2011AMC10美国数学竞赛A卷附中文翻译和答案

第10届美国数学邀请赛(AIME)试题及解答
刘智
【期刊名称】《中等数学》
【年(卷),期】1992(000)005
【摘要】1.既约分数的分母为30,求小于10的所有这样的正有理数的和. 2.一个最少为两位数的十进制正整数,它的每一位数总比其右边位置上的数小,我们把这样的正整数叫做“上升数”.求上升数有多少个? 3.把一个球员赢球的场数除以他参赛的总场数叫做他的赢球率.一个球员在第一阶段的比赛中赢球率为0.5,第二阶段他比赛
【总页数】5页(P23-26,34)
【作者】刘智
【作者单位】北京大学;学生
【正文语种】中文
【中图分类】G633.6
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美国数学竞赛AMC8 -- 2010年真题解析(英文解析+中文解析)Problem 1Answer: CSolution:Given that these are the only math teachers at Euclid Middle School and we are told how many from each class are taking the AMC 8, we simply add the three numbers to find the total.11+8+9=28.中文解析:参加竞赛的学生总人数是：11+8+9=28. 答案是C。

Problem 2Answer: DSolution:Substitute a=5, b=10 into the expression for a@b to get: 5@10=(5*10)/(5+10)=50/15=10/3.中文解析：（5*10）/（5+10）=50/15=10/3. 答案是D。

Problem 3Answer: CSolution:The highest price was in Month 1, which was $17. The lowest price was in Month 3, which was $10. 17 is 17/10 =170% of 10, and is 170-100=70% more than 10. Therefore, the answer is 70. 中文解析：最高价是1月，17美元。

最低价格是3月10美元。

最高价比最低价多：（17-10）/10=70%。

答案是C。

Problem 4Answer: CSolution:Putting the numbers in numerical order we get the list 0,0,1,2,3,3,3,4 The mode is 3, The median is (2+3)/2=2.5. The average is 2. is The sum of all three is 3+2.5+2=7.5.中文解析：这组数按照从小到大的顺序排列是：0，0，1，2，3，3，3，4. 中位数Median是2.5；mode 是3，mean是16/8=2. 因此mean,median,mode的和是: 2.5+3+2=7.5. 答案是C。

卤味面 起司肉燥面 水饭 意大利面人 数 A 3 BC DE 36 2007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 。

