019Pick two consecutive positive integers whose sum is less than100.Square both of those integers and thenfind the difference of the squares.Which of the following could be the difference?(A)2(B)64(C)79(D)96(E)13120Before district play,the Unicorns had won45%of their basketball games.During district play,they won six more games and lost two,tofinish the season having won half their games.How many games did the Unicorns play in all?(A)48(B)50(C)52(D)54(E)6021Two cards are dealt from a deck of four red cards labeled A,B,C,D and four green cards labeled A,B,C,D.A winning pair is two of the same color or two of the same letter.What is the probability of drawing a winning pair?(A)27(B)38(C)12(D)47(E)5822A lemming sits at a corner of a square with side length10meters.The lemming runs6.2 meters along a diagonal toward the opposite corner.It stops,makes a90◦right turn and runs 2more meters.A scientist measures the shortest distance between the lemming and each side of the square.What is the average of these four distances in meters?(A)2(B)4.5(C)5(D)6.2(E)723What is the area of the shaded pinwheel shown in the5×5grid?2007(A)4(B)6(C)8(D)10(E)1224A bag contains four pieces of paper,each labeled with one of the digits 1,2,3or 4,with norepeats.Three of these pieces are drawn,one at a time without replacement,to construct a three-digit number.What is the probability that the three-digit number is a multiple of 3?(A)14(B)13(C)12(D)23(E)3425On the dart board shown in the figure,the outer circle has radius 6and the inner circle hasradius 3.Three radii divide each circle into the three congruent regions,with point values shown.The probability that a dart will hit a given region is proportional to to the area of the region.What two darts hit this board,the score is the sum of the point values in the regions.What is the probability that the score is odd?1112222007(A)1736(B)3572(C)12(D)3772(E)1936The problems on this page are copyrighted by the Mathematical Association of America’s American Mathematics Competitions.。
AMC12 2007
~ 1 ~2007年 第58屆 AMC 12考試須知1. 未經監考人員宣佈打開測驗卷之前，不可先行打開試卷作答。
2. 本測驗為選擇題共有25題，每一題各有A 、B 、C 、D 、E 五種選項，其中祇有一種選項是正確的答案。
3. 請將正確答案用2B 鉛筆在「答案欄」上適當的圓圈內塗黑，請檢查所圈選的答案是否正確，並將錯誤及模糊不清部分擦拭乾淨。
請注意，祇有將答案圈選清楚在答案卡上才得以計分。
4. 計分方式：每一題答對可得6分，不作答得1.5分，答錯得0分。
5. 除了考試所准許使用的尺、圓規、量角器、橡皮擦、計算器、方格紙及計算紙外，請勿攜帶任何東西進入考場，考卷上所有的題目均不需使用計算機便可作答。
6. 考試之前，監考人員會指示你填寫一些基本資料於答案卡上，待監考人員給予指示後開始作答，你作答時間共75分鐘。
7. 當你完成作答後，請在答案卡的簽名空格內簽名。
8. AMC 12考生考100分以上，或者成績名列前5%者，將會受邀參加2007年3月31日星期六所舉行的第25屆American Invitational Mathematics Examination (AIME)考試。
