-年amc 10a和b竞赛真题及答案(英文版)演示教学

2014 AMC 10B Problems Problem 1 Leah has 13 coins^ all of which are pennies and nickels. If 5h已 had one more nickel than she has now, then she would h^ve the same number of pennies and nickels, lr )匚ents, how much are Leah's coins worth?(A) 33 (B) 35 (C) 37 (D) 39 (E) 41Problem 223 4 2s 2-彳 + 2-^ Problem 3Randy drove the first third of his trip on a gravel road^ the next 20 miles on pavement, and ths remaining □ne -fifth on a dirt road . In miles how long was Randy's tripi?(A) 30 (B)等 (C)粤(D) 40 (E)竽Problem 4Susie pays for 4 muffins and 3 bananas. Calvin spends twice as much paying for 2 muffins and 16 bananas. A muffin is. how many times as expensive as a banana?m 吨 了(A) | (B)彩 (C) f (D) 2 (E)-Problem 5Doug constructs a square window using 8 equal-size panes of glass, as shown ・ The ratio of the height to width ft3『 each pane is 5 : 2f and the borders around and between the panes are 2 inches wide, in inches, what 谄 the sid& length of th© square wind 口w?(A) 26 (B) 28 (C) 30 (D) 32 (E) 34Problem 6Orvin went to the store with just enough rrnoney to buy 30 balloons. When h@ arrived^ he discovered that the store had a special sale on b a I loons ：: buy 1 balloon at the regular p 『i 匚已 and get a second at g off the regular price. What is the greatest number of balloons Orvin could buy?(A) 33 (B) 34 (C) 36 (D) 38 (E) 39what im(A) 16 (B) 24 (C) 32 (D) 48 (E) 64Problem 7Suppose A> U > 0 and A is 近% greater than B. what is £?⑷100(#) (B) 1 叫字)(C) 100(字)(D) 1 叫乎)(E) 100(和SolutionProblem 8A truck tr^ve>l5 f feet every t seconds. There ars 3 feet in a yard i How many yards dDss the truck travel6in 3 rninute-s?b frn 血m10i 1 口⑷廖(X (①〒⑴)石㈣〒Solutionproblem 9For real numbers w and z,丄+丄予二于=2014,ut ww + zWhat is ----------- ?w ― z、一1 J 1(A) - 2014 (B)艄(C) —(D) 1 (E) 2014Problem 10In the addition shown below A. [3. C.and D are distinct digits. How many different values are po^ible ForD?ABBCB+ BCADADI3DDD(A) 2 (E) 4 (C) 7 (D) 8 (E) 9Problem 1111. For the consume^ a single discount of n%is more adv^ntagsous than any of the following discounts:(1)two ^ucces^ive 15% discounts(2)three successive 10% discounts(3) a 25f/J discount Fellow sd by s 5xu discountWhat is the- smallest possibls positive intsgsr valus uf n?(A) 27 (B) 2S (C) 29 (D) 31 (E) 33Problem 1212. The largest divisor of 2, 014h 000* 000 is rtsslf. What is its fifth largest divisor?(A) 125t875.000 (B) 20L400. 000 (C) 251, 750.000 (D) 402： 800.000 (E) 503.500.000Problem 13Sin regular hexagons surround 召regular hexagon of ^ide length 1 as shown. What is the ar?a of A J4BC?(A) 2V5 {B) 3V3 (C) 1+3辺(D) 2 + 2“$(E) 3+2\/3Problem 14Oariica drove her new car on a trip for a whole number of hour勺averaging 55 miles per hour. At the beginning of the trip」abc miles was displayed on thm Dclannstsr^ where abc is 耳3-digit number with ti > 1 and « + 6 —c < 7. At the end of the trip，the odometer showed eba miles. What is(i~ + fc2+ c~ ?(A) 26 (B) 27 (C) 36 (D) 37 (E) 41Problem 15In rectangle 打<7 = 2GR and points E3nd F lie on AF£□ that ED and FD tnsect Z.ADCas shown, what is the ratio of the area of ADEF to the ares of rectangle ABCD?S)晋(B)^ ©瞬㈣笫Problem 16Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the sarme value? (A)吉(B)12(C) I {D)島㈣|Problem 17What is thm greatest power of 2 t hat if a fmutor of 101— 4U：，01?Problem 18(A) 21DO2(B) 21003(C) 21004(D) 21C05(E) 21D0C A list of 11 positive! integers has a mean of 10；a median of 9j and a uiniqus> mads af 8- What is the largest passible value of an integer in the list?(A) 24 {B) 30 (C) 31 (D) 33 (E) 35Problem 19Two ccrncentric circles have radii 1 and 2. Two points an the outer circle are chosen independently and uniformly at random. What iw the probabi^it/ that the chord joining the two points in tersects the inner circle?(A) g (B) i (C)匕詳(D) i (E) +Problem 20For how many integers 工is the number —51J:2 + 50 negative?