廣東省華南師范大學(xué)數(shù)學(xué)科學(xué)學(xué)院(510631) 李湖南

本試題是2019年美國中學(xué)數(shù)學(xué)競賽(高一年級)在中國考試統(tǒng)一使用的B 卷.

1.Alicia had two containers.The first wasfull of water and the second was empty.She poured all the water from the first container into the second container, at which point the second container wasfull of water.What is the ratio of the volume of the smaller container to the volume of the larger container?

解: 設(shè)第一、二個容器的容積分別為V1和V2,依題意有從而故(D)正確.

2.Consider the statement,”If n is not prime,then n?2 is prime.”Which of the following values of n is a counterexample to this statement?

(A) 11 (B) 15 (C) 19 (D) 21 (E) 27

譯文: 考慮陳述“若n 不是素數(shù),則n?2 是素數(shù).”以下哪個值n 是這個陳述的一個反例?

解: 反例是指符合命題的條件但不符合命題的結(jié)論的例子,即滿足“若n 不是素數(shù),則n?2 是素數(shù).”選項中只有27滿足,故(E)正確.

3.In a high school with 500 students, 40% of the seniors play a musical instrument, while 30%of the non-seniors do not play a musical instrument.In all, 46.8% of the students do not play a musical instrument.How many non-seniors play a musical instrument?

(A) 66 (B) 154 (C) 186 (D) 220 (E) 266

譯文: 在一所有500 名學(xué)生的高中里,40%的高年級學(xué)生會演奏樂器,而30%的非高年級學(xué)生不會演奏樂器.總的來說,46.8%的學(xué)生不會演奏樂器.問有多少名非高年級學(xué)生會演奏樂器?

解: 設(shè)非高年級學(xué)生有x 名, 則有(1?40%)(500?x)+30%·x=500×46.8%, 解得x=220, 所求學(xué)生為220×(1?30%)=154 名,故(B)正確.

4.All lines with equation ax+by =c such that a,b,c form an arithmetic progression pass through a common point.What are the coordinates of that point?

(A) (?1,2) (B) (0,1) (C) (1,?2) (D) (1,0) (E) (1,2)

譯文: 所有方程為ax+by=c 的直線經(jīng)過一個公共點(diǎn),其中a,b,c 構(gòu)成一個等差數(shù)列.試給出這個公共點(diǎn)的坐標(biāo).

解: 依題意有a+c=2b,即?a+2b=c,因此直線過點(diǎn)(?1,2),故(A)正確.

5.Triangle ABC lies in the first quadrant.Points A,B and C are reflected across the line y =x to points A′,B′and C′,respectively.Assume that none of the vertices of the triangle lie on the line y=x.Which of the following statements is not always true?

(A) Triangle A′B′C′lies in the first quadrant.

(B) Triangle ABC and A′B′C′have the same area.

(C) The slope of line AA′is-1.

(D) The slopes of lines AA′and CC′are the same.

(E) Lines AB and A′B′are perpendicular to each other.

譯文: 三角形ABC 位于第一象限.點(diǎn)A,B,C 關(guān)于直線y =x 的對稱點(diǎn)分別為A′,B′,C′.假設(shè)三角形的頂點(diǎn)都不在直線y =x 上,以下哪個命題是不正確的?

解: 對于第一象限中任意一點(diǎn)P(x0,y0), 有x0>0,y0>0,關(guān)于直線y =x 的對稱點(diǎn)P′(x0,y0)顯然也在第一象限,(A)正確;△ABC△A′B′C′,面積當(dāng)然也相等,(B)正確;設(shè)A(x1,y1),則A′(y1,x1),斜率(C) 正確; 同理(D) 也正確; 取A(4,1), B(5,3), 則A′(1,4),B′(3,5),斜率直線AB 與A′B′不垂直,(E)不正確,故答案為(E).

6.A positive integer n satisfies the equation(n+1)!+(n+2)!=440·n!.What is the sum of the digits of n?

