2010AMC10美国数学竞赛A卷

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) 8 (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 freethrows 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 all different 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 is30. 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 of tangency. 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 three 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 the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra?(A) 112 (B) 12 (C) 12 (D)16 (E) 625. 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）。

2009 AMC10美国数学竞赛A 卷1. One can holds 12 ounces of soda. What is the minimum number of cans needed to provide a gallon (128 ounces) of soda? (A) 7 (B) 8 (C) 9 (D) 10 (E) 112. Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes and quarters. Which of the following could not be the total value of the four coins, in cents? (A) 15 (B) 25 (C) 35 (D) 45 (E) 553. Which of the following is equal to 111111+++?(A) 54(B)32(C)53(D) 2 (E) 34. Eric plans to compete in a triathlon. He can average 2 miles per hour in the 1/4-mile swim and 6 miles per hour in the 3-mile run. His goal is to finish the triathlon in 2 hours. To accomplish his goal what must his average speed in miles per hour, be for the 15-mile bicycle ride? (A) 12011(B) 11 (C)565(D)454(E) 125. What is the sum of the digits of the square of 111,111,111? (A) 18 (B) 27(C) 45(D) 63(E) 816. A circle of radius 2 is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle’s area is shaded?(A) 12 (B)6π(C) 2π(D) 23(E) 3π7. A carton contains milk that is 2% fat, an amount that is 40% less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?(A) 125(B) 3 (C) 103(D) 38 (E) 428. Three Generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a 50% discount as children. The two members of the middle generation receive no discount. Grandfather Wen, whose senior ticket costs $6.00, is paying for everyone. How many dollars must he pay?(A) 34 (B) 36 (C) 42 (D) 46 (E) 489. Positive integers a, b and 2009, with 2009a b<<, from a gometric sequence with an integer ratio. What is a?(A) 7 (B) 41 (C) 49 (D) 289 (E) 200910. Triangle ABC has a right angle at B. Point D is the foot of the altitude from B, AD=3, and DC=4. What is the area of △ABC?43DCBA(A)(B)(C) 21(D) 1 (E) 4211. One dimension of a cube is increased by 1, another is decreased by 1, and the third is left unchanged. The volume of the new rectangular solid is 5 less than that of the cube. What was the volume of the cube? (A) 8 (B) 27 (C) 64 (D) 125 (E) 21612. In quadrilateral ABCD, AB=5, BC=17, CD=5, DA =9, and BD is an integer. What is BD?(A) 11 (B) 12 (C) 13 (D) 14 (E) 1513. Suppose that P=2m and Q=3n . Which of the following is equal to 12mn for every pair of integers (m,n)? (A) P 2Q(B) P n Q m(C) P n Q 2m(D) P 2m Q n (E) P 2n Q m14. Four congruent rectangles are placed as shown. The area of the outer square is 4times that of the inner square. What is the ratio of the length of the longer side ofBeach rectangle to the length of its shorter side? (A) 3 (B)(C)2+(D)(E) 415. The figures F 1, F 2, F 3, and F4 shown are the first in a sequence of figures. For3n ≥. F n is constructed from F n-1 by surrounding is with a square and placing onemore diamond on each side of the new square than F n-1 had on each side of its outside square. For example, figure F 3 has 13 diamonds. How many diamonds are there in figure F 20? (A) 401 (B) 485 (C) 585(D) 626(E) 76116. Let a, b, c, and d be real numbers with 2a b -=,3b c -= and4c d -=. What isthe sum of all possible values of a d-.?(A) 9 (B) 12(C) 15(D) 18 (E) 2417. Rectangle ABCD has AB=4 and BC=3. Segment EF is constructed through B so that BF is perpendicular to DB, and A and C lie on DE and DF, respectively. What is EF? (A) 9(B) 10(C) 125/12(D) 103/9(E) 12F4F3F2F118. At Jefferson Summer Camp, 60% of the children play soccer, 30% of the children swim, and 40% of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer?(A) 30% (B) 40% (C) 49% (D) 51% (E) 70%19. Circle A has radius 100. Circle B has an integer radius r<100 and remains internally tangent to circle A as it rolls once around the circumference of circle A. The two circles have the same points of tangency at the beginning and end of circle B’s trip. How many possible values can r have?(A) 4 (B) 8 (C) 9 (D) 50 (E) 9020. Andrea and Lauren are 20 kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of 1 kilometer per minute. After 5minites, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea?(A) 20 (B) 30 (C) 55 (D) 65 (E) 8021. Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a larger circle. In the figure shown, the number of smaller circles is four. What is the ratio of the sum of the areas of the foursmaller circles to the area of the larger circle? (A)3- (B)2-(C)43-（ (D)132-（ (E)2-22. Two cubical dice each have removable numbers 1 through 6. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is 7? (A) 19 (B)18(C)16(D)211(E)1523. Convex quadrilateral ABCD has AB=9 and CD=12. Diagonals AC and BD intersect at E, AC=14, and △ABD and △BEC have equal areas. What is AE? (A) 92 (B)5011(C)214(D)173(E) 624. Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube? (A) 14 (B)38(C)47(D)57(E)3425. For k>0, let I k =10…064, where there are k zeros between the 1 and the 6. Let N(k) be the number of factors of 2 in the prime factorization of I k . What is the maximum value of N(k)? (A) 6(B) 7(C) 8(D) 9 (E) 10。

