2019年美國數學競賽(AMC12B)的試題與解答

華南師范大學數學科學學院(510631) 李湖南

這是今年美國數學競賽(高三年級)在中國統一使用的考試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?

譯文艾麗西婭有兩個容器.第一個裝滿了的水，第二個是空的.她把第一個容器里的水全部倒進第二個容器里，這時第二個容器里充滿了的水.問小容器與大容器的容積之比是多少？

解設第一、二個容器的容積分別為V1和V2，依題意有故(D)正確.

2.Consider the statement，“Ⅰfnis not prime，thenn-2 is prime.”Which of the following values ofnis 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，則n-2 是素數.”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論的例子，即滿足“若n不是素數，則n-2 不是素數.”選項中只有27 滿足，故(E)正確.

3.Which one of the following rigid transformations(isometries)maps the line segmentonto the line segmentA′B′so that the image ofA(-2,1)isA′(2,-1)and the image ofB(-1,4)isB′(1,-4)?

(A)reflection in the-axis

(B)counterclockwise rotation around the origin by 90°

(C)translation by 3 units to the right and 5 units down

(D)reflection in the-axis

(E)clockwise rotation about the origin by 180°

譯文以下哪個剛體變換(等距變換)將線段映成線段使得點A(-2,1)對應A′(2,-1)且點B(-1,4)對應B′(1,-4)?

解由于點A(-2,1)和A′(2,-1)的兩個坐標均互為相反數，可知原點O為線段AA′的中點，同理O也為線段BB′的中點.因此，將線段繞原點O順時針旋轉180°可得到，故(E)正確.

4.A positive integernsatisfies the equation (n+ 1)! +(n+2)!=440·n!.What is the sum of the digits ofn?

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

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

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

5.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，ornpieces of purple candy.A piece of purple candy costs 20 cents.What is the least possible value ofn?

(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美分.卡斯帕剛好夠錢買12 塊紅糖，或14 塊綠糖，或15 塊藍糖，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[12,14,15]= 420，卡斯帕的錢一定是420 美分的整數倍，因此最小值n= 420/20=21，故(B)正確.

6.Ⅰn a given plane，pointsAandBare 10 units apart.How many pointsCare there in the plane such that the perimeter of?ABCis 50 units and the area of ?ABCis 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A,B相距10 個單位，有多少個點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A(0,0),B(10,0),由于?ABC的面積為100，則點C到AB的高為20，可設C(x,20)，此時，?ABC的周長為10+20+20=50，因此符合條件的點C不存在，故(A)正確.

7.What is the sum of all real numbersxfor which the median of the numbers 4,6,8,17,andxis equal to the mean of those five numbers?

譯文所有使得五個數4,6,8,17 和x的中位數和平均數相等的實數x之和是多少？……