(A) 2 (B) 5 (C) 10 (D) 12 (E) 15

譯文: 正整數(shù)n 滿足方程(n+1)!+(n+2)!=440·n!,則n 的各位數(shù)字之和是多少?

解: 原方程等價于(n+1)+(n+1)(n+2)=440,解得n=19,故(C)正確.

7.Each piece of candy in a shop costs a whole number of cents.Casper has exactly enough money to buy either 12 pieces of red candy,14 pieces of green candy,15 pieces of blue candy,or n pieces of purple candy.A piece of purple candy costs 20 cents.What is the least possible value of n?

(A) 18 (B) 21 (C) 24 (D) 25 (E) 28

譯文: 商店里的每一塊糖果的價格都是整數(shù)美分.卡斯帕的錢剛好夠買12 塊紅糖,或14 塊綠糖,或15 塊藍(lán)糖,或n塊紫糖.1 塊紫糖值20 美分.問n 的最小可能值是多少?

解: 最小公倍數(shù)[12,14,15]=420, 卡斯帕的錢一定是420 美分的整數(shù)倍,因此最小值為n=420/20=21,故(B)正確.

8.The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square,such that each triangle has side length 2 and the third vertices of the triangles meet at the center of the square.The region inside the square but outside the triangles is shaded.What is the area of the shaded region?

圖1

譯文: 下圖是一個正方形和四個等邊三角形,每個三角形的邊位于正方形的一條邊上,且邊長均為2,三角形的第三個頂點(diǎn)在正方形的中心相交.正方形內(nèi)部但三角形的外部被涂成陰影.問陰影部分的面積是多少?

9.The function f is defined byfor all real numbers x,wheredenotes the greatest integer less than or equal to the real number r.What is the range of f?

(A) {?1,0} (B) the set of nonpositive integers

(C) {?1,0,1} (D) {0} (E) the set of nonnegative integers

10.In a given plane, points A and B are 10 units apart.How many points C are there in the plane such that the perimeter of △ABC is 50 units and the area of △ABC is 100 square units?

(A) 0 (B) 2 (C) 4 (D) 8 (E) infinitely many

譯文: 在一個給定的平面上,點(diǎn)A、B 相距10 個單位,則平面上有多少個點(diǎn)C 使得△ABC 的周長是50 個單位且面積是100 個平方單位?

解: 不妨設(shè)A(0,0), B(10,0), 由于△ABC 的面積為100, 則點(diǎn)C 到AB 的高為20, 可設(shè)C(x,20), 此時,△ABC 的周長為10+20+20=50,故符合條件的點(diǎn)C 不存在,(A)正確.

11.Two jars each contain the same number of marbles,and every marble is either blue or green.In Jar 1 the ratio of blue to green marbles is 9: 1, and the ratio of blue to green marbles in Jar 2 is 8: 1.There are 95 green marbles in all.How many more blue marbles are in Jar 1 than in Jar 2?

(A) 5 (B) 10 (C) 25 (D) 45 (E) 50

譯文: 兩個罐子里裝有相同數(shù)量的彈珠,每個彈珠都是藍(lán)色或綠色的.1 號罐中藍(lán)綠色彈珠的比例為9 : 1,2 號罐中藍(lán)綠色彈珠的比例為8:1.總共有95 顆綠色彈珠.問1 號罐子比2 號罐子多幾顆藍(lán)色彈珠?

解: 設(shè)1 號罐子里的綠色彈珠有x 顆,藍(lán)色彈珠有9x 顆,則2 號罐子里的綠色彈珠有顆,于是解得x=45,從而為所求,故(A)正確.

12.What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than 2019?

(A) 11 (B) 14 (C) 22 (D) 23 (E) 27

譯文: 小于2019 的正整數(shù)用七進(jìn)制表示出來的各位數(shù)字和的最大值是多少?

解: 由于2018=5×73+6×72+1×7+2, 即2018=(5612)7, 在七進(jìn)制的表示中, 小于等于5612 的數(shù)之?dāng)?shù)字和最大的為5566 或4666,均為22,故(C)正確.