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 。

2011AMC10美国数学竞赛A卷1. A cell phone plan costs $20 each month, plus 5¢ per text message sent, plus 10¢ for each minute used over 30 hours. In January Michelle sent 100 text messages and talked for 30.5 hours. How much did she have to pay?(A) $24.00 (B) $24.50 (C) $25.50 (D) $28.00 (E) $30.002. A small bottle of shampoo can hold 35 milliliters of shampoo, Whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?(A) 11 (B) 12 (C) 13 (D) 14 (E) 153. Suppose [a b] denotes the average of a and b, and {a b c} denotes the average of a, b, and c. What is {{1 1 0} [0 1] 0}?(A) 29(B)518(C)13(D) 718(E) 234. Let X and Y be the following sums of arithmetic sequences: X= 10 + 12 + 14 + …+ 100.Y= 12 + 14 + 16 + …+ 102.What is the value of Y X?(A) 92 (B) 98 (C) 100 (D) 102 (E) 1125. At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of 12, 15, and 10 minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?(A) 12 (B) 373 (C) 887 (D) 13 (E) 146. Set A has 20 elements, and set B has 15 elements. What is the smallest possible number of elements in A ∪B, the union of A and B?(A) 5(B) 15 (C) 20 (D) 35 (E) 3007. Which of the following equations does NOT have a solution?(A)2(7)0x +=(B) -350x += (C) 20= (D)80= (E) -340x -=8. Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese?(A) 20(B) 30 (C) 40 (D) 50 (E) 609. A rectangular region is bounded by the graphs of the equations y=a, y=-b, x=-c, and x=d, where a, b, c, and d are all positive numbers. Which of the following represents the area of this region?(A) ac + ad + bc + bd (B) ac – ad + bc – bd (C) ac + ad – bc – bd(D) –ac –ad + bc + bd (E) ac – ad – bc + bd10. A majority of the 20 students in Ms. Deameanor’s class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than 1. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $17.71. What was the cost of a pencil in cents?(A) 7(B) 11 (C) 17 (D) 23 (E) 7711. Square EFGH has one vertex on each side of square ABCD. Point E is on AB with AE=7·EB. What is the ratio of the area of EFGH to the area of ABCD?(A)4964 (B) 2532 (C) 78 (D) (E)12. The players on a basketball team made some three-point shots, some two-point shots, some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team’s total score was 61 points. How many free throws did they make?(A) 13(B) 14 (C) 15 (D) 16 (E) 1713. How many even integers are there between 200 and 700 whose digits are alldifferent and come from the set {1, 2, 5, 7, 8, 9}?(A) 12(B) 20 (C) 72 (D) 120 (E) 20014. A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle’s circumference? (A)136 (B) 112 (C) 16 (D) 14 (E) 51815. Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first 40 miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of 0.02 gallons per mile. On the whole trip he averaged 55 miles per gallon. How long was the trip in miles?(A) 140(B) 240 (C) 440 (D) 640 (E) 84016. Which of the following in equal to(A)(B) (C) 2 (D) (E) 617. In the eight-term sequence A, B, C, D, E, F, G , H, the value of C is 5 and the sum of any three consecutive terms is 30. What is A + H?(A) 17(B) 18 (C) 25 (D) 26 (E) 4318. Circles A, B, and C each have radius 1. Circles A and B share one point oftangency. Circle C has a point of tangency with the midpoint of AB. What is the area inside Circle C but outside Circle A and Circle B? (A) 32π- (B) 2π (C) 2 (D) 34π (E) 12π+19. In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town’s popu lation during this twenty-year period?(A) 42(B) 47 (C) 52 (D) 57 (E) 6220. Two points on the circumference of a circle of radius r are selected independently and at random. From each point a chord of length r is drawn in a clockwise direction. What is the probability that the two chords intersect? (A)16 (B) 15 (C) 14 (D) 13 (E) 1221. Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine?(A) 711(B) 913(C) 1115(D) 1519(E) 151622. Each vertex of convex pentagon ABCDE is to be assigned a color. There are 6 colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?(A) 2500 (B) 2880 (C) 3120 (D) 3250 (E) 375023. Seven students count from 1 to 1000 as follows:·Alice says all the numbers, except she skips the middle number in each consecutive group of thre e numbers. That is Alice says 1, 3, 4, 6, 7, 9, …, 997, 999, 1000.·Barbara says all of the numbers that Alice doesn’t say, except she also skips the middle number in each consecutive grope of three numbers.·Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers. ·Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.·Finally, George says the only number that no one else says.What number does George say?(A) 37 (B) 242 (C) 365 (D) 728 (E) 99824. Two distinct regular tetrahedra have all their vertices among the vertices of thesame unit cube. What is the volume of the region formed by the intersection of the tetrahedra?(A)112 (B) (C) (D) 16 (E)25. Let R be a square region and 4n an integer. A point X in the interior of R is called n-ray partitional if there are n rays emanating from X that divide R into N triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?(A) 1500(B) 1560 (C) 2320 (D) 2480 (E) 25002011AMC10美国数学竞赛A 卷1. 某通讯公司手机每个月基本费为20美元, 每传送一则简讯收 5美分（一美元=100 美分）。

AMC 中的数论问题1：Remember the prime between 1 to 100：2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 91 2：Perfect number:Let is the prime number.if 21p - is also the prime number. then 1(21)2pp --is the perfectnumber. For example:6，28，496。