2020年美国数学竞赛(AMC10A)的试题与解答广东省广州市华南师范大学(510631)李湖南1.What value of x satisfies x−34=512−13?A.−23B.736C.712D.23E.56译文方程x−34=512−13的解x是多少?解化简可得x=34+112=56,故(E)正确.2.The numbers3,5,7,a,and b have an average(arithmetic mean)of15.What is the average of a and b?A.0B.15C.30D.45E.60译文数字3,5,7,a和b的平均值(算术平均值)是15.则a和b的平均值是多少?解依题意有3+5+7+a+b=15×5,则a+b=60,平均值为30,故(C)正确.3.Assuming a=3,b=4and c=5,what is the value insimplest form of the following expression a−35−c·b−43−a·c−54−bA.−1B.1C.abc60D.1abc−160E.160−1abc译文设a=3,b=4,c=5,则表达式a−35−c·b−43−a·c−54−b的最简形式是什么?解约分即得a−35−c·b−43−a·c−54−b=−1,故(A)正确.4.A driver travels for2hours at60miles per hour,during which her car gets30miles per gallon of gasoline.She is paid $0.50per mile,and her only expense is gasoline at$2.00per gal-lon.What is her net rate of pay,in dollars per hour,after this expense?A.20B.22C.24D.25E.26译文一位司机以60英里/小时的速度驾车2小时,她的车每跑30英里需要消耗1加仑汽油.她能获得0.50美元/英里的报酬,唯一的花费就是2美元/加仑的汽油.问她每小时除去消耗之后的净收益是多少美元?解1个小时她能跑60英里,获得60×0.50=30美元,汽油费为60÷30×2=4美元,故净收益为26美元,(E)正确.5.What is the sum of all real numbers x for whichx2−12x+34=2?A.12B.15C.18D.21E.25译文满足方程x2−12x+34=2的所有实数x之和是多少?解分别解方程x2−12x+34=2和x2−12x+34=−2,可得x1,2=4,8和x3,4=6,故和为18,(C)正确.6.How many4-digit positive integers(that is,integers be-tween1000and9999,inclusive)having only even digits are di-visible by5?A.80B.100C.125D.200E.500译文有多少个四位的正整数(也就是在1000和9999之间的整数)能被5整除且所有数字均为偶数?解依题意,符合条件的四位数的个位数只能是0,十位数和百位数可以是0,2,4,6,8,千位数只能是2,4,6,8,共有1×5×5×4=100种选择,故(B)正确.7.The25integers from−10to14,inclusive,can be ar-ranged to form a5-by-5square in which the sum of the numbers in each row,the sum of the numbers in each column,the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?A.2B.5C.10D.25E.50译文将25个整数分别是从−10到14,放入5×5的格子中,使得格子里的每行、每列和两条对角线的数字和均相等.问这个数字和是多少?解这是一个5阶幻方问题,25个数字之和是(−10)+(−9)+···+13+14=50,分别放入5行,故每行的数字和是10,(C)正确.8.What is the value of1+2+3−4+5+6+7−8+···+197+198+199−200?A.9800B.9900C.10000D.10100E.10200译文1+2+3−4+5+6+7−8+···+197+198+199−200的值是多少?解原式=1+(2+3−4)+5+(6+7−8)+···+197+(198+199−200)=2×(1+5+···+197)=2×(1+197)×502=9900,故(B)正确.9.A single bench section at a school event can hold either7adults or11children.When N bench sections are connected end to end,an equal number of adults and children seated together will occupy all the bench space.What is the least possible posi-tive integer value of N?A.9B.18C.27D.36E.77译文在某学校的活动中,一条长凳可以坐7个成人或者11个儿童.当N条长凳首尾相接的时候,刚好坐满了相同数量的成人和儿童.问N的最小正整数值是多少?解设有x条长凳坐了儿童,则有N−x条长凳坐了成人,依题意有11x=7(N−x),因而xN−x=711⇒xN=718.故N min=18,(B)正确.10.Seven cubes,whose volumes are1,8,27,64,125, 216,and343cubic units,are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube,the bottom face of each cube lies completely on top of the cube below it.What is the total surface area of the tower(including the bottom)in square units?A.644B.658C.664D.720E.749译文七个立方体,体积分别是1,8,27,64,125,216,343个立方单位,依次按照体积大小由底到顶垂直地堆积成一座塔.除了最底部的立方体,每个立方体的底面都完全被下面的立方体的顶面覆盖.问这座塔的表面积(包括底面)是多少个平方单位?解这七个数都是立方数,则这七个立方体的棱长分别是1,2,3,4,5,6,7,从而塔的侧面积为4×(12+22+...+72)= 560,而上、下底面积之和为2×72=98,共658,故(B)正确.11.What is the median of the following list of4040num-bers?1,2,3,...,2020,12,22,32,...,20202A.1974.5B.1975.5C.1976.5D.1977.5E.1978.5译文下列4040个数:1,2,3,...,2020,12,22,32,..., 20202的中位数是多少?解由于442=1936,452=2025,从而以上数列按递增排列的话,就成为:1,12,...,4,22,...,1936,442,...,1976,1977,...,2020,452,462, (20202)此时,1976成为第2020个数,所求中位数为1976+19772=1976.5,故(C)正确.12.Triangle∆AMC is isosceles with AM=AC.Me-dians MV and CU are perpendicular to each other,and MV=CU=12.What is the area of∆AMC?A.48B.72C.96D.144E.192译文等腰∆AMC中,AM=AC,中线MV和CU互相垂直,且MV=CU=12.则∆AMC的面积是多少?解如图示,设MV交UC于点E,过A作AD⊥MC于D,交UV于F,则AD是MC的中垂线.依题意,U,V分别是AM,AC的中点,则UV是∆AMC的中位线,且UM=V C=12AM,从而∆UMC =∆V CM(SSS),可得∠UCM=∠V MC=45◦,因此图1EM=EC,点E在AD上.于是∆UV E ∆CME,且均是等腰直角三角形,因而UECE=UVCM=12⇒CE=23CU=8,进而CD=DE=4√2,EF=12DE=2√2,MC=2DC=8√2;又∆AUV ∆AMC,得AFAD=UVMC=12⇒AD=2DF=12√2.故S∆AMC=12MC·AD=8√2×12√22=96,(C)正确.13.A frog sitting at the point(1,2)begins a sequence ofjumps,where each jump is parallel to one of the coordinate ax-es and has length1,and the direction of each jump(up,down,left,right)is chosen independently at random.The sequence endswhen the frog reaches a side of the square with vertices(0,0),(0,4),(4,0),and(4,4).What is the probability that the sequence ofjumps ends on a vertical side of the square?A.12B.58C.23D.34E.78译文一只青蛙坐在点(1,2)上,开始一系列的跳跃,每次跳跃都平行于坐标轴且长度为1,方向(上、下、左、右)是随机的且独立,当青蛙到达由点(0,0),(0,4),(4,0),(4,4)构成的正方形的一条边的时候,跳跃终止.问跳跃终止于正方形竖直的两条边上的概率是多少?解如图示,青蛙在点F 1处,它可以向四个方向跳跃,概率均为14,向左跳跃,立刻达成目标;向上、向右、向下分别跳跃到点A 1,C,A 3处,再通过其它跳跃达成目标.根据对称性,青蛙由点A 1,A 2,A 3,A4图2出发达成目标的概率是一样的,设为a ;青蛙由点B 1,B 2出发达成目标的概率是一样的,设为b ;青蛙由点F 1,F 2出发达成目标的概率是一样的,设为x ;青蛙由点C 出发达成目标的概率设为c .因此,P (青蛙由F 1出发达成目标)=P (青蛙向左)+P (青蛙向上)×P (青蛙由A 1出发达成目标)+P (青蛙向右)×P (青蛙由C 出发达成目标)+P (青蛙向下)×P (青蛙由A 3出发达成目标),即有x =14+a 2+c4;同理,可得方程组x =14+a 2+c 4a =14+x 4+b 4b =a 2+c 4c =x 2+b 2成立,解得x =58,a =12,b =38,c =12.故(B)正确.14.Real numbers x and y satisfies x +y =4and xy =−2.What is the value of x +x 3y 2+y 3x2+y ?A.360B.400C.420D.440E.480译文实数x,y 满足方程x +y =4和xy =−2.则x +x 3y 2+y 3x2+y 的值是多少?解依题意可得x 2+y 2=(x +y )2−2xy =20,x 3+y 3=(x +y )(x 2−xy +y 2)=88,原式=x +y 3x 2+y +x 3y 2=x 3+y 3x 2+x 3+y 3y 2=88(x 2+y 2)x 2y 2=440.故(D)正确.15.A positive integer divisor of 12!is chosen at random.The probability that the divisor is a perfect square can be ex-pressed as mn,where m and n are relatively prime positive in-tegers.What is m +n ?A.3B.5C.12D.18E.23译文随机选取12!的一个正整数因子,该因子是一个完全平方数的概率可以表示为mn,其中m,n 为互素的正整数.则m +n 是多少?解由于12!=210×35×52×7×11,则12!有11×6×3×2×2=792个正因子;设k 是12!的平方因子,则k =2a ×3b ×5c ×7d ×11i ,其中a 10,b 5,c 2,d 1,i 1,且a,b,c,d,i 均为非负偶数,即a =0,2,4,6,8,10,b =0,2,4,c =0,2,d =i =0,此时k 有6×3×2=36种选择.从而所求概率为36792=122.故m +n =23,(E)正确.16.A point is chosen at random within the square in the co-ordinate plane whose vertices are (0,0),(2020,0),(2020,2020),and (0,2020).The probability that the point lies within d unitsof a lattice point is 12.(A point (x,y )is a lattice point if x and yare both integers.)What is d to the nearest tenth?A.0.3B.0.4C.0.5D.0.6E.0.7译文坐标平面上有一个以(0,0),(2020,0),(2020,2020)和(0,2020)为顶点的正方形.在正方形内随机选择一个点,该点位于格点的d 个单位内的概率是12.(点(x,y )称为格点,若x 和y 均为整数.)图3则d 精确到十分位是多少?解如图示,以格点为圆心,d 为半径作一些圆,则正方形内的圆内部分就是符合条件的点集.因此,该点落在此区域的概率为P =该区域的面积正方形面积,即12=(20192+2019×2+1)πd 220202=πd 2,求得d =√12π≈0.4,故(B)正确.17.Define P (x )=(x −12)(x −22)···(x −1002).How many integers n are there such that P (n ) 0?A.4900B.4950C.5000D.5050E.5100译文定义P (x )=(x −12)(x −22)···(x −1002).则有多少个整数n 使得P (n ) 0?解解不等式P (n )=(n −12)(n −22)···(n −1002) 0,得12 n 22,32 n 42,···,992 n 1002,因此符合条件的n 有(22−12+1)+(42−32+1)+···+(1002−992+1)=2×(2+4+···+100)=5100个.故(E)正确.18.Let (a,b,c,d )be an ordered quadruple of not necessar-ily distinct integers,each one of them in the set {0,1,2,3}.For how many such quadruples is it true that ad −bc is odd?