~ 2 ~58th AMC 12 20071. 某場表演一張票的原價為美金20元, 蘇珊用折價券買了4張票, 可以少付25%. 小潘用折價券買了5張票, 可以少付30%. 小潘比蘇珊總共多付了美金多少元？ (A) (B) (C) (D) (E) 251015202. 一個水族箱內的底部是100 cm ×40 cm 的矩形且其高為50cm. 將水族箱裝水至40 cm 高, 並將底部是4020高度是10的一個長方體磚塊放到水族箱內. 水面會上升多少公分(cm)？cm ×cm cm 0.51 1.52 2.548121620(A) (B) (C) (D) (E)3. 兩個連續奇數中, 較大的數等於較小的數的3倍. 這兩數的和是多少？(A) (B) (C) (D) (E)4. 凱特以每小時16公里的速率騎腳踏車30分鐘後, 接著以每小時4公里的速率走路90分鐘. 他全部行程的平均速率是每小時多少公里？(A) (B) (C) (D) (E) 79101214~ 3 ~5. 去年陳先生繼承了遺產. 他必須付遺產的20%為中央稅,付完了中央稅後剩下金額的10%為地方稅.這兩筆稅他總共付了10,500元. 他原來繼承的遺產是多少元？(A) 30,000 (B) 32,500 (C) 35,000 (D) 37,500 (E) 40,0006. 與ABC ∆ADC ∆均為等腰三角形, 其中AB BC =,AD DC =ABC ∆40ABC , 且D 點在的內部, ∠=°140ADC ∠=,°. 問BAD ∠是多少度？ (A) 20 (B) 30 (C) 40 (D) 50 (E) 607. 設a , b , c , d , e 為一等差數列中的連續五項, 並設. 則必可算出下列哪一項？30a b c d e ++++=a b c d e 2024303660~ 4 ~(A) (B) (C) (D) (E)9. 小言站在他家與運動場之間. 要到運動場, 他可以直接走路到運動場, 或他走路回家再循原路徑騎腳踏車到運動場. 騎腳踏車的速率是走路速率的7倍, 兩種方法所花的時間都相同. 小言所站位置到他家的距離與到運動場的距離之比值是多少？(A) 23 (B) 344556 (C) (D) (E) 673:4:58.6412510. 一個三角形的外接圓半徑是3, 且其三邊長之比是.此三角形的面積是多少？(A) (B) (C) π (D) (E) 17.28183713374311. 考慮有限數列, 每項均為三位數且具有下列性質： 每一項的十位數字與個位數字分別是下一項的百位數字與十位數字, 最後一項的十位數字與個位數字是第一項的百位數字與十位數字. 例如, 247, 475, 756,…, 824就是一個這種的數列. 用S 表示這種數列各項的和. 下面哪一數是一定可以整除S 的最大質數？(A) (B) (C) (D) (E)8. 在鐘面上從每一個數畫直線到順時針方向的第五個數, 形成一個多角星形. 例如從12畫線到5, 從5畫線到10, 從10畫線到3, 如此繼續劃下去, 直到回到12為止. 這個多角星形每一個頂點的角度是多少度？(A) (B) (C) (D) (E)~ 5 ~ad bc −12. 從0到2007中任意的選出整數a , b , c , d , 它們不必都相異.問為偶數的機率是多少？(A) 38 (B) 716 (C) 12 (D) 916 (E) 580)1y x13. 已知一塊乳酪放在坐標平面上處. 現有一隻老鼠沿著直線8(12,15=−+(4a 從處往上跑. 老鼠跑到點)後, 即開始遠離乳酪而不是更靠近乳酪. 問b ,2)−(,a b +之值是多少？(A) (B) (C) (D) (E) 610141822(6)(6)(6(6)45c d e −−−a b c d ++++51725273079192426~ 6 ~9610411212025616. 有多少個三位數, 它是由三個相異的阿拉伯數字所組成,且其中一個位數是其它兩個位數數字的平均值？ (A) (B) (C) (D) (E)sin sin a b +=cos cos 1a b +=cos()a b −17. 設且. 問之值是多少？(A) 11312 (B) (C) (D) 231432() (E)14. 設a , b , c , d , e 為相異的整數且滿足18. 多項式函數f x x ax bx cx d =++++(2)(2)0f i f i 的係數均為實數,且)(6)a b −−=.問e 之值是多少？(A) (B) (C) (D) (E)=+=a b c d . 問+++014916ABC 之值是多少？ (A) (B) (C) (D) (E)19. 設∆與的面積分別為2007和7002, 其中ADE ∆15. 在3, 6, 9, 10中加入第五個數n , 它與其它的四個數都不同.這五個數的中位數等於它們的平均值. 問n 的所有可能值之和是多少？(A) (B) (C) (D) (E), , , (0,0)B =(223,0)C =(680,380)D =(689,389)E =.則所有可能的A 點之x 坐標的和是多少？(A) (B) (C) (D) (E) 2823006009001200~ 7 ~20. 將單位正立方體的角落切掉, 使得原來的六個面的每一個面都變成正八邊形. 所有切下來的四面體之體積的總和是多少？(A) 73−(B) 103−(C) 33−(D) 113(E) 63−2()21. 已知函數f x ax bx c =++()0f x 若=各根之和、各根之積及()f x 所有係數之和都相等. 則這個共同的數值必須與下列何者相同？