2023年AMC10美国数学竞赛A卷附中文翻译和答案.doc

2023AMC10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) $30.00

2. 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) 15

3. 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) (B) (C) (D) (E)

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 ?
A) 92 (B) 98 (C) 100 (D) 102 (E) 112

5. 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) (C) (D) 13 (E) 14

6. 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) 300

7. Which of the following equations does NOT have a solution?
(A) (B) (C)
(D) (E)

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) 60

9. 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 + bd

10. 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) 77

11. 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) (B) (C) (D) (E)

12. The players on a basketball team made some three-point shots, some two-point shots, some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team’s total score was 61 points. How many free throws did they make?
(A) 13 (B) 14 (C) 15 (D) 16 (E) 17

13. 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) 200

14. 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) (B) (C) (D) (E)

15. 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) 840

16. Which of the following in equal to ?
(A) (B) (C) (D) (E)

17. In the eight-term sequence A, B, C, D, E, F, G, H, the value of C is 5 and the sum of any three consecutive terms is 30. What is A + H?
(A) 17 (B) 18 (C) 25 (D) 26 (E) 43

18. 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) (B) (C) (D) (E)

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 2023, 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 population during this twenty-year period?
(A) 42 (B) 47 (C) 52 (D) 57 (E) 62

20. 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) (B) (C) (D) (E)

21. 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) (B) (C) (D) (E)

22. 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) 3750

23. 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) 998

24. 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) (B) (C) (D) (E)

25. Let R be a square region and 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) 2500

2023AMC10美国数学竞赛A卷
1. 某通讯公司手机每月基本费为20美元, 每传送一则简讯收 5美分（一美元=100 美分）。 若通话超过30小时，超过的时间每分钟加收10美分。已知小美一月份共传送了100条简讯及通话30.5小时，则她需要付多少美元
(A) $24.00 (B) $24.50 (C) $25.50 (D) $28.00 (E) $30.00

2.小瓶装有35毫升的洗发液，大瓶可装500毫升的洗发液。小华至少要买多少瓶小瓶的洗发液才干装满一个大瓶的洗发液?
(A) 11 (B) 12 (C) 13 (D) 14 (E) 15

3. 若以 [a b]表达 a ， b两数的平均数, 以 {a b c} 表达a, b, c三数的平均数，则{{1 1 0} [0 1] 0}之值为什么?
(A) (B) (C) (D) (E)

4. 设 X 和 Y 为下列等差级数之和:
X= 10 + 12 + 14 + …+ 100.
Y= 12 + 14 + 16 + …+ 102.
则之值为什么?
(A) 92 (B) 98 (C) 100 (D) 102 (E) 112

5. 在某小学三年级，四年级及五年级的学生，天天分别平均跑12, 15, 及10 分钟, 已知三年级的学生人数是四年级人数的两倍，四年级的学生人数是五年级学生人数的两倍。试问所有这些学生天天平均跑几分钟？
(A) 12 (B) (C) (D) 13 (E) 14

6. 已知集合A中有20个元素, 集合B 中有 15 个元素. A∪B是集合A和集合B的联集,它是由集合A与集合B中所有元素所形成的集合，则集合A∪B中至少有多少个元素?
(A) 5 (B) 15 (C) 20 (D) 35 (E) 300

7.下列哪个方程式没有解?
(A) (B) (C)
(D) (E)

8.去年夏季保护区里有 30%是鹅 ，25%是鸳鸯, 10%是苍鹰, 35% 是鸭子. 试问不是鸳鸯的鸟类中鹅占多少比例？
(A) 20 (B) 30 (C) 40 (D) 50 (E) 60

9. 某个矩形是由y=a, y=-b, x=-c, 与x=d,的圆形所围成的，其中a, b, c, ， d 均为正数。试问下列何者可以表达这个矩形的面积？
(A) ac + ad + bc + bd (B) ac – ad + bc – bd (C) ac + ad – bc – bd
(D) –ac –ad + bc + bd (E) ac – ad – bc + bd

10.戴老师班上 30位学生中超过半数的学生买了同一种铅笔，这些学生所买的铅笔都多于1支，且每个人所买的数目相同。以美分计，每支铅笔的价格都是整数，且比每位同学所买的支数多。若买铅笔共花了17.71美元（1美元=100美分），则每支铅笔的价格为多少美分？
(A) 7 (B) 11 (C) 17 (D) 23 (E) 77

11. 已知正方形 EFGH 的顶点分别在正方形 ABCD的四边上.若 E点 在 AB 上且 AE=7·EB. 试问 EFGH 的面积与ABCD的面积比值多少?
(A) (B) (C) (D) (E)