(For ex-ample,(0,3,1,1)is one such quadruple,because 0·1−3·1=−3is odd.)A.48B.64C.96D.128E.192译文设(a,b,c,d)是一个四元数,其中a,b,c,d∈{0,1,2,3}且可以相同.则有多少个这样的四元数使得ad−bc是奇数?(例如,(0,3,1,1)就是一个符合条件的四元数,因为0·1−3·1=−3是奇数.)解要使得ad−bc是奇数,ad和bc必一奇一偶,分两种情况:(1)ad是奇数,bc是偶数:此时a,d∈{1,3},b,c可以是一奇一偶、一偶一奇、两个偶数,共有2×2×(2×2×3)=48种选择;(2)ad是偶数,bc是奇数:同理可得48种选择.故共有96个,(C)正确.19.As shown in the figure below,a regular dodecahedron (the polyhedron consisting of12congruent regular pentagonal faces)floats in space with two horizontal faces.Note that there is a ring of five slanted faces adjacent to the top face,and a ring of five slanted faces adjacent to the bottom face.How many ways are there to move from the top face to the bottom face via a se-quence of adjacent faces so that each face is visited at most once and moves are not permitted from the bottom ring to the top ring?A.125B.250C.405D.640E.810译文如下图左所示,一个正十二面体(由12个完全相同的正五边形面组成的多面体)放置在两个水平面的空间中.注意到顶面附近有一个由五个斜面组成的圆环,底面附近有一个由五个斜面组成的圆环.问有多少种方法可以通过一系列相邻面从顶面移动到底面,使得每个面最多经过一次,并且不允许从底环移动到顶环?图4图5解将该图简化成平面图:顶面为点T,底面为点B,顶环依次为点T1,T2,T3,T4,T5,底环依次为点B1,B2,B3,B4,B5,相邻的两个面用线段连接,如上图右所示.则原问题相当于:有多少条从点T出发,经过顶环,再经过底环,最后到达点B的不重复路径?(1)点T到顶环,有5种选择,即T1,T2,T3,T4,T5;(2)不妨设先到T1,又有9种选择到达底环,分别是:T1→,T1→T2→,T1→T2→T3→,T1→T2→T3→T4→,T1→T2→T3→T4→T5→,T1→T5→,T1→T5→T4→,T1→T5→T4→T3→,T1→T5→T4→T3→T2→;(3)每一个顶环上的点有2种方式到达底环,如T1→B1或T1→B5;(4)以到达B1为例,有以下9条路径到达点B,分别是:B1→B,B1→B2→B,B1→B2→B3→B,B1→B2→B3→B4→B,B1→B2→B3→B4→B5→B,B1→B5→B,B1→B5→B4→B,B1→B5→B4→B3→B,B1→B5→B4→B3→B2→B.综上分析可得,共有5×9×2×9=810条路径,故(E)正确.20.Quadrilateral ABCD satisfies∠ABC=∠ACD=90◦,AC=20and CD=30.Diagonals AC and BD inter-sects at point E and AE=5.What is the area of QuadrilateralABCD?A.330B.340C.350D.360E.370译文四边形ABCD满足∠ABC=∠ACD=90◦,AC=20,CD=30.对角线AC和BD交于点E,且AE=5.求四边形ABCD的面积是多少?解如图示,以AC为直径作一个圆,交BD与点F,图6依题意可得EC=15,ED=√EC2+CD2=15√5.设BE=x,依据相交弦定理AE·EC=BE·EF,则得EF=75x,DF=15√5−75x,DB=15√5+x;再由切割线定理DC2=DF·DB,得900=(15√5−75x)·(15√5+x),解得x=3√5或x=−5√5(舍去).而S∆ACD=12×20×30=300,S∆ABCS∆ACD=BEED=15,可得S∆ABC=60,故S ABCD=360,(D)正确.21.There exists a unique strictly increasing sequence ofnonnegative integers a1<a2<···<a k such that2289+1217+1=2a1+2a2+···+2a k.What is k?A.117B.136C.137D.273E.306译文存在唯一严格递增的非负整数列a1<a2<···<a k使得2289+1217+1=2a1+2a2+···+2a k,则k是多少?解令217=x ,则2289+1217+1=x 17+1x +1=x 16−x 15+x 14−x 13+···+x 2−x +1,而x 16−x 15=2272−2255=2271+2270+···+2255,同理x14−x13=2238−2221=2237+2236+···+2221,···,x 2−x =234−217=233+232+···+217,从而2189+1217+1=20+(217+···+232+233)+···+(2255+···+2270+2271)共8×17+1=137项,故(C)正确.22.For how many positive integers n 1000is ⌊998n ⌋+⌊999n ⌋+⌊1000n⌋not divisible by 3?(recall that ⌊x ⌋is the great-est integer less than or equal to x .)A.22B.23C.24D.25E.26译文有多少个正整数n 1000使得⌊998n ⌋+⌊999n ⌋+⌊1000n ⌋不被3整除?(注意⌊x ⌋表示小于等于x 的最大整数)解当n 不是998,999或1000的因子时,易得⌊998n ⌋=⌊999n ⌋=⌊1000n ⌋,从而N =⌊998n ⌋+⌊999n ⌋+⌊1000n⌋能被3整除;由于998=2×499,999=33×37,1000=23×53,只有1和2两个公因子,共有4+8+16−2−1=25个不同的因子,分情况讨论:(1)公因子:n =1时,N =2997能被3整除;n =2时,N =1498不能被3整除;(2)998的非公因子(2个):⌊998n ⌋=⌊999n ⌋=⌊1000n ⌋,N 能被3整除;(3)999的非公因子:N =⌊998n ⌋+⌊999n ⌋+⌊1000n⌋=3·999n −1不能被3整除;(4)1000的非公因子:N =⌊998n ⌋+⌊999n ⌋+⌊1000n⌋=3·1000n−2不能被3整除.综上可得,符合条件的n 有25−3=22个,故(A)正确.23.Let T be the triangle in the coordinate plane with ver-tices (0,0),(4,0),and (0,3).Consider the following five isome-tries (rigid transformations)of the plane:rotation of 90◦,180◦,and 270◦counterclockwise around the origin,reflection across the x -axis,and reflection across the y -axis.How many of the 125sequences of three of these transformations (not necessarily dis-tinct)will return T to its original position?(For example,a 180◦rotation,followed by a reflection across the x -axis,followed bya reflection across the y -axis will return T to its original position,but a 90◦rotation,followed by a reflection across the x -axis,fol-lowed by another reflection across the x -axis will not return T to its original position.)A.12B.15C.17D.20E.25译文设T 是坐标平面上以(0,0),(4,0)和(0,3)为顶点的三角形.考虑以下五种平面上的等距变换(刚体变换):绕原点作90◦,180◦和270◦的逆时针旋转,关于x 轴或y 轴的反射.任选三种变换(不必不同)可以组成125种组合,有图7多少种组合将使得T 变回起始位置?(例如,一个关于y 轴的反射,接着一个关于x 轴的反射,再接着一个180◦的旋转,将会使得T 变回起始位置;但一个关于x 轴的反射,接着另一个关于x 轴的反射,再接着一个90◦的旋转,将不会使得T 变回起始位置.)解分两种情况:(1)全部由旋转组成:只要三次旋转的角度和为360◦或720◦即可满足要求,因此有90◦+90◦+180◦,90◦+180◦+90◦,180◦+90◦+90◦,270◦+270◦+180◦,270◦+180◦+270◦,180◦+270◦+270◦共6种组合;(2)由旋转和反射组合而成:有y 轴+x 轴+180◦,y 轴+180◦+x 轴,180◦+x 轴+y 轴,180◦+y 轴+x 轴,x 轴+180◦+y 轴,x 轴+y 轴+180◦,也是6种组合.故(A)正确.24.Let n be the least positive integer greater than 1000for which gcd (63,n +120)=21and gcd (n +63,120)=60.What is the sum of digits of n ?A.12B.15C.18D.21E.24译文设n 是大于1000的使gcd (63,n +120)=21,gcd (n +63,120)=60成立的最小正整数.则n 的数字和是多少?解由gcd (63,n +120)=21,可得n ≡6(mod 21);由gcd (n +63,120)=60,可得n ≡57(mod 60).联立解得n ≡237(mod 420),于是n =1077,1497,1917,···.当n =1077时,gcd (63,n +120)=63,不符;当n =1497时,gcd (n +63,120)=120,也不符;当n =1917时,验证后符合条件,此时数字和为18,故(C)正确.例谈伸缩变换在解高考题中的应用广东省佛山市高明区教师发展中心(528500)张文玲一、定义引出在高中数学选修4-4第一讲中,有如下定义:设点P (x,y )是平面直角坐标系中的任意一点,在变换φ: x ′=λx (λ>0)y ′=µy (µ>0)的作用下,点P (x,y )对应到点P ′(x ′,y ′),称φ为平面直角坐标系中的坐标伸缩变换,简称伸缩变换.在它的作用下,可以实现平面图形的伸缩.对于椭圆x 2a 2+y 2b 2=1(a >b >0),在变换φ:x ′=xy ′=my(简言之,横坐标不变,纵坐标拉伸,m 称为伸缩率)的作用下,变为圆x ′2+y ′2=a 2,其中m =ab[1].在此变换下,原来直角坐标系xOy 中的点A (x 1,y 1),B (x 2,y 2),C (x 3,y 3)对应变为直角坐标系x ′O ′y ′中的点A ′(x 1′,y 1′),B ′(x 2′,y 2′),C ′(x 3′,y 3′),且有如下结论:1⃝若点A,B,C 三点共线,则点A ′,B ′,C ′三点也共线[1].2⃝若A,B,C 三点共线且|AB |=λ|BC |(λ>0),则变换之后|A ′B ′|=λ|B ′C ′|(λ>0).故若点B 为线段AC 的中点,则点B ′为线段A ′C ′的中点[1].3⃝若直线AB 的斜率为k ,则直线A ′B ′的斜率为mk [1].故两条平行直线经变换后仍然平行.4⃝两封闭图形的面积之比在变换前后不变.在圆中有很多优美的性质,将椭圆伸缩变换为圆之后,就可以利用圆的性质来解决一些问题,简化了计算,使学生不再“望椭圆而生畏”.下面从圆的四个常见性质出发来解决一些涉及到椭圆的高考题目.二、应用举例(一)直径所对的圆周角是直角例1(2019年高考全国Ⅱ卷理科第21题)已知点A (−2,0),B (2,0),动点M (x,y )满足直线AM 与BM 的斜率之积为−12.记M 的轨迹为曲线C .25.Jason rolls three fair standard six-sided dice.Then he looks at the rolls and chooses a subset of the dice (possibly emp-ty,possibly all three dice)to reroll.After rerolling,he wins if and only if the sum of the numbers faces up on the three dice is exactly 7.Jason always plays to optimize his chances of winning.What is the probability that he chooses to reroll exactly two of the dice?A.736B.524C.29D.1772E.14译文詹森掷3颗标准、均匀的骰子,他看了结果之后会选择若干(可能是0,也可能是3)颗重掷.当3颗骰子正面朝上的数字和为7点的时候,他就赢了.詹森总是按照朝着他赢的最优策略去掷.问他刚好选择2颗骰子重掷的概率是多少?解掷1颗骰子得1,2,3,4,5,6点的概率均为16;掷2颗骰子得3点只有两种情况:12和21,概率为236,···;掷3颗骰子得7点有15种情况:115,151,511,124,142,214,241,412,421,133,313,331,223,232,322,概率为15216,···.经过计算,所有结果如下表所示:分类/概率/结果1234567掷1颗161616161616掷2颗136236336436536掷3颗1216321632161021615216因此,詹森要选择2颗骰子重掷,则上次掷的结果中,任意两颗骰子的数字和不能小于7点,否则他将选择重掷1颗骰子;且不能3颗骰子都是4点或者以上,要不然他将选择重掷3颗骰子.根据以上分析,满足条件的情况有:(1)掷出1点、6点、6点,3种情况;(2)掷出2点、5点、5点,3种情况;(3)掷出2点、5点、6点,6种情况;(4)掷出2点、6点、6点,3种情况;(5)掷出3点、4点、4点,3种情况;(6)掷出3点、4点、5点,6种情况;(7)掷出3点、4点、6点,6种情况;(8)掷出3点、5点、5点,3种情况;(9)掷出3点、5点、6点,6种情况;(10)掷出3点、6点、6点,3种情况.由加法原理,共42种情况,故所求概率为42216=736,(A)正确.。