13.What is the sum of all real numbers x for which the median of the numbers 4,6,8,17,and x is equal to the mean of those five numbers?

譯文: 所有使得五個數(shù)4,6,8,17 和x 的中位數(shù)和平均數(shù)相等的實(shí)數(shù)x 之和是多少?

解: 中位數(shù)只可能是6,8 或x: (1) 若中位數(shù)是6, 則解得x =?5;(2)若中位數(shù)是8,則解得x=5,不符合要求; (3)若中位數(shù)是x,則解得也不符合要求.故(A)正確.

14. The base-ten representation for 19! is 121,6T5,100,40M,832,H00 where T,M and H denote digits that are not given.What is T +M +H?

(A) 3 (B) 8 (C) 12 (D) 14 (E) 17

譯文:19!的十進(jìn)制表示為121,6T5,100,40M,832,H00,其中T,M,H 是未知數(shù)字.則T +M +H 是多少?

解: 在19!的標(biāo)準(zhǔn)分解式中,有8 個3,3 個5,2 個7,1 個11,1 個13,從而能被9 整除,也能被7×11×13=1001 整除,且尾數(shù)上有3 個連續(xù)的0,于是H=0;又?jǐn)?shù)字和為33+T+M 能被9 整除,且121+100+832=6T5+40M+H00,解得T =4,M =8.因此T +M +H =12,故(C)正確.

15.Right triangle T1and T2have areas 1 and 2,respectively.A side of T1is congruent to a side of T2,and a different side of T1is congruent to a different side of T2.What is the square of the product of the lengths of the other(third)sides of T1and T2?

譯文: 兩個直角三角形T1和T2的面積分別為1 和2,T1的一條邊等于T2的一條邊,T1的另一條邊等于T2的另一條邊,問T1和T2的第三條邊長度之積的平方是多少?

圖2

解: 如圖示, T1的一條直角邊等于T2的一條直角邊b, T1的斜邊等于T2的另一條直角邊c, 則T1的另一條直角邊等于的斜邊等于依題意有解得所求值為故(A)正確.

16.In △ABC with a right angle at C, point D lies in the interior ofand point E lies in the interior ofso that AC=CD, DE=EB, and the ratio AC : DE=4 : 3.What is the ratio AD :DB?

圖3

解: 如圖示, 過點(diǎn)E 作EF⊥AB 于F, 依題意有∠ADC=∠A, ∠EDB=∠B, 從而∠CDE=180??∠ADC?∠EDB =180??∠A?∠B =90?,不妨設(shè)AC =CD=4, DE=EB=3, 則BC =CE+EB =8,AB =又由于△BEF△BAC,可得即BF=于是AD :故(A)正確.

17.A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball,the probability that it is tossed into bin k is 2?kfor k =1,2,3,···.What is the probability that the red ball is tossed into a higher-numbered bin than the green ball?

譯文: 一個紅球和一個綠球被隨機(jī)地、獨(dú)立地拋入用正整數(shù)編號的箱子中,每個球被拋入箱子k 的概率為2?k,其中k=1,2,3,···.則紅球被拋入比綠球編號更高的箱子的概率是多少?

解: 紅球被拋入比綠球編號更高的箱子的概率與綠球被拋入比紅球編號更高的箱子的概率是相等的,而紅球與綠球被拋入同一個箱子的概率是結(jié)果為因此所求概率為(C)正確.

18.Henry decides one morning to do a workout, and he walksof the way from his home to his gym.The gym is 2 kilometers away from Henry’s home.At that point, he changes his mind and walksof the way from where he is back toward home.When he reaches that point,he changes his mind again and walksof the distance from there back toward the gym.If Henry keeps changing his mind when he has walkedof the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point A kilometers from home and a point B kilometers from home.What is|A?B|?

譯文: 一天早上亨利決定去鍛煉,健身房離亨利家2 公里.當(dāng)他從家里往健身房走到路程的時候,他改變了主意,然后往回走.當(dāng)他往家里又走了路程的時候,他再次改變主意,然后又往健身房走,而且又走了的路程.如果亨利從他上次改變主意的地方向健身房或家走了距離的時候,不斷地改變主意,那么他就可以非常接近地在離家A 公里的一點(diǎn)和離家B 公里的一點(diǎn)之間來回走動了.請問|A?B|是多少?