3: Let ,0n abc a =≠ is three digital integer 。

if 333n a b c =++Then the number n is called Daffodils number. There are only four numbers ： 153 370 371 407Let ,0n abcd a =≠ is four digital integer 。

if 4444d n a b c +=++ Then the number n is called Roses number. There are only three numbers: 1634 8208 94744:The Fundamental Theorem of ArithmeticEvery natural number can be written as a product of primes uniquely up to order.5：Suppose that a and b are integers with b =0. Then there exists unique integers q and r such that 0 ≤ r< ｜b ｜ and a = bq + r.6:（1)Greatest Common Divisor: Let gcd (a, b) = max {d ∈ Z： d | a and d | b}。

一、填空题（每小题8分，共64分）1．设集合},,,{4321a a a a A =，若A 中所有三元子集的三个元素之和组成的集合为}8,5,3,1{-=B ，则集合=A ．2．函数11)(2-+=x x x f 的值域为 ． 3．设b a ,为正实数，2211≤+ba ，32)(4)(ab b a =-，则=b a log ． 4．如果)cos (sin 7sin cos 3355θθθθ-<-，)2,0[πθ∈，那么θ的取值范围是 ．二、解答题（本大题共3小题，共56分）9．（16分）设函数|)1lg(|)(+=x x f ，实数)(,b a b a <满足21()(++-=b b f a f ，2lg 4)21610(=++b a f ，求b a ,的值．10．（20分）已知数列}{n a 满足：∈-=t t a (321R 且)1±≠t ，121)1(2)32(11-+--+-=++nn n n n n t a t t a t a ∈n (N )*． （1）求数列}{n a 的通项公式；（2）若0>t ，试比较1+n a 与n a 的大小．11．（本小题满分20分）作斜率为31的直线l 与椭圆C ：143622=+y x 交于B A ,两点（如图所示），且)2,23(P 在直线l 的左上方．（1）证明：△PAB 的内切圆的圆心在一条定直线上；（2）若︒=∠60APB ，求△PAB 的面积．加 试1. （40分）如图，Q P ,分别是圆内接四边形ABCD 的对角线BD AC ,的中点．若DPA BPA ∠=∠，证明：CQB AQB ∠=∠．2. （40分）证明：对任意整数4≥n ，存在一个n 次多项式0111)(a x a x a x x f n n n ++++=--具有如下性质：4.（50分）设A 是一个93⨯的方格表，在每一个小方格内各填一个正整数．称A 中的一个)91,31(≤≤≤≤⨯n m n m 方格表为“好矩形”，若它的所有数的和为10的倍数．称A 中的一个11⨯的小方格为“坏格”，若它不包含于任何一个“好矩形”．求A 中“坏格”个数的最大值。