(A) 8 (B) 10 (C) 12 (D) 14(E) 16Problem 21Trapezoid ABCD has parallel sides AB af length 33 and CD of length 21. The other two sides are of lengths 1(' and 14. The angles at A and B are acute. What js the Imn gth of the shorter diagonal of ABCD?{A) 10?6 (B) 25 (C) S\/10 (D) 18“(E) 26SolutiariProblem 22Eight semicircles line the insid目af a ^qu^re with 吕id日length 2 as shown. What is the radium Q F the circle tangent tc ell of these semicircles?A sphere is inscribed in a truncated right circular cons as shown. The volume of thm truncated cone is twice that of the sphere. What is the ratro of the radius af the bottam base of the truncated cone to the radius of the tap base of the truncated cone?Problem 23(A) | ©近(D)2 (E)^^ Problem 24The numbers 1, 2} 3}斗』5 are to be arranged in a circle. An arrangement rs bad if rt is not true that fcr every n from 1 to 15 one can find a subset of the numberm that appear consecutively on the circle that sum to J ?. Arrangements that differ only by a rotation or a reflection mne considered the same. Haw many different bad arrangements are there?(A) 1 (B) 2 (C) 3 (D) 4 (E) 5 Problem 25In a small pond there are eleven lily pads in a row labeled 0 through 10, A frog is sitting on pad 1. When the frog is on pad N } 0 < N < 10, it 呷illju 叩 to pad N-l with probability — and to pad N +1 with probability 1 - Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake, If the frog reaches pad 10 it will exit the pond> never to return, what is the probability that the frog will escape being eaten by the snake?| (B)曙(C)磊(D)纟(E) \答案：1. C2. E3. E4. B5.A6. C7. A8. E9. A10. C11. C12. C13. B14. D15. A16. B17. D18. E19. D20. C21. B22. B23. E24. B25. Cwwt 冷10诲+ * +矿叮(A) 3 (B) & (C) y (D)爭 (E) 170Problem 2Roy's 亡吕t m吕t 百 of a can cf eat food ev&ry morning and y of a ean of eat food every evening. Before 怡目ding hiis 匚 a )t □“ Mlonday morn in Rciy ope me d a bck containing 6 cans of eat feotl*. On what day of the wisek dbd the 匚已t finish eating all the cat food in the box? (A) Tuesday (U) Wednesday (C) Thur 吕d 暫(U) Friday (E) Saturday Problem 3 Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for 32,501 each. In the aftemaon she sells two thirds of what she has left 3 and bscauss they are not freshshe charges only half priice. Tn the late aftern 口口n shethe r^mainiiing loaves -at B dollar leach. Each loaf 匚口百上鼻S (J_75 farher to make, In dollarSj what is her profit for the day? (A) 24 (B) 36 (C) 44 (D) 4S (E) 52Problem 4Walking down Jane Street, Ralph passed four houses in 吕 row, each painted a different 匚olcr- He passed th® orange house before the red housSi and the passed the blue house before ths yellow house. Ths blue house was not n&^t to t.h& ysillow house. How many orderings of the colored fhouses 囂「白 possible?(A) 2 (B) 3 (C) 4 (D) 5 (E) 6Problem 5On an 3l^eb rs quiz d 10% of Elis students scored 70 point 勺 35% scored 80 points^ 30% scored 90 pointSj and the rest scored 100 points. What is the differsincc between ths mean and median score cf ths students' scares on this quis?(A) 1 (B) 2 (C) 3 (D) 4 (E) 5Problerri 6SuppoBis that o 匚OWE ; give 6- gallons of milk in n days. At thus rate, how marTpr gallanE 口f nnilk will d cows give in c days?Probfem 7IN 口 r^zSrd 『自 £l nurfib&f£ ir. 些、ci f and b S-Stiisfy J : < LL <L b. Hdw marhy Gf th€i Follow in g id 白口u£ lit i 自弓 mustba true?(I) H + 空 V ci + b(II) Ji — y < a — 6(III) xy < ab(A) 0 (B) 1 (C) 2 (D) 3 (E) 4(C)cibdc (D) bcdeProblem 8 Which of the following niumbers Is a perfect square?Problem 9(A) 14115!~^T~(D)17fl81~2~18?193~2~Tfie two legs of a right trimngl已which are altitudes, ha\/e lengths 2\/3 and 6. How long is the third altitude of the triangte?(A) 1 (B) 2 (C) 3 (D) 4 (E) 5Problem 10Five positive consecutive integers starting with a have average b. what is the average of 5 consecutive integers that start with b?