AMC8模考0718

*1. What is the value of below expression? 

以下表达式的值是多少？

11×16−16

*2. As shown in the figure, place a 45° right triangle between parallel lines a and b. If ∠1 = 15°, find the size of ∠2. 

如图，把一块45°直角三角板放在平行线a和b之间，如果∠1=15°，求∠2的大小。

*3. What is the value of 199 + 298 + 400 + 2 + 1 ? 

计算 199 + 298 + 400 + 2 + 1 的结果是多少？

*4. Two prime numbers add up to 2027. How many such pairs of primes exist? 

两个质数的和为2027。这样的质数对有多少组？

*5. A wetland contains 180 egrets, along with various other water birds. When an ecologist catches and releases a sample of 240 water birds from the wetland, 36 are identified as egrets. Assume that the ratio of egrets to the total number of water birds is the same in both the sample and the wetland. How many water birds are there in the wetland? 

湿地中有180只白鹭，以及其他多种水鸟。当一位生态学家从湿地中捕捉并放飞240只水鸟的样本时，其中36只被识别为白鹭。假设样本中白鹭与水鸟总数的比例，与湿地中白鹭占所有水鸟的比例相同。问湿地里一共有多少只水鸟？

*6. If =32,=9,and =64,find the value of abc

如果=32，=9，=64，求abc的值

*7. Perform the following three transformations on point A (-3, 4), and where will the final coordinate be? First, move downwards by 4 units, then reflect it about the y-axis, and finally rotate it counterclockwise by 90 degrees around the origin. 

点 A(-3,4) 做以下变换三步，最后的坐标在哪里？先向下移动 4 个单位，然后关于 y 轴对称，最后根据原点逆时针旋转90度。

*8. There are 10 lottery tickets in a box, 4 of them are first prize, the rest have no prize. 10 students line up to draw tickets in order one by one without replacement. Mary is the 5th to draw. After the first 4 students have drawn, Mary notices that half of the first-prize tickets are already gone. What is the probability that Mary draws a first prize ticket now? 

一个盒子里有10张奖券，其中4张是一等奖，其余无奖。10个同学按排队顺序抽奖，一人抽一张，不放回。玛丽排在第5位。当前面4个同学抽完后，玛丽发现一等奖已经被抽走了一半。请问：现在轮到玛丽抽，她抽到一等奖的概率是多少？

*9. Emma went to her grandfather's house and came back. The journey-time graph shows the information about her trip. How far was Emma from home at 11am? 

Emma去了她爷爷家又回来，路程-时间图显示了她旅程的信息，11am的时候Emma离家多远？（Distance from home：离家的距离，Time：时间）

*10. A bottle of juice is filled to 2/3 capacity. After Alex drinks 1/2 of the juice in the bottle, his mother adds 90 ml of juice, and the bottle is just filled. What is the total capacity of the bottle. 

一瓶果汁装满了2/3瓶。Alex 喝掉瓶中果汁的1/2后，他的妈妈加入 90 ml 果汁，瓶子刚好装满。求瓶子的总容量。

*11. NASA's Europa Clipper was launched on October 14, 2024. After traveling 108,000,000 miles, it entered a preliminary cruise orbit about 3.75 months later. Which of the following is closest to the Clipper’s average speed in miles per hour? 

美国国家航空航天局的“欧罗巴快船”探测器于2024年10月14日发射。在飞行了108,000,000英里后，它在大约3.75个月后进入了初步巡航轨道。以下哪个选项最接近“欧罗巴快船”的平均速度（单位：英里/小时）？

*12. Given a 3×3 grid table filled with numbers, how many triangles can be formed by selecting three cells from it, provided that they can constitute the three vertices of a triangle (connecting the cells according to their grid positions forms a triangle)? 

现有一个3×3的方格表格，其内的格子填有数字。从中选取 3 个格子，如果它们能构成一个三角形的三个顶点（按方格位置连接后图形为三角形），那么可以得到多少个三角形？

*13. Johnny bought some candies. He found that he could divide them equally over 30 days (a month) and also equally over 365 days (a year). What is the least number of candies he bought? 乔尼买了一些糖果，他发现按一个月30天可以平分这些糖果，按一年365天也可以平分这些糖果。乔尼最少买了多少糖果？

*14. Let N be the greatest five-digit number whose digits have a product of 270. What is the sum of the digits of N? 

N表示各个位上数字之积为270的最大的五位数。那么N的各个位上数字之和是多少？

*15. Amy and Ben are cycling along a circular path with a circumference of 15 kilometers. They start from the same point and depart at the same time, but they are riding in opposite directions. Amy rides clockwise at a constant speed of 35 kilometers per hour, while Ben rides counterclockwise at a constant speed of 25 kilometers per hour. How long will it take for them to meet? 