2010 AMC 12-AProblem 1What is ?Solution.Problem 2A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day?SolutionIt is easy to see that the ferry boat takes trips total. The total number of people taken to the island isProblem 3Rectangle , pictured below, shares of its area with square . Square shares of its area with rectangle . What is ?SolutionIf we shift to coincide with , and add new horizontal lines to divide into fiveequal parts:This helps us to see that and , where . Hence .Problem 4If , then which of the following must be positive?Solutionis negative, so we can just place a negative value into each expression and find the onethat is positive. Suppose we use .Obviously only is positive.Problem 5Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8, 4, 2, and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea's next shots are bullseyes she will be guaranteed victory. What is the minimum value for ?SolutionLet be the number of points Chelsea currently has. In order to guarantee victory, we must consider the possibility that the opponent scores the maximum amount of points by getting only bullseyes.The lowest integer value that satisfies the inequality is .Problem 6A , such as 83438, is a number that remains the same when its digits are reversed. The numbers and are three-digit and four-digit palindromes, respectively. What is the sum of the digits of ?。

AMC10美国数学竞赛讲义全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.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 a d such that 0 ≤ < |b| a d a = bq + .6：(1)G eatest Commo D v so : Let gcd (a, b) = max {d ∈ Z: d | a a d 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 commo mult le:Let lcm(a,b)=m {d∈Z: a | d a d 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 a b c d m Z 0 m 0 .If a b mod m c d mod m then we havemod a c b d m ±≡± , mod ac bd m ≡ , mod k k a b m ≡(2) The equat o ax ≡ b (mod m) has a solut o f a d o ly f gcd(a, m) d v des 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 n a . For example, when 100,3n a ==9：Palindrome, such as 83438, is a number that remains the same when its digits are reversed.There are some number not only palindrome but 2 2 ，222 ， (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 pp p p p p p p δ==+++++++++∑(3) {}11221111122()#:,gcd(,)1()()()t t r r rr rr 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 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) 36 8. What is the sum of all integer solutions to 21<(x-2)<25? (A) 10 (B) 12(C) 15(D) 19(E) 5(A) 6 (B) 7 (C) 8 (D) 9 (E) 10(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. For3n ≥, 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 with replacement from the set {1, 2, 3,…, 2010}. What is the probability that abc 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 and222+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) 224 23. What is the hundreds digit of 20112011?(A) 1 (B) 4 (C) 5 (D) 6 (E) 99. 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= 6 6-22= 2 2-12= 1 1-12= 0Let N be the smallest umbe fo wh ch J m’s seque ce has umbe s. What is the units digit of N?(A) 1 (B) 3 (C) 5 (D) 7 (E) 9 21．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) 81ABC 100643a S 2(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 yP 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 nroots 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 , , 2, 3, 5, ,… 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 ?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 and222+3b +3c -3-2-2=-1997a ab ac bc . What is a ?(A) 249 (B) 250(C) 251(D) 252(E) 2531. What is246135135246++++-++++? (A) -1 (B) 536(C) 712(D)14760(E)43310. Consider the set of numbers {1, 10, 102, 103(010)}. 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) 101=(A) -64 (B) -24 (C) -9 (D) 24 (E) 5764. Let X and Y be the following sums of arithmetic sequences: X= 10 + 12 + + …+ 00. Y= 2 + + + …+ 02. 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 += 20=80= (E) -340x -=(A)(B)(C)2(D) (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… 9 , 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 ≤+≤ is 10.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) 118 15．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 integers16．Let ,,, and be real numbers with,, and. What is the sum of all possible values of5. 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 0°；④三⾓形三个外⾓的和等于3 0°；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。