(A) 2x 項的係數 (B) 項的係數 x ()y f x =(C) 圖形的y 截距 (D) 圖形中的一個截距 ()y f x =x ()y f x =x S n ()(())2007n S n S S n ++=12345~ 8 ~(E) 圖形截距的平均值23. 正方形ABCD 的面積為36, 且AB log a y x 平行於x 軸. 頂點A , B及C 分別在=, 2log a y x =及x 3log a y =的圖形上. 問等於多少？ a (A) 22. 對每一個正整數n , 以表示n 各位數字的和. 問使得的n 有多少個？()(A) (B) (C) (D) (E)(B)61n >()F n sin sin x (E)24. 對每一個整數, 令為方程式nx =在區間20072()n F n =∑[0,]π中解的個數.之值是多少？(A) (B) (C)2,014,5242,015,0282,015,0332,016,5322,017,033(D) (E)25. 如果一個整數集合至多只包含所有三個連續整數中的其中一個數, 則稱此集合為古怪的. 則集合有多少個子集合（含空集合）, 是古怪的{1,2,3,,12}L ？(A) (B) (C) (D) (E) 121123125127129。
美国高中数学测验 AMC12之机率问题(下)
美国高中数学测验 AMC12之机率问题(下)洪伟诚 . 李俊贤 . 蔡诚祐 . 何家兴 . 张福春关键词：高中数学几何机率前述两节的问题皆建立在样本点个数为可数时的情况,接下来将介绍一不可数的无穷样本空间S且利用此空间的一些几何测量m(S),例如长度、面积、或者体积,来求A事件的机率。
而A事件的机率可用A事件之几何测量与样本空间S之几何测量的比例来计算,其形式有下列三种:P(A)=A的长度S的长度或P(A)=A的面积S的面积或P(A)=A的体积S的体积注:我们必须假设一不可数无穷样本空间S满足均匀性质,这样才能做以上的几何机率。
几何测量—长度所谓长度的几何测量,表示其无穷样本空间可用一线段、数线或是时间轴⋯等表示,则考虑某事件的机率时,只需探讨此事件所占的线段(或数线)与样本空间相对的长度比值即可,下列为利用长度测量来求的机率问题。
例1.(1972 AMC12 #17) 随机将一条线切为两段,试问较长的一段至少是较短的一段的x倍(其中x¸1)的机率为多少?•(B)2x•(C)1x+1•(D)1x•(E)2x+1解:(E)假设线段AB被切为两段,若较长一段(标记为l1,长度为sx)为较短一段(标记为l2,长度为s)的x倍,则线段AB总长度为(x+1)s,因此切点会落于l2中的机率为\dfrac{1}{x+1}。
但因线段的切法可能为(l1,l2)或是(l2,l21)两种情况,如下图故所求机率为2x+1。
例2.(2007 AMC12B #13)有一交通号志以下列的循环重复的运作:绿灯30秒,然后黄灯3秒,之后再转红灯30秒。
利亚随机挑选三秒钟的区间去注视号志灯,试问号志灯在转换颜色时,利亚正在注视的机率为多少?•(A)163•(B)121•(C)110•(D)17解:(D)由题意知,交通号志运作一循环需时63秒,而若利亚所注视的时间在当绿灯转变为黄灯、黄灯转变为红灯或红灯转变为绿灯的前三秒钟内,则利亚会看到号志颜色正在转变,如下图所示故所求机率为3+3+363=963=17例3.(2009 AMC12B #18) 瑞吉儿与罗伯特在一圆形的跑道上跑步,其中瑞吉儿以逆时钟方向跑且跑完一圈需时90秒,而罗伯特以顺时钟方向跑且跑完一圈需时80秒。
美国数学竞赛概况概要
http://bjmaths.blog.sohu.c om/entry/
AMC的历史概况
1950年Mathematics Association of America 的一个分支机构在创办了针对高中学生的数学竞 赛活动——American High School Mathematics Examination （AHSME） （最 初是50个选择题，1974年改为30个选择题， 2000年以后是25个选择题） 1985年又增加了针对初中学生的活动—— American Junior High School Mathematics Examination（AJHSME） 这个模式一直持续到1999年
六年级及六年级以下的同学成绩达 到一定的水平可以有一个鼓励证书
ห้องสมุดไป่ตู้
成绩位于所有参赛学生（15万人）的前 1%，则可以到证书（Honor Roll of Distinction）
成绩位于所有参赛学生（15万人） 的前5%，则可以到证书（Honor Roll）
在参赛学校中，成绩最高的学生 （可以并列）会得到一个证章 （Winner Pin）
分数排在参赛学校前三位的同学 （可以并列）将得到金、银、 铜证书： First Place Second Place Third Place
中国地区参加AMC大事记（1）
1980年代中期北京、上海、天 津等地都曾断断续续地参加过 美国的中学生数学竞赛—— AHSME活动，不过那时候是把 题目翻译成中文，那样题目对 中国学生们来讲就显得很容易 了。
中国地区参加AMC大事记（2）
2004年7月中旬，参加在希腊雅典举行的第45届 国际数学奥林匹克（IMO）活动期间，中国数学 会普及工作委员会（Popularization Committee of The Chinese Mathematical Society ）的代表与美国数学协会数学竞赛委员 会（Committee on American Mathematics Competitions of The Mathematical Association of America）的代表交流了各自国 内数学竞赛以及IMO代表队选拔的情况，并就组 织部分地区的中国学生参加美国数学协会（MAA） 主办的AMC（American Mathematics Competitions）系列活动达成了初步意向。 2005年8月下旬，美国数学协会（MAA）数学竞 赛委员会的代表来京出席会议期间，将以上意向 具体化。