圖4

解: 如圖示, 亨利無限地行走下去, 他基本上就會在固定的兩個點(diǎn)之間來回.此時有解得從而故(C)正確.

19.Let S be the set of all positive integer divisors of 100,000.How many numbers are the product of two distinct elements of S?

(A) 98 (B) 100 (C) 117 (D) 119 (E) 121

譯文: 設(shè)S 是100,000 的所有正整數(shù)因子的集合,則S中兩個不同元素的乘積共有多少個?

解: 100000=25×55, 則S={2i×5j: i,j =0,1,2,3,4,5}, 于是S 中兩個元素乘積的集合為{2i×5j:i,j=0,1,2,3,··· ,10},共121 個數(shù).注意到題目要求兩個元素不同,所以乘積中要去掉1,210,510,210×510這四個數(shù),剩下117 個,故(C)正確.

20.As showing in the figure,line segmentis trisected by points B and C so that AB=BC=CD=2.Three semicircles of radius 1,AEB,BFC and CGD,have their diameters onlie in the same halfplane determined by line AD,and are tangent to line EG at E,F and G, respectively.A circle of radius 2 has its center at F.The area of the region inside the circle but outside the three semicircles,shaded in the figure,can be expressed in the formwhere a,b,c and d are positive integers and a and b are relatively prime.What is a+b+c+d?

(A) 13 (B) 14 (C) 15 (D) 16 (E) 17

圖5

圖6

解: 如圖示, 設(shè)⊙F 與線段AD 相交于點(diǎn)H,I, 取線段BC,CD 的中點(diǎn)分別為J,K, 連結(jié)FH,FI,FJ,GK, 則FJKG 是個矩形, 且FJ=GK=1, 從而∠IFJ=60?,所求陰影部分的面積S=S1+S2+S3,其中S1為⊙F 的上半圓,S2為⊙F 被HI 所截的弓形,S3為其余部分.于是,因此S =即有a=7,b=3,c=3,d=4,得a+b+c+d=17,故(E)正確.

21.Debra flips a fair coin repeatedly,keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row,at which point she stops flipping.What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head?

譯文: 黛布拉反復(fù)地擲一枚均勻的硬幣,記錄下她總共看到了多少個正面和多少個反面,直到她連續(xù)擲出兩個正面或兩個反面,這時她就停止.請問她以連續(xù)擲出兩個正面結(jié)束, 但在出現(xiàn)第二個正面之前出現(xiàn)第二個反面的概率是多少?

解: 她至少需要擲5 次才能結(jié)束, 此時擲出的結(jié)果是01011, 其中0 代表反面, 1 代表正面, 這個結(jié)果的概率是當(dāng)然, 她擲出的結(jié)果還可以是0101011, 010101011,……, 概率分別是因此, 所求概率為正確.

22.Raashan,Sylvia,and Ted play the following game.Each starts with$1.A bell rings every 15 seconds,at which time each of the players who currently has money simultaneously chooses one of the other two players independently and at random and gives$1 to that player.What is the probability that after the bell has rung 2019 times, each player will have$1? (For example,Raashan and Ted may each decide to give$1 to Sylvia,and Sylvia may decide to give her dollar to Ted,at which point Raashan will have$0,Sylvia will have$2,and Ted will have$1,and that is the end of the first round of play.In the second round Raashan has no money to give,but Sylvia and Ted might choose each other to give their$1 to, and the holdings will be the same at the end of the second round.)