Problem 1The ratio is closest to which of the following numbers?SolutionProblem 2Given that a, b, and c are non-zero real numbers, define. Find.SolutionProblem 3According to the standard convention for exponentiation,.If the order in which the exponentiations are performed is changed, how manyother values are possible?SolutionProblem 4For how many positive integers is there at least 1 positive integer such that ?infinitely manySolutionProblem 5Each of the small circles in the figure has radius one. The innermost circle istangent to the six circles that surround it, and each of those circles is tangent tothe large circle and to its small-circle neighbors. Find the area of the shaded region.SolutionProblem 6From a starting number, Cindy was supposed to subtract 3, and then divide by 9,but instead, Cindy subtracted 9, then divided by 3, getting 43. If the correct instructions were followed, what would the result be?SolutionProblem 7A arc of circle A is equal in length to a arc of circle B. What is the ratio of circle A's area and circle B's area?SolutionProblem 8Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let be the total area of the blue triangles,the total area of the white squares, and the area of the red square. Which of the following is correct?SolutionThere are 3 numbers A, B, and C, such that, and. What is the average of A, B, and C?Not uniquely determined SolutionProblem 10What is the sum of all of the roots of?SolutionProblem 11Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB each, 12 of the files take up 0.7 MB each, and the rest take up 0.4 MB each. It is not possible to split a file onto 2 different disks. What is thesmallest number of disks needed to store all 30 files?SolutionProblem 12Mr. Earl E. Bird leaves home every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time?SolutionProblem 13Given a triangle with side lengths 15, 20, and 25, find the triangle'ssmallest height.SolutionBoth roots of the quadratic equation are prime numbers. The number of possible values of isSolutionProblem 15Using the digits 1, 2, 3, 4, 5, 6, 7, and 9, form 4 two-digit prime numbers, usingeach digit only once. What is the sum of the 4 prime numbers?SolutionProblem 16Let. What is?SolutionProblem 17Sarah pours 4 ounces of coffee into a cup that can hold 8 ounces. Then she pours4 ounces of cream into a second cup that can also hold 8 ounces. She then pours half of the contents of the first cup into the second cup, completely mixes the contents of the second cup, then pours half of the contents of the second cup back into the first cup. What fraction of the contents in the first cup is cream?SolutionProblem 18A 3x3x3 cube is made of 27 normal dice. Each die's opposite sides sum to 7.What is the smallest possible sum of all of the values visible on the 6 faces of the large cube?SolutionProblem 19Spot's doghouse has a regular hexagonal base that measures one yard oneach side. He is tethered to a vertex with a two-yard rope. What is the area, insquare yards, of the region outside of the doghouse that Spot can reach?SolutionProblem 20Points and lie, in that order, on , dividing it into fivesegments, each of length 1. Point is not on line . Point lies on ,and point lies on . The line segments and are parallel.Find .SolutionProblem 21The mean, median, unique mode, and range of a collection of eight integersare all equal to 8. The largest integer that can be an element of this collection isSolutionProblem 22A set of tiles numbered 1 through 100 is modified repeatedly by the followingoperation: remove all tiles numbered with a perfect square , and renumber theremaining tiles consecutively starting with 1. How many times must the operationbe performed to reduce the number of tiles in the set to one?SolutionProblem 23Points and lie on a line, in that order, with and. Point is not on the line, and. The perimeter ofis twice the perimeter of. Find.SolutionProblem 24Tina randomly selects two distinct numbers from the set {1, 2, 3, 4, 5}, and Sergio randomly selects a number from the set {1, 2, ..., 10}. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?SolutionProblem 25, we have , In trapezoid, , andwith bases and(diagram notto scale). The area of is Solution。