AMC 中的数论问题1：Remember the prime between 1 to 100：2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 91 2：Perfect number: Letis the prime number.if21p - is also the prime number. then1(21)2p p --is the perfect number. For example:6,28,496.3: Let ,0n abc a =≠ is three digital integer .if 333n a b c =++Then the number n is called Daffodils number . There are only four numbers: 153 370 371 407 Let ,0n abcd a =≠ is four digital integer .if 4444d n a b c +=++Then the number n is called Roses number . There are only three numbers: 1634 8208 94744：The Fundamental Theorem of ArithmeticEvery natural number can be written as a product of primes uniquely up to order.5：Suppose that a and b are integers with b =0. Then there exists unique integers q and r such that 0 ≤ r< |b| and a = bq + r.6：(1)Greatest Common Divisor: Let gcd (a, b) = max {d ∈ Z: d | a and d | b}. For any integers a and b, we havegcd(a, b) = gcd(b, a) = gcd(±a, ±b) = gcd(a, b − a) = gcd(a, b + a). For example: gcd(150, 60) = gcd(60, 30) = gcd(30, 0) = 30(2)Least common multiple:Let lcm(a,b)=min{d∈Z: a | d and b | d }. (3)We have that: ab= gcd(a, b) lcm(a,b) 7：Congruence modulo nIf ,0a b mq m -=≠,then we call a congruence b modulo m and we rewritemod a b m ≡.(1)Assume .Ifthen we havemod a c b d m ±≡± , mod ac bd m ≡ , mod k k a b m ≡(2) The equation ax ≡ b (mod m) has a solution if and only if gcd(a, m) divides b.8：How to find the unit digit of some special integers (1)How many zero at the end of !nFor example, when 100n =, Let N be the number zero at the end of 100!then10010010020424525125N ⎡⎤⎡⎤⎡⎤=++=+=⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦(2) ,,a n Z ∈Find the unit digit na . For example, when 100,3n a ==9：Palindrome, such as 83438, is a number that remains the same when its digits are reversed.Therearesomenumber not only palindrome but(1)Some special palindrome n that 2n is also palindrome. For example :222221111121111123211111123432111111111112345678987654321=====(2)How to create a palindrome? Almost integer plus the number of its reverseddigits and repeat it again and again. Then we get a palindrome. For example:87781651655617267266271353135335314884+=+=+=+=But whether any integer has this Property has yet to prove(3) The palindrome equation means that equation from left to right and right to left it all set up.For example ：1242242112231132211121241388888831421211⨯=⨯⨯=⨯⨯==⨯Let ab and cde are two digital and three digital integers. If the digits satisfy the,,9a c b e d c e d ⨯=⨯=+≤, then ab cde edc ba ⨯=⨯.10: Features of an integer divisible by some prime number If n is even ，then 2|n一个整数n 的所有位数上的数字之和是3（或者9）的倍数，则n 被3（或者9）整除一个整数n 的尾数是零， 则n 被5整除一个整数n 的后三位与截取后三位的数值的差被7、11、13整除， 则n 被7、11、13整除一个整数n 的最后两位数被4整除，则n 被4整除 一个整数n 的最后三位数被8整除，则n 被8整除一个整数n 的奇数位之和与偶数位之和的差被11整除，则n 被11整除 11. The number Theoretic functions If 312123t rr rrt n p p p p =(1) {}12()#0:|(1)(1)(1)t n a a n r r r χ=>=+++(2)12222111222|()(1)(1)(1)t r r r t t t a nn a p p p p p p p p p δ==+++++++++∑(3){}11221111122()#:,gcd(,)1()()()ttr r r r r r t t n a N a n a n p p p p p p φ---=∈≤==---For example: 2(12)(23)(21)(11)6χχ=⋅=++= 22(12)(23)(122)(13)28δδ=⋅=+++=22(12)(23)(22)(31)4φφ=⋅=--=Exercise1. The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?(A) 4 (B) 5 (C) 6 (D) 7 (E) 83. For the positive integer n, let <n> denote the sum of all the positive divisors of n with the exception of n itself. For example, <4>=1+2=3 and <12>=1+2+3+4+6=16. What is <<<6>>>?(A) 6 (B) 12 (C) 24 (D) 32 (E) 368. What is the sum of all integer solutions to 21<(x-2)<25? (A) 10 (B) 12(C) 15(D) 19(E) 510 How many ordered pairs of positive integers (M,N) satisfy the equation6=6M N(A) 6(B) 7(C) 8(D) 9(E) 101. Let a and b be relatively prime integers with >>0a b and 333-73=(-)3a b a b . What is -a b ?