AMC中文试题和答案
卤味面 起司肉燥面水饭 意大利面 人 數 A 3BCDE3 62007年 美国AMC8 (2007年11月 日 时间40分钟)1. 如果希瑞莎能够持续6周，平均每周花10小时帮忙照顾房子，她的父母就帮她买她喜爱乐 团的入场券。
在前五周她分别花了8、11、7、12及10小时照顾房子。
在最后一周，她必须 要花多少小时去照顾房子才能获得入场券？(A) 9 (B) 10 (C) 11 (D) 12 (E) 13 。
2. 调查650位学生对面食种类的偏好。
选项包含：卤味面、起司 肉燥面、水饺、意大利面，调查结果如长条图所示。
试问偏好 意大利面的学生数与偏好起司肉燥面的学生数之比值为多少？ (A) 52 (B) 21 (C) 45 (D) 35 (E) 25。
3. 250的最小两个质因子之和为多少？(A) 2 (B) 5 (C) 7 (D) 10 (E) 12 。
4. 某间鬼屋有六个窗子。
小精灵乔治从一个窗子进入屋内，而从不同的另一个窗子出来的方法 共有多少种？(A) 12 (B) 15 (C) 18 (D) 30 (E) 36 。
5. 姜德想买一辆价值美金500元的越野脚踏车。
在他生日时，祖父母给他美金50元，姑姑给 他美金35元，表哥给他美金15元。
他送报纸每周可赚美金16元。
若用他生日得到的所有礼 金及送报纸所有赚得的钱去买越野脚踏车，他需要送几周的报纸才能有足够的钱？ (A) 24 (B) 25 (C) 26 (D) 27 (E) 28 。
6. 在1985年美国的长途电话费是每分钟41分钱，在2005年的长途电话费是每分钟7分钱。
试求每分钟长途电话费下降的百分率最接近下列哪一项？ (A) 7 (B) 17 (C) 34 (D) 41 (E) 80 。
7. 房间内5个人的平均年龄为30岁。
若其中一位18岁的人离开了房间，则剩下四个人的平均 年龄是几岁？ (A) 25 (B) 26 (C) 29 (D) 33 (E) 36 。
美国数学邀请赛2007AMC10试题及中文翻译
What is
(E)
6@2 中文：定义 a @ b=ab-b2 ，a # b=a + b - ab2。求 的值。 6#2 3. An aquarium has a rectangular base that measures 100cm by 40cm and has a height of 50cm. It is filled with water to a height of 40cm. A brick with a rectangular base that measures 40cm by 20cm and a height of 10cm is placed in the aquarium. By how many centimeters does that water rise? (A) 0.5 (B) 1 (C) 1.5 (D) 2 (E)2.5 中文：一个养鱼缸有 100cm×40cm 的底，高为 50cm。它装满水到 40cm 的高度。把一 个底为 40cm×20cm，高为 10cm 的砖块放在这个养鱼缸里。鱼缸里的水上升了多少厘 米？ 4. The larger of two consecutive odd integers is three times the smaller. What is their sum? (A) 4 (B) 8 (C) 12 (D) 16 (E) 20 中文：2 个连续的奇整数中较大的数是较小的数的 3 倍。它们的和是多少？ 5. A school store sells 7 pencils and 8 notebooks for $4.15. It also sells 5 pencils and 3 notebooks for $1.77. How much do 16 pencils and 10 notebooks cost? (A) $4.76 (B) $5.84 (C) $6.00 (D) $6.16 (E) $6.32 中文：在一个学校商店花 4.15 美元可以买到 7 支铅笔和 8 个笔记本，花 1.77 美元可以 买到 5 个铅笔和 3 个笔记本。问 16 支铅笔和 10 个笔记本需要多少钱？ 6. At Euclid High School, the number of students taking the AMC10 was 60 in 2002, 66 in 2003, 70 in 2004, 76 in 2005, and 78 in 2006, and is 85 in 2007. Between what two consecutive years was there the largest percentage increase? (A) 2002 and 2003 (B) 2003 and 2004 (C) 2004 and 2005
精选美国amc中文版试题及答案
2000到2012年A M C10美国数学竞赛P 0 A 0 B 0 CD 0 全美中学数学分级能力测验(AMC 10)2000年 第01届 美国AMC10 (2000年2月 日 时间75分钟)1. 国际数学奥林匹亚将于2001年在美国举办，假设I 、M 、O 分别表示不同的正整数，且满足I ?M ?O =2001，则试问I ?M ?O 之最大值为 。
(A) 23 (B) 55 (C) 99 (D) 111 (E) 6712. 2000(20002000)为 。
(A) 20002001 (B) 40002000 (C) 20004000 (D) 40000002000 (E) 200040000003. Jenny 每天早上都会吃掉她所剩下的聪明豆的20%，今知在第二天结束时，有32颗剩下，试问一开始聪明豆有 颗。