Amy和Ben沿着周长为15千米的环形路径骑行。他们同一起点且同时出发，但背向而行。Amy沿顺时针方向以每小时35千米的速度匀速前进，而Ben则沿逆时针方向以每小时25千米的速度匀速前进。问他们多久可以相遇？

*16. In the following cryptarithm, all the different letters stand for different digits. Find the smallest possible value of the 4-digit number MATH. 

在下面的数字谜中，所有不同的字母代表不同的数字。求出四位数MATH的最小可能值。

*17. A football is a three-dimensional figure composed of regular pentagons and regular hexagons, with the specific structure shown below. Each black patch is adjacent to 5 white patches; each white patch is alternately adjacent to 3 black patches and 3 white patches. If there are 12 regular pentagons in a football, how many more regular hexagons are needed to make the football satisfy the following structure? 

足球是由正五边形和正六边形组成的一个三维图形，具体的结构如下所示。每个黑色皮块邻接5个白色皮块；每个白色皮块相间地与3个黑色皮块及3个白色皮块相邻接。如果一个足球的正五边形有12个，正六边形还需要有多少个才能让足球满足如下的结构？

*18. A small robot named Rollo is standing on a number line, starting at position 0. It has two possible moves: Move 7 units to the right, Move 4 units to the left. What is the minimum number of moves Rollo must make to land exactly on position 1500 on the number line? 

小机器人 Rollo 站在数轴上，初始位置为 0。它有两种移动方式：向右移动 7 个单位，向左移动 4 个单位。求Rollo 至少要移动多少次，才能刚好到达数轴上位置 1500？

*19. As shown in the figure, the quadrilateral ABCD is a rectangle. Given that EG=16, and point B is the midpoint of line segment EC. Find the length of line segment DG. 

如图，四边形ABCD是一个长方形。已知EG=16，并且B点是线段EC的中点。求线段DG的长度。

*20. Mr. Page wants to distribute 100 candies to 10 children in the class, and requires that each child gets at least one candy. Let x and y be the unique mode and the median of the number of candies each child receives, respectively. What is the maximum possible value of y - x? 

佩奇先生要将100颗糖果分给班里的10个孩子，并要求每个孩子至少得到1颗。令x和y分别是所有孩子得到糖果数量的唯一众数和中位数。问y-x的最大可能值是多少？

*21. Person A and B play a game: they take turns writing a number on a blank paper to form the first two digits of a multi-digit number. Starting from the third digit, each number is exactly the sum of the two preceding numbers, and the game ends when no more digits can be written. For example, if A and B write 1 and 4, the multi-digit number is 1459. How many such multi-digit numbers are there in total? 

A与B做游戏，两人轮流在一张白纸上写出一个数字，组成一个多位数的前2位，而这个多位数从第三个数字开始，每个数字都恰好是它前面两个数字之和，直至不能再写为止.例如A与B写了1和4,那么这个多位数就是1459,则这类多位数共有多少个?

*22. In a sequence of positive integers, each term from the third term onwards is the product of the two preceding terms. Given that the sixth term is 864, what is the first term? 

在一个正整数数列中，从第三项起每一项都是前两项的乘积。已知第6项是864。求第一项是多少？

*23. Al, Bill, and Cal will each randomly be assigned a whole number from 1 to 10, inclusive, with no two of them getting the same number. What is the probability that Al’s number will be a whole number multiple of Bill’s and Bill’s number will be a whole number multiple of Cal’s? 

Al, Bill 和Cal每个人被随机分配一个1到10之间的整数（包含1和10），且每个人的数字不一样。Al的数正好是Bill的整数倍，同时Bill的数也是Cal的数的整数倍的概率是？

*24. The diagram shows three touching semicircles with radius 1 inside an equilateral triangle, which each semicircle also touches. The diameter of each semicircle lies along a side of the triangle. What is the length of each side of the equilateral triangle? 

这个图展示了三个半径为 1 的半圆，它们彼此相切且都位于一个等边三角形内部，每个半圆都与三角形相切。每个半圆的直径都沿着三角形的一条边摆放。求这个等边三角形的边长是多少？

*25. In the triangle, each of the number 1, 2, 3, 4, 5, 6, 7, 8 is placed into a different circle. The sums of the numbers on each of the three sides of the triangle are equal to the same number, S. What is the sum of all the different possible values of S? 

在三角形中，数字1、2、3、4、5、6、7、8分别被放在不同的圆圈中。三角形三条边上每个数字的和都等于同一个数S。则S的所有不同的可能值之和是多少？