(A) 1 (B) 2(C) 3(D) 4(E) 515．The figures 123,,F F F and 4F shown are the first in a sequence of figures. For 3n ≥, n F is constructed from -1n F by surrounding it with a square and placing one more diamond on each side of the new square than -1n F had on each side of its outside square. For example, figure 3F has 13 diamonds. How many diamonds are there in figure 20F ?18. Positive integers a, b, and c are randomly and independently selected withreplacement from the set {1, 2, 3,…, 2010}. What is the probability thatabc ab a ++ is divisible by 3?(A)13(B)2981(C)3181(D)1127(E)132724. Let ,a b and c be positive integers with >>a b c such that 222-b -c +=2011a ab and 222+3b +3c -3-2-2=-1997a ab ac bc . What is a ?(A) 249 (B) 250 (C) 251 (D) 252 (E)2535. In multiplying two positive integers a and b, Ron reversed the digits of the two-digit number a. His erroneous product was 161. What is the correct value of the product of a and b?(A) 116 (B) 161 (C) 204 (D) 214 (E) 22423. What is the hundreds digit of 20112011?(A) 1 (B) 4 (C) 5 (D) 6 (E) 9 9. A palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers x and x+32 are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x?(A) 20 (B) 21 (C) 22 (D) 23 (E) 2421. The polynomial 322010x ax bx -+- has three positive integer zeros. What is the smallest possible value of a?(A) 78 (B) 88 (C) 98 (D) 108 (E) 11824. The number obtained from the last two nonzero digits of 90! Is equal to n. What is n?(A) 12 (B) 32 (C) 48 (D) 52 (E) 6825. Jim starts with a positive integer n and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with n=55, then his sequence contain 5 numbers:55 55-72= 66-22= 22-12= 11-12= 0Let N be the smallest number for which Jim’s sequence has 8 numbers. What is the units digit of N?(A) 1 (B) 3 (C) 5 (D) 7 (E) 921．What is the remainder when 01220093+3+3++3is divided by 8?(A) 0(B) 1(C) 2(D) 4(E) 65．What is the sum of the digits of the square of 111,111,111? (A) 18 (B) 27(C) 45(D) 63(E) 8110064(A) 6 (B) 7 (C) 8 (D) 9 (E) 1024. Let 2200820082k =+. What is the units digit of 222k +? (A) 0(B) 1(C) 4(D) 6(E) 8AMC about algebraic problems一、Linear relations(1) Slope y-intercept form: y kx b =+ (k is the slope, b is the y-intercept) (2)Standard form: 0Ax By C ++=(3)Slope and one point 0000(,),()()P x y k slope y y k x x -=- (4) Two points 1122(,),(,)P x y P x y12121212y y y y y y x x x x x x ---==--- (5)x,y-intercept form: (,0),(0,),(0,0)1x y P a Q b a b a b≠≠+= 二、the relations of the two lines 111222:0,:0l A x B y C l A x B y C ++=++= (1) 1l ∥2l 122112210,0A B A B C B C B ⇔-=-≠ (1) 1l ⊥2l 12120A A B B ⇔-=三、Special multiplication rules:222223322332212211222112222()()()2()()()()()()(2)()((1)(1))(1)()n n n n n n n n n n n n n n a b a b a b a b a ab b a b a b a ab b a b a b a ab b a b a b a a b ab b n a b a b a a b ab b n is odd n a b c ab bc ac a b -----------=-+±=±+-=-+++=+-+-=-++++≥+=+-++-+->++=++⇔-22()()0b c c a a b c+-+-=⇔==四、quadratic equations and PolynomialThe quadratic equations 2(0)y ax bx c a =++≠ has two roots 12,x x then we has1212b c x x x x a a+=-=More generally, if the polynomial 121210nn n n n x a x a x a x a ---+++++=has n roots 123,,,,n x x x x ,then we have:1231122312123(1)n n n n n nx x x x a x x x x x x a x x x x a -++++=-++==-开方的开方、估计开方数的大小 绝对值方程Arithmetic Sequence123(1)(2)(3)()n m a a n d a n d a n d a n m d =+-=+-=+-==+-121321()()()()2222n n n m n m n n a a n a a n a a n a a s ---+++++=====1(1)2n n n ds na -=+If n=2k, then we have 1()n k k s k a a +=+ If n=2k+1, then we have 1n k s na += Geometric sequence123123n n n n m n m a a q a q a q a q ----=====1(1)1,1n n a q q s q-≠=-Some special sequence 1, 1, 2, 3, 5, 8,… 