2023 AMC 10A卷解析1.2023年AMC 10A卷是美国数学竞赛（American MathematicsCompetition）中的一套试卷。

此次考试是美国高中生之间一场激烈而受欢迎的数学竞赛，旨在培养学生的逻辑思维和问题解决能力。

本文将对该卷的题目进行解析和讨论，帮助读者更好地理解和应对类似的数竞赛。

2.第一题要求计算一个几何序列的和。

这是一道比较基础的题目，只需要按照公式进行计算即可得出结果。

这道题目主要考察了学生对于数列的理解和运算能力。

3.第二题是一道概率题。

题目给出了一个箱子中红球和蓝球的比例，并要求计算从箱子中取出两个后，至少有一个红球的概率。

这个题目需要学生应用概率的知识，并通过计算可能性来得出答案。

4.第三题涉及到了平面几何和三角函数的知识。

题目给出了一个等腰直角三角形，要求计算某个角的正弦值。

学生需要运用正弦函数的定义和特性来解答这道题目。

5.第四题是一道代数题。

题目给出了一个等差数列，并要求计算该数列的第n项。

这道题目需要学生应用数列的知识和公式，进行运算得出结果。

6.第五题是一道几何问题，涉及到圆的性质和角度的计算。

题目给出了一个圆，并要求计算其中一个弧所对应的圆心角的大小。

这道题目考察了学生对于圆的基本概念和角度的计算能力。

7.第六题是一道数论题。

题目给出了一个数列，并要求计算其中满足特定条件的数字个数。

学生需要运用数论的知识，确定满足条件的数字范围，并进行计数得出结果。

8.第七题是一道概率问题，涉及到了排列组合的知识。

题目给出了一组数字，并要求计算其中两个数字相加为奇数的概率。

学生需要运用排列组合的知识，确定满足条件的数字组合数量，并进行计算得出结果。

9.第八题是一道代数问题，需要学生解方程并求解变量的值。

题目给出了一个方程，并要求求出方程中某个变量的值。

学生需要运用代数的知识，将方程进行化简，得出变量的值。

10.第九题是一道几何问题，涉及到三角形的知识和比例关系。

AMC10美国数学竞赛真题 xx年Problem 1The median of the lististhe mean?Solution. What isProblem 2A number is more than the product of its reciprocal and its additive inverse. In which interval does the number lie?SolutionProblem 3The sum of two numbers is . Suppose is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?SolutionProblem 4What is the maximum number for the possible points of intersection of a circle and a triangle?SolutionProblem 5How many of the twelve pentominoes pictured below have at least one line of symmetry?SolutionProblem 6Letanddenote the product and the sum, respectively, of the digits ofand. Supposeis athe integer . For example,two-digit number such that . What is the units digit ofSolutionProblem 7When the decimal point of a certain positive decimal number is moved four places to the right, thenew number is four times the reciprocal of the original number. What is the original number?SolutionProblem 8Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Theirschedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab?SolutionProblem 9The state income tax where Kristin lives is levied at the rate ofof annual income plusof any amount aboveof the first. Kristin of hernoticed that the state income tax she paid amounted to annual income. What was her annual income?Problem 10If ,, and are positive withis,, andSolution, thenProblem 11Consider the dark square in an array of unit squares, part of which is shown. The ?rst ring of squares around this center square contains unit squares. The second ring contains unit squares. If we continue this process, the number of unit squares in the ring isSolutionProblem 12Suppose that is the product of three consecutive integers and that by . Which of the following is not necessarily a divisor of ?is divisibleProblem 13A telephone number has the form , where each letter represents a different digit. The digits in each partof the numbers are in decreasing order; that is, , , and . Furthermore, , , and are consecutive even digits; , , , and are consecutive odd digits; and . Find . SolutionProblem 14A charity sells benefit tickets for a total of . Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?SolutionProblem 15A street has parallel curbs feet apart. A crosswalk bounded by two parallel stripes crosses the street atan angle. The length of the curb between the stripesis feet and each stripe is feet long. Find the distance,in feet, between the stripes?SolutionProblem 16The mean of three numbers is more than the least of the numbers and less than the greatest. The median of the three numbers is . What is their sum?SolutionProblem 17Which of the cones listed below can be formed froma radius by aligning the two straight sides?sector of a circle ofSolutionProblem 18The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest toSolutionProblem 19Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible? SolutionProblem 20A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length . What is the length of each side of the octagon?SolutionProblem 21A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter and altitude , and the axes of the cylinder and cone coincide. Find the radius of the cylinder.SolutionProblem 22In the magic square shown, the sums of the numbersin each row, column, and diagonal are the same. Five of these numbers are represented by , , , , and . Find .SolutionProblem 23A box contains exactly five chips, three red and two white. Chips are randomly removed one at a timewithout replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?SolutionProblem 24In trapezoid, ,andare perpendicular to , and . What is, withSolutionProblem 25How many positive integers not exceeding ?are multiples of or but notSolutionProblem 1The median of the lististhe mean?Solution. What isProblem 2A number is more than the product of its reciprocal and its additive inverse. In which interval does the number lie?SolutionProblem 3The sum of two numbers is . Suppose is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?SolutionProblem 4What is the maximum number for the possible points of intersection of a circle and a triangle?SolutionProblem 5How many of the twelve pentominoes pictured below have at least one line of symmetry?SolutionProblem 6Letanddenote the product and the sum, respectively, of the digits ofand. Supposeis athe integer . For example,two-digit number such that . What is the units digit ofSolutionProblem 7When the decimal point of a certain positive decimal number is moved four places to the right, the new number is four times the reciprocal of the original number. What is the original number?SolutionProblem 8Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Theirschedule is as follows: Darren works every thirdschool day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working in the math lab. In how many school days from today will they next be together tutoring in the lab?SolutionProblem 9The state income tax where Kristin lives is levied at the rate ofof annual income plusof any amount aboveof the first. Kristin of hernoticed that the state income tax she paid amounted to annual income. What was her annual income?SolutionProblem 10If ,, and are positive withis,, and, thenProblem 11Consider the dark square in an array of unit squares, part of which is shown. The ?rst ring of squares around this center square contains unit squares. The second ring contains unit squares. If we continue this process, the number of unit squares in the ring isSolutionProblem 12Suppose that is the product of three consecutive integers and that by . Which of the following is not necessarily a divisor of ?is divisibleSolutionProblem 13A telephone number has the form , where each letter represents a different digit. The digits in each partof the numbers are in decreasing order; that is, , , and . Furthermore, , , and are consecutive even digits; , , , and are consecutive odd digits; and . Find .Problem 14A charity sells benefit tickets for a total of . Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets?SolutionProblem 15A street has parallel curbs feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is feet and each stripe is feet long. Find the distance, in feet, between the stripes?SolutionProblem 16The mean of three numbers is more than the least of the numbers and less than the greatest. The median of the three numbers is . What is their sum?SolutionProblem 17Which of the cones listed below can be formed from a radius by aligning the two straight sides?sector of a circle ofSolutionProblem 18The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest toSolutionProblem 19Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible? SolutionProblem 20A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length . What is the length of each side of the octagon?。