譯文: 拉珊、西爾維婭和特德在玩下列游戲: 開始每人持有1 美元,鈴每15 秒響一次,然后每個手上有錢的玩家同時獨(dú)立、隨機(jī)地選擇另外兩個玩家中的一個,并給該玩家1 美元.請問鈴響2019 次后,每個玩家仍然持有1 美元的可能性是多少? (例如,拉珊和特德各自決定給西爾維婭1 美元,西爾維婭決定給特德1 美元,第一輪游戲結(jié)束,此時拉珊將得到0 美元,西爾維婭將得到2 美元,特德得到1 美元.在第二輪中,拉珊沒有錢可給,但西爾維婭和特德可能會選擇給對方1 美元,那么第二輪結(jié)束時,他們的持有量將不變.)

解: 記拉珊、西爾維婭和特德分別持有的現(xiàn)金m,n,k 美元為(m,n,k), 則在任何時候都有(m,n,k) ∈{(1,1,1),(2,1,0),(2,0,1),(1,2,0),(1,0,2),(0,1,2),(0,2,1)}.

(1)當(dāng)(m,n,k)=(1,1,1)時,鈴響之后,可能出現(xiàn)以下8 種情況:

此時的結(jié)果分別為(1,1,1),(1,1,1),(2,1,0),(1,2,0),(0,2,1),(0,1,2),(2,0,1),(1,0,2),即一輪過后結(jié)果仍然是(1,1,1)的概率為

(2)當(dāng)三個人持有現(xiàn)金不一樣的時候,不妨設(shè)(m,n,k)=(2,1,0),鈴響之后,可能出現(xiàn)以下4 種情況:

此時的結(jié)果分別為(2,1,0),(1,1,1),(2,0,1),(1,0,2), 即一輪過后結(jié)果是(1,1,1)的概率為

因此,不管哪種情況,2019 輪過后,結(jié)果回到(1,1,1)的概率就是故(B)正確.

23.Points A(6,13) and B(12,11) lie on a circle ω in the plane.Suppose that the tangent lines to ω at A and B intersect at a point on the x-axis.What is the area of ω?

譯文: A(6,13)和B(12,11)是平面上的一個圓ω 上的兩點(diǎn).設(shè)ω 上的過A 和B 兩點(diǎn)的切線相交于x 軸上的同一個點(diǎn),則圓ω 的面積是多少?

圖7

解: 如圖示,設(shè)圓ω 的圓心為E,半徑為r,兩條切線交點(diǎn)為C,點(diǎn)C 的坐標(biāo)為(x,0),連結(jié)EA,EB,EC,AB,且AB交EC 于D,依題意有CA=CB,EA=EB,則點(diǎn)E,C 均在AB 的中垂線上,即EC⊥AB,且有AD =DB,于是點(diǎn)D的坐標(biāo)為(9,12),另外

24.Define a sequence recursively by x0=5 and xn+1=for all nonnegative integers n.Let m be the least positive integer such thatIn which of the following intervals does m lie?

(A) [9,26] (B) [27,80] (C) [81,242] (D)[243,728] (E) [729,+∞)

譯文: 定義遞歸數(shù)列如下: x0=5, xn+1=?n ∈N.令m 是使得成立的最小正整數(shù),則m 在以下哪個區(qū)間?

解:

由于x0>4,可得xn+1>4,?n ∈N;又

可得數(shù)列{xn}是單調(diào)下降的,則4 < xn5,進(jìn)一步可得

25.How many sequences of 0s and 1s of length 19 are there that begin with a 0,end with a 0,contain no two consecutive 0s,and contain no three consecutive 1s?

(A) 55 (B) 60 (C) 65 (D) 70 (E) 75

譯文: 有多少個長度為19 的只含0 和1 的序列: 以0 開頭,以0 結(jié)尾,不包含兩個連續(xù)的0,也不包含三個連續(xù)的1?

解: 0 后面只能接1 或11, 記01 為A, 011 為B, 原問題就相當(dāng)于將一些A 和B 排列在長度為18 的序列上, 可能出現(xiàn)(1) 6 個B: 只有1 種排列; (2) 4 個B, 3 個A: 共有種排列; (3) 2 個B, 6 個A: 共有種排列;(4)9 個A: 只有1 種排列.因此,符合條件的序列共有1+35+28+1=65 個,故(C)正確.