9,99,999,9999，… 1，11，111，1111，… Exercise4 .When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over?7 For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?21. Four distinct points are arranged on a plane so that the segments connecting them have lengths ,,,,, and . What is the ratio of to ?6. The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the two numbers?8. In a bag of marbles,35of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?13. An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, and then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?16. Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?24. Let ,a b and c be positive integers with >>a b c such that 222-b -c +=2011a ab and 222+3b +3c -3-2-2=-1997a ab ac bc . What is a ?(A) 249 (B) 250(C) 251(D) 252(E) 2531. What is 246135135246++++-++++? (A) -1(B) 536(C) 712(D)14760(E)43310. Consider the set of numbers {1, 10, 102, 103 (1010)}. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer?(A) 1 (B) 9 (C) 10 (D) 11 (E) 10125816x x +=-(A) -64 (B) -24 (C) -9 (D) 24 (E) 5764. Let X and Y be the following sums of arithmetic sequences: X= 10 + 12 + 14 + …+ 100. Y= 12 + 14 + 16 + …+ 102. What is the value of Y X -?(A) 92 (B) 98 (C) 100 (D) 102 (E) 1127. Which of the following equations does NOT have a solution? (A) 2(7)0x +=(B) -350x += (C) 20x --=(D) 80x -= (E) -340x -=16. Which of the following in equal to 962962-++? (A) 32(B) 26(C)722(D) 33 (E) 613. What is the sum of all the solutions of 2602x x x =--?(A) 32 (B) 60 (C) 92 (D) 120 (E) 12414. The average of the numbers 1, 2, 3… 98, 99, and x is 100x. What is x?(A)49101 (B)50101(C)12 (D)51101(E)509911. The length of the interval of solutions of the inequality 23a x b ≤+≤ is10. What is b-a?(A) 6 (B) 10 (C) 15 (D) 20 (E) 3013. Angelina drove at an average rate of 80 kph and then stopped 20 minutes for gas. After the stop, she drove at an average rate of 100 kph. Altogether she drove 250 km in a total trip time of 3 hours including the stop. Which equation could be used to solve for the time t in hours that she drove before her stop?(A) 880100()2503t t +-=(B) 80250t = (C) 100250t =(D) 90250t =(E) 880()1002503t t -+=21. The polynomial 32-2010x ax bx +- has three positive integer zeros. What is the smallest possible value of a?(A) 78 (B) 88 (C) 98 (D) 108 (E) 11815．When a bucket is two-thirds full of water, the bucket and water weigh kilograms. When the bucket is one-half full of water the total weight is kilograms. In terms of and , what is the total weight in kilograms when the bucket is full of water?13．Suppose thatand. Which of the following is equal tofor every pair of integers?16．Let ,,, and be real numbers with,, and. What is the sum of all possible values of?5. Which of the following is equal to the product?81216442008............481242004n n + (A) 251(B) 502(C) 1004(D) 2008 (E) 40167. The fraction 20082200622007220052(3)(3)(3)(3)-- simplifies to which of the following?(A) 1 (B) 9/4 (C) 3 (D) 9/2 (E) 913. Doug can paint a room in 5 hours. Dave can paint the same room in 7 hours. Doug and Dave paint the room together and take a one-hour break for lunch. Let t be the total time, in hours, required for them to complete the job working together, including lunch. Which of the following equations is satisfied by t ?(A) 11()(1)157t ++=(B) 11()1157t ++= (C) 11()157t +=(D) 11()(1)157t +-=(E) (57)1t +=15. Yesterday Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Han drove 70 miles more than Ian. How many more miles did Jan drive than Ian?(A) 120 (B) 130 (C) 140 (D) 150 (E) 160AMC 中的几何问题一、三角形有关知识点1.三角形的简单性质与几个面积公式①三角形任何两边之和大于第三边； ②三角形任何两边之差小于第三边； ③三角形三个内角的和等于180°； ④三角形三个外角的和等于360°；⑤三角形一个外角等于和它不相邻的两个内角的和； ⑥三角形一个外角大于任何一个和它不相邻的内角。