2000 AMC 10 ProblemsProblem 1In the year 2001, the United States will host the International Mathematical Olympiad. Let , , and be distinct positive integers such that the product 2001=••O M I . What is the largest possible value of the sum O M I ++ ?（A ）23 （B ）55 （C ）99 （D ）111 （E ）671Problem 22000 （20002000）＝（A ）20012000 （B ）20004000 （C ）40002000 （D ）2000000,000,4 （E ）000,000,42000Problem 3Each day, Jenny ate 20% of the jellybeans that were in her jar at the beginning of that day. At the end of the second day, 32 remained. How many jellybeans were in the jar originally?（A ）40 （B ）50 （C ）55 （D ）60 （E ）75Problem 4Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was , but in January her bill was because she used twice as much connect time as in December. What is the fixed monthly fee?（A ）2.53 （B ）5.06 （C ）6.24 （D ）7.42 （E ）8.77Problem 5Points M and N are the midpoints of sides PA and PB of △PAB. As P moves along a line that is parallel to side AB, how many of the four quantities listed below change? (a) the length of the segment MN (b) the perimeter of △PAB(c) the area of △PAB (d) the area of trapezoid ABNM（A ）0 （B ）1 （C ）2 （D ）3 （E ）4Problem 6The Fibonacci sequence 1，1，2，3，5，8，13，21，…… starts with two s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence?（A ）0 （B ）4 （C ）6 （D ）7 （E ）9Problem 7 In rectangle , , is on , and and trisect .What is the perimeter of ?（A ）333+ （B ）3342+ （C ）222+ （D ）2533+ （E ）3352+ Problem 8At Olympic High School, 52 of the freshmen and 54 of the sophomores took the AMC-10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true?（A ）There are five times as many sophomores as freshmen.（B ）There are twice as many sophomores as freshmen.（C ）There are as many freshmen as sophomores.（D ）There are twice as many freshmen as sophomores.（E ）There are five times as many freshmen as sophomores.Problem 9If , where , then（A ）－2 （B ）2 （C ）2－2p （D ）2p-2 （E ）｜2p-2｜Problem 10The sides of a triangle with positive area have lengths , , and . The sides of a second triangle with positive area have lengths , , and . What is the smallest positive number that is not a possible value of ｜x-y ｜?（A ）2 （B ）4 （C ）6 （D ）8 （E ）10Problem 11Two different prime numbers between 4 and 18 are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?（A ）21 （B ）60 （C ）119 （D ）180 （E ）231Problem 12Figures 0, 1, 2 and 3 consist of 1, 5, 13, and 25 nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be infigure 100?Figure 0 Figure 1 Figure 2 Figure 3（A ）10401 （B ）19801 （C ）20201 （D ）39801 （E ）40801 Problem 13There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?Problem 14Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were 71, 76, 80, 82, and 91. What was the last score Mrs. Walter entered?（A ）71 （B ）76 （C ）80 （D ）82 （E ）91Problem 15Two non-zero real numbers, and , satisfy. Find a possible value of ab ab b a -+ . （A ）-2 （B ）-21 （C ）31 （D ）21 （E ）2 Problem 16The diagram shows 28 lattice points, each one unit from its nearest neighbors. Segment AB meets segment CD at E. Find the length of segment AE.（A ）354 （B ）355 （C ）7512 （D ）52 （E ）9565Boris has an incredible coin changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?（A ）<dollar/>3.63 （B ）<dollar/>5.13 （C ）<dollar/>6.30 （D ）<dollar/>7.45 （E ）<dollar/>9.07Problem 18Charlyn walks completely around the boundary of a square whose sides are each 5 km long. From any point on her path she can see exactly km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?（A ）24 （B ）27 （C ）39 （D ）40 （E ）42Problem 19Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is times the area of the square. The ratio of the area of the other small right triangle to the area of the square is（A ）121 m （B ）m （C ）1-m （D ）m 41 （E ）281mProblem 20Let A, M, and C be nonnegative integers such that A+M+C=10. What is the maximum value of A ·M ·C + A ·M + M ·C +C ·A?（A ）49 （B ）59 （C ）69 （D ）79 （E ）89Problem 21If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true?I. All alligators are creepy crawlers.II. Some ferocious creatures are creepy crawlers.III. Some alligators are not creepy crawlers.（A ）I only （B ）II only （C ）III only （D ）II and III only （E ）None must be true Problem 22One morning each member of Angela's family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?（A ）3 （B ）4 （C ）5 （D ）6 （E ）7When the mean, median, and mode of the list 10, 2, 5, 2, 4, 2, x are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of ? （A ）3 （B ）6 （C ）9 （D ）17 （E ）20Problem 24Let be a function for which. Find the sum of all values of for which. （A ）-31 （B ）-91 （C ）0 （D ）95 （E ）35 Problem 25In year , the day of the year is a Tuesday. In year, the day is also a Tuesday. On what day of the week did the day of year occur?2000 AMC 10 SolutionProblem 1 The following problem is from both the 2000 AMC 12 #1 and 2000 AMC 10 #1, so both problems redirect to this page.The sum is the highest if two factors are the lowest.So, 1·3·776=2001 and 1+3+667=671 (E)Problem 2 The following problem is from both the 2000 AMC 12 #2 and 2000 AMC 10 #2, so both problems redirect to this page.2000 （20002000）＝12000（20002000）=20012000 (A)Problem 3 The following problem is from both the 2000 AMC 12 #3 and 2000 AMC 10 #3, so both problems redirect to this pageSince Jenny eats 20% of her jelly beans per day, 80%=4/5 of her jelly beans remain after one day. Let be the number of jelly beans in the jar originally.325454=••x x=50 (B) Problem 4Let be the fixed fee, and be the amount she pays for the minutes she used in the first month.x+y=12.48 x+2y=17.54 y=5.06 x=7.42 We want the fixed fee, which is (D) Problem 5(a) Clearly AB does not change, and MN=0.5AB, so MN doesn't change either. (b) Obviously, the perimeter changes.(c) The area clearly doesn't change, as both the base AB and its corresponding height remain the same.(d) The bases AB and MN do not change, and neither does the height, so the area of the trapezoid remains the same.Only quantity changes, so the correct answer is .Problem 6 The following problem is from both the 2000 AMC 12 #4 and 2000 AMC 10 #6, so both problems redirect to this page.Note that any digits other than the units digit will not affect the answer. So to make computation quicker, we can just look at the Fibonacci sequence in :The last digit to appear in the units position of a number in the Fibonacci sequence is 6 (C).Problem 7AD=1 Since ∠ADC is trisected, ∠ADP=∠PDB=∠BDC=30º Thus, PD=332 BD=2 BP=332333=- Adding, 3342+Problem 8Let be the number of freshman and be the number of sophomores.s f 5452= f=2s There are twice as many freshmen as sophomores. Problem 9 The following problem is from both the 2000 AMC 12 #5 and 2000 AMC 10 #9, so both problems redirect to this page.When x<2, x-2 is negative so ∣x -2∣=2-x =p and x=2-pThus x-p=(2-p)-p=2-2p (C)Problem10From the triangle inequality, 2<x <10 and 2<y <10. The smallest positive number not possible is 10-2, which is . (D)Problem 11 The following problem is from both the 2000 AMC 12 #6 and 2000 AMC 10 #11, so both problems redirect to this page.All prime numbers between 4 and 18 have an odd product and an even sum. Any odd number minus an even number is an odd number, so we can eliminate B andD. Since the highest two prime numbers we can pick are 13 and 17, the highest number we can make is (13) (17) – (13+17) = 221-30=191. Thus, we can eliminateE. Similarly, the two lowest prime numbers we can pick are 5 and 7, so the lowest number we can make is (5) (7) – (5+7) =23. Therefore, A cannot be an answer. So, the answer must be .Problem 12 The following problem is from both the 2000 AMC 12 #8 and 2000 AMC 10 #12, so both problems redirect to this page.Solution 1We have a recursion: .I.E. we add increasing multiples of each time we go up a figure. So, to go from Figure 0 to 100, we addWe then add to the number of squares in Figure 0 to get, which ischoice Solution 2We can divide up figure to get the sum of the sum of the first odd numbers and the sum of the first odd numbers. If you do not see this, here is the example for :The sum of the first odd numbers is 2n , so for figure , thereare 22)1(n n ++ unit squares. We plug in n=100 to get 20201, which is choice Solution 3Using the recursion from solution 1, we see that the first differences of 4,8,12, … form an arithmetic progression, and consequently that the second differences are constant and all equal to . Thus, the original sequence can be generated from a quadratic function.If c bn an n f ++=2)(, and f(0)=1, f(1)=5, and f(2)=13, we get a system of three equations in three variables:f(0)=1 gives c=1; f(1)=5 gives a+b+c=5; f(2)=13 gives 4a+2b+c=13 Plugging in into the last two equations gives a+b=4 4a+2b=12 Dividing the second equation by 2 gives the system: a+b=4 2a+b=6Subtracting the first equation from the second gives , and hence . Thus, our quadratic function is: 122)(2++=n n n fCalculating the answer to our problem, f(100)=20201Problem 13In each column there must be one yellow peg. In particular, in the rightmost column, there is only one peg spot, therefore a yellow peg must go there.In the second column from the right, there are two spaces for pegs. One of them is in the same row as the corner peg, so there is only one remaining choice left forthe yellow peg in this column.By similar logic, we can fill in the yellow pegs as shown:After this we can proceed to fill in the whole pegboard, so there is only arrangement of the pegs. The answer is (B)Problem 14 The following problem is from both the 2000 AMC 12 #9 and 2000 AMC 10 #14, so both problems redirect to this page.Solution 1The first number is divisible by 1.The sum of the first two numbers is even.The sum of the first three numbers is divisible by 3.The sum of the first four numbers is divisible by 4.The sum of the first five numbers is 400.Since 400 is divisible by 4, the last score must also be divisible by 4. Therefore, the last score is either 76 or 80.Case 1: 76 is the last number entered.Since 400≡76≡1 (mod 3), the fourth number must be divisible by 3, but none of the scores are divisible by 3. Case 2: 80 is the last number entered. Since 80≡2 (mod 3), the fourth number must be 2 (mod 3). Thatnumber is 71 and only 71. The next number must be 91, since the sum of the first two numbers is even.So the only arrangement of the scores 76, 82, 91,71,80Solution 2We know the first sum of the first three numbers must be divisible by 3, so we write out all 5 numbers (mod 3), which gives 2,1,2,1,1, respectively. Clearly the only way to get a number divisible by 3 by adding three of these is by adding the three ones. So those must go first. Now we have an odd sum, and since the next average must be divisible by 4, 71 must be next. That leaves 80 for last, so the answer is .Problem 15 The following problem is from both the 2000 AMC 12 #11 and 2000 AMC 10 #15, so both problems redirect to this page.22)()()(22222=-=----+=--+=-+ba ab b a b a b a b a b a ab b a ab a b b a (E)Alternatively, we could test simple values, like )21,1(),(=b a , which would yield 2=-+ab ab b a Another way is to solve the equation for giving 1+=a a b , then substituting this into the expression and simplifying gives the answer ofProblem 16Solution 1Let be the line containing A and B and let be the line containing C and D. If we set the bottom left point at (0,0), then A=(0,3), B=(6,0), C=(4,2), and D=(2,0) . The line is given by the equation 11b x m y +=. The -intercept is A=(0,3), so 1b =3. We are given two points on , hence we can compute the slope, to be 210630-=-- , so is the line 321+-=x y Similarly, is given by 22b x m y +=. The slope in this case is 12402=--, so 2b x y +=. Plugging in the point (2,0) gives us 2b =-2, so is the line 2-=x y . At E, the intersection point, both of the equations must be true,2-=x y , 321+-=x y so 3212+-=-x x SO 310=x 34=y We have the coordinates of and , so we can use the distance formula here: 355)334()0310(22=-+- which is answer choiceSolution 2Draw the perpendiculars from andto , respectively. As it turns out, . Let be the point onfor which . , and, so by AA similarity,By the Pythagorean Theorem, wehave, ,。