(A) 40 (B) 50 (C) 55 (D) 60 (E) 754. Candra 每月要付给网络公司固定的月租费及上网的拨接费，已知她12月的账单为12.48元，而她1月的账单为17.54元，若她1月的上网时间是12月的两倍，试问月租费是 元。
5. 如图M ，N 分别为PA 与PB 之中点，试问当P 在一条平行AB 的直 在线移动时，下列各数值有 项会变动。
(a) MN 长 (b) △PAB 之周长 (c) △PAB 之面积 (d) ABNM 之面积 (A) 0项 (B) 1项 (C) 2项 (D) 3项 (E) 4项 6. 费氏数列是以两个1开始，接下来各项均为前两项之和，试问在费氏数列各项的个位数字中， 最后出现的阿拉伯数字为 。
(A) 0 (B) 4 (C) 6 (D) 7 (E) 97. 如图，矩形ABCD 中，AD =1，P 在AB 上，且DP 与DB 三等分图3 图2 图1 图0 ?ADC ，试问△BDP 之周长为 。
(A) 3?33 (B) 2?334 (C) 2?2 (D) 2533+ (E) 2?335 8. 在奥林匹克高中，有52的新生与54的高二生参加AMC 10年级测验。
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What is
(E)
6@2 中文：定义 a @ b=ab-b2 ，a # b=a + b - ab2。求 的值。 6#2 3. An aquarium has a rectangular base that measures 100cm by 40cm and has a height of 50cm. It is filled with water to a height of 40cm. A brick with a rectangular base that measures 40cm by 20cm and a height of 10cm is placed in the aquarium. By how many centimeters does that water rise? (A) 0.5 (B) 1 (C) 1.5 (D) 2 (E)2.5 中文：一个养鱼缸有 100cm×40cm 的底，高为 50cm。它装满水到 40cm 的高度。把一 个底为 40cm×20cm，高为 10cm 的砖块放在这个养鱼缸里。鱼缸里的水上升了多少厘 米？ 4. The larger of two consecutive odd integers is three times the smaller. What is their sum? (A) 4 (B) 8 (C) 12 (D) 16 (E) 20 中文：2 个连续的奇整数中较大的数是较小的数的 3 倍。它们的和是多少？ 5. A school store sells 7 pencils and 8 notebooks for $4.15. It also sells 5 pencils and 3 notebooks for $1.77. How much do 16 pencils and 10 notebooks cost? (A) $4.76 (B) $5.84 (C) $6.00 (D) $6.16 (E) $6.32 中文：在一个学校商店花 4.15 美元可以买到 7 支铅笔和 8 个笔记本，花 1.77 美元可以 买到 5 个铅笔和 3 个笔记本。问 16 支铅笔和 10 个笔记本需要多少钱？ 6. At Euclid High School, the number of students taking the AMC10 was 60 in 2002, 66 in 2003, 70 in 2004, 76 in 2005, and 78 in 2006, and is 85 in 2007. Between what two consecutive years was there the largest percentage increase? (A) 2002 and 2003 (B) 2003 and 2004 (C) 2004 and 20and 125 =5 . What is ab? (E) 60
b a-3
b
a-3
和 125 =5 ，问 ab 的值是多少？
10. The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is 20, the father is 48 years old, and the average age of the mother and children is 16. How many children are in the family? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 中文：Dunbar 一家的家庭成员包括爸爸，妈妈和几个孩子。他们的平均年龄是 20 岁， 其中爸爸 48 岁，妈妈和几个孩子的平均年龄是 16 岁。问家中有多少个孩子？ 11. The numbers from 1 to 8 are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum? (A) 14 (B) 16 (C) 18 (D) 20 (E) 24 中文：数字 1 到 8 被放在立方体的 8 个顶点上。使每面上的四个顶点代表的数字和相 同。问每面上的四个顶点的数字和是多少？ 12. Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?