(A) d + 3 (B) ti + 4 (C) a 4^5 (D) u 4- 6 {E) <i + 7Problem 11A cusEomer who intends to purchase art appliance has three coupon5^ only ore of which may be uw总d;Coupon 1: .10/( off the listed price if the listed price rs at least S50Coupon 2. $20 off the listed price if the listed pnce is at Itfast 1100coupon 3： 18*X off the amount by which the listed price e^teeds $100For which of the following listed prices will coupari 1 offer a greater price reduction than either coupon 2 or coupon 3?(A) S179.95 (B) S199.95 (C) S219.95 (D) $239.95 (E) $259.95Problem 12A regular hexagon has side length 6. Ccngruerit arcs with radius 3 are drawn with the center at each of the vertical, creating circular sac tors 启弓shown. The region inside the ha^agon but outside the sectors is shaded as shown Whe t is thm^rea of the shaded regian?(A)幫価—跖(B) 27V3-&r (C) 54?5-18TT(D) &4^3-12^(E) 伽Problem 13Equilateral A>1R(7 hms side length 1』and squmrms AB DE, BOH I, CAFG bs outbids the triangle. What is the area of hexagon DEFGH I?Problem 14Ths y-intencBptSj P目nd Q, of two perp&ndicjldr lines intersecting mt the point j4(6h 8)have m sunn of zero. What is the area of AAPQ?(A) 45 (B) 4S (c) 54 (D) M (E) 72Problem 15Oavidl drives from his home to the airport to cat匚h a flight. He drives 35 milES in th^ first hour, but realizes that he will be 1 hour late if he continues at this speed. He increases his speed by 15 miles per hour for the rest of the way to the airport and arrives 30 minutes early. How many miles is the airport from his home?(A) 140 (B) 175 (C) 210 (D) 245 (E) 280Problem 16Fn rectangle A拜C0 A/J =L OC7 = 2』and points E, F、and <7 are midpoints of f 门』-and respectively・Point H is the midpoint of GE. What is the arsa of the shaded region?12 + 3>/3-4-(C) 3 + v5 (D)(E) 6/ Z?⑷吉㈣兽©害 (D)卷(E) 111 / 12Problem 17Three fair six-sided dn^e are rolled. What 祐 the prob ability that the values shown on two of the dice sum to the value shown an the rematning die?(A) j (B)第(C)籟(D)备(E) | Problem 18A square in the coordinate plane has vertices whose y-co ordinates are 0』1』4, and 5. What is the area of thm square?(A) 16 (B) 17 (C) 25 (D) 26 (E) 27 Problem 19Four cubes with edge lengths 1, 2, 3, and 4 are stacked as shown. What is the length of the portion of XV contdin^d in th© cub@ with edge length 3?(A)警 (B) 2^/3 (C)警 (D) 4 ㈣ 3闪Problem 20The product (8)(888 . .. 8), where ttie second factor has k digits, is an integer whose digits have a sum of 1000. What is fc?(A) 901 (B) 911 (C) 919 (D) 991 (E) 999Problem 21Positive integers a and b are such that the grsplis of = EAX + & and y = 驻+ b intersect this ir-a^is at ths same point. What is the sum erf all possible jr-coondinat&s of ths SB points of intersection?⑷-20 (B) -18 (C) 一苗(D) -12 (E) —8Problem 22[n r直亡tanglm AnC'D r AB= 20 and BG =10. Let £? be a point Ort CD such that Z.CBE= 15°. What is AE?(A) (B) IO A/3(C) 18 (D) 11 辺(E) 20Problem 23A rectmngubr piece of paper whose length is V3 times the width has area A, The paper is divided into three equal sections along the opposite I eng th Sj artd then a dotted fine is drawn from the first divider to the 亡end divider cn the opposite side shown. The paper 左then fbldEd flat along thiim dotted line to cnemtm a. new whmp日with area /?. What is the ratio J? : A?(A) 1 : 2 (B) 3 : &Problem 24A ssquanc^ cf natural nuimber^ i£cori£trut?ted by listing the first 4H then skipping one^ fci sting the newt 5, skipping 2. listing 6』skipping 3d andj on the nth itEration, listing n + 3 and skipping n. The sequence bsgins L 2h 3h 4, 6, 7. 8. 9.10. 13. What is ths 500- QOOth number in the sequence?(A) 996,506 (B) 996507 (C) 996508 (D) 996509 (E) 996510Problem 25The number 5B67is between 22011 and 2Z014r now many pairs or integers ^re there such thmt(C) 2 : 3 (D) 3 : 4 (E) 4 : 5Problem 201 < m < 2012 and(A) 278 (B) 279 (C) 280 (D) 281 (E) 282答案：1.C 2.D 3.D 4.B 5.E 6.B 7.B 8.B 9.D 10.C 11.C 12.C 13.C 14.C 15.B 16.B 17.D 18.E 19.D 20.B 21.C 22.A 23.C 24.B 25.A。

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.。