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 P is the prime number.if21p - is also the prime number. then 1(21)2p p --is theperfect 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 n can be written as a product of primes uniquely up to order.n =∏p i ri ki=15：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 rewrite mod a b m ≡. (1)Assume a,b,c,d,m ,k ∈Z (k >0,m ≠0).If a ≡b mod m,c ≡d mod m then 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. There are some number not only palindrome but 112=121，222=484，114=14641(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 reversed digits 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) 36 8. What is the sum of all integer solutions to 21<(x-2)<25? (A) 10(B) 12(C) 15(D) 19(E) 5(A) 6(B) 7(C) 8(D) 9(E) 10(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 with replacement from the set {1, 2, 3,…, 2010}. What is the probability that abc 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 and222+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) 224 23. What is the hundreds digit of 20112011?(A) 1 (B) 4 (C) 5 (D) 6 (E) 99. 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) 24 21. 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:5555-72= 6 6-22= 2 2-12= 1 1-12= 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) 9 21．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 xa x a x a ---+++++= has n roots123,,,,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 the324. Let ,a b and c be positive integers with >>a b c such that 222-b -c +=2011a ab and222+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) 101=(A) -64 (B) -24 (C) -9 (D) 24 (E) 576 4. 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) 112 7. Which of the following equations does NOT have a solution?(A) 2(7)0x += (B) -350x += (C)20=(D)80=(E) -340x -=(A)(B)(C)2(D) (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 ≤+≤ is 10. 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 thetotal weight in kilograms when the bucket is full of water?13．Suppose that and. Which of the following is equal tofor everypair of integers?16．Let ,,, andbe 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。