12. 某篮球队投进一些三分球、两分球及一分的罚球。他们三分球所得的分数与两分球所得的分数相同，且罚球投进的球数比两分球投进的球数多一球。若此球队总共得到61分，则此球队罚球共投进了多少球？
(A) 13 (B) 14 (C) 15 (D) 16 (E) 17 。
13. 在200至700中有多少个偶数，其各位数字都不相同，且各位数字是取自{1, 2, 5, 7, 8, 9}？
(A) 12 (B) 20 (C) 72 (D) 120 (E) 200 。
14. 投掷两个有六面的公正骰子一次，用两个骰子出现点数的和作为一个圆的直径。试问圆面、积的数值小于圆周长的数值之机率为多少？
(A) (B) (C) (D) (E) 。
15. 罗先生买了一部新型的油电车。在某路程中，这部车子在前40公里只使用电池；在之后的、旅途中只使用汽油，而此时每公里需用0.02加仑的汽油。若用同量的这些汽油恰可跑完所有路程，则平均每加仑须跑55公里。试问此路程是多少公里？
(A) 140 (B) 240 (C) 440 (D) 640 (E) 840 。
16. 下列何者等于+？
(A) 3 (B) 2 (C) (D) 3 (E) 6
17. 在八项的数列A、B、C、D、E、F、G、H中，C的值是5，且任何连续三项的和都是30。试问A+H的值是多少？
(A) 17 (B) 18 (C) 25 (D) 26 (E) 43 。
18. 如图所示，A, B, C三圆的半径均为1，圆A与圆B外切，圆C与相切于的中点。试问在圆C的内部且在圆A与圆B外部阴影区域的面积是多少？
A
B
C

(A) 3- (B) (C) 2 (D) (E) 1+ 。
19. 某城乡1991年的人口数是一个完全平方数；十年后，增长了150人，当时人口数比一个完全平方数多9；到了2023年，人口又再增长了150人，这时人口数又是一个完全平方数。试问这二十年间此城乡人口成长率的比例最接近下列何者？
(A) 42 (B) 47 (C) 52 (D) 57 (E) 62 。
20. 在一个半径为r的圆周上随意的取两点，从这两点依顺时针方向各画一条长度为r的弦，则这两弦会相交的机率是多少？
(A) (B) (C) (D) (E) 。
21. 两枚重量相等的伪币与8枚相同的真币混在一起，伪币的重量与真币的重量不同。从这10枚钱币中任取两枚，不再放回去；再从剩下的8枚钱币中任取两枚。已知第一次取出两枚钱币的重量和，与第二次取出两枚钱币的重量和相等。试问取出的4枚钱币均为真币的机率为多少？
(A) (B) (C) (D) (E) 。
22. 将凸五边形ABCDE的每一个顶点涂一种颜色，共有6种颜色可供涂色，若规定每一条对角线两端点的颜色必须不同，则共有多少种不同的涂色方法？
(A) 2520 (B) 2880 (C) 3120 (D) 3250 (E) 3750 。
23. 七位学生按下列的方式念出从1到1000中的某些数：
· A生的念法是每三个连续的数一组，跳过中间的数不念，即A生念出：
1, 3, 4, 6, 7,9 ,…, 997, 999, 1000。
· B生的念法是由小而大念出所有A生没有念的数，但他也跳过每接续三个数中间的数。
· C生的念法是由小而大念出所有A生与B生都没有念的数，但他也跳过每接续三个数中间的数。
· 再依序由D生、E生、F生来念，他们的念法也是由小而大念出前面所有同学都没有念的数，但他们也跳过每接续三个数中间的数。
· 最后，G生念出前面所有同学都没有念的数。
试问G生念出的数为什么？
(A) 37 (B) 242 (C) 365 (D) 728 (E) 998 。
24. 两个相异的正四周体，它们的顶点都在同一个单位正方体的顶点上。试问这两个四周体相交区域的体积为多少？
(A) (B) (C) (D) (E) 。
25. 设R为一正方形，n³4为一整数。在正方形R内部的一点X，假如以点X为起点，画出n条射线可以将正方形R分割成n个等面积的三角形，则称X为n-射线分割点。试问有多少个点是100-射线分割点，但它不是60-射线分割点？
(A) 1500 (B) 1560 (C) 2320 (D) 2480 (E) 2500 。
答　案：
1 ( D )
2 ( E )
3 ( D )
4 ( A )
5 ( C )
6 ( C )
7 ( B)
8 ( C )
9 ( A)
10 ( B )
11 ( B )
12 ( A )
13 ( A )
14 ( B )
15 ( C )
16 ( B )
17 ( C )
18 ( C )
19 ( E )
20 ( C )
21 ( D )
22 ( C )
23 ( C )
24 ( D )
25 ( C )