共 6 页 第 1 页
8th AMC 10 A
2007
(D) 2005 and 2006 (E) 2006 and 2007 中文：在 Euclid 高中，2002 年参加 AMC10 的学生有 60 人，2003 年有 66 人，2004 年 有 70 人，2005 年有 76 人，2006 年有 78 人，2007 年有 85 人。在相邻的哪两年中人数 增长的百分率最大？ 7. Last year Mr. John Q. Public received an inheritance. He paid 20% in federal taxes on the inheritance, and paid 10% of what he had left in state taxes. He paid a total of $10,500 for both taxes. How many dollars was the inheritance? (A) 30,000 (B) 32,500 (C) 35,000 (D) 37,500 (E) 40,000 中文：去年 Mr. John Q. Public 收到了一笔遗产。其中 20%用于交联邦税，剩下钱中的 10%用于交州税。交这两种税款一共需要 10,500 美元。问这笔遗产共多少美元？ 8. Triangles ABC and ADC are isosceles with AB=AC and AD=DC. Point D is inside △ABC, ∠ABC=40°, and ∠ADC=140°. What is the degree measure of ∠BAD? (A) 20 (B) 30 (C) 40 (D) 50 (E) 60 中文：△ABC 和△ADC 是等腰三角形且 AB=AC，AD=DC。点 D 在△ABC 内，∠ ABC=40°, ∠ADC=140°。问∠BAD 的度数是多少？ 9. Real numbers a and b satisfy the equations 3 =81 (A)－60 (B)－17 (C) 9 (D) 12 中文：实数 a 和实数 b 满足等式 3 =81
(A) 32
(B) 22+12 2
(C) 16+16 3
(D) 48
(E) 36+16 2
中文：如图所示，四个半径是 1 的圆分别与正方形的两边相切且与一个半径是 2 的圆 外切，这个正方形的面积是多少？ 16. Integers a, b, c, and d, not necessarily distinct, are chosen independently and at random
8th AMC 10 A 8th AMC 10 A
2007 2007
1. One ticket to a show costs $20 at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan? (A) 2 (B) 5 (C) 10 (D) 15 (E) 20 中文：一张展览票全价为 20 美元。Susan 用优惠券买 4 张票打七五折。Pam 用优惠券 买 5 张票打七折。Pam 比 Susan 多花了多少美元？ 2. Define a@b=ab - b2 and a#b=a + b - ab2. 1 (A)－ 2 1 (B)－ 4 (C) 1 8 (D) 1 4 6@2 ? 6#2 1 2
中文：Yan 站在他家和运动场之间，要到运动场。他可以直接走路到运动场，或他走路 回 家再沿原路径骑自行车到运动场。骑自行车的速率是走路速率的 7 倍，两种方法所花 的时间都相同。问 Yan 站的位置到他家的距离与到运动场的距离之比值是多少？ 14. A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 3. What is the area of the triangle? (A) 8.64 (B) 12 (C) 5π (D) 17.28 (E) 18 中文：一个三角形的外接圆半径是 3，且其三边长之比是 3:4:5。问此三角形的面积是 多 少？ 15. Four circles of radius 1 are each tangent to two sides of a square and externally tangent to a circle of radius 2, as shown. What is the area of the square?