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AMC8 2023

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AMC8 · 2023

Q1
What is the value of $(8 \times 4 + 2) - (8 + 4 \times 2)$?
$(8 \times 4 + 2) - (8 + 4 \times 2)$ 的值为多少？
Q2
A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?
一张正方形纸被折叠两次成四个相等的四分之一，如下图所示，然后沿虚线剪裁。展开后，该纸将与下列哪个图形匹配？
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Q3
Wind chill is a measure of how cold people feel when exposed to wind outside. A good estimate for wind chill can be found using this calculation \[(\text{wind chill}) = (\text{air temperature}) - 0.7 \times (\text{wind speed}),\] where temperature is measured in degrees Fahrenheit $(^{\circ}\text{F})$ and the wind speed is measured in miles per hour (mph). Suppose the air temperature is $36^{\circ}\text{F}$ and the wind speed is $18$ mph. Which of the following is closest to the approximate wind chill?
风寒是衡量人们在室外暴露于风中时感觉多冷的一种度量。风寒的一个良好估计可以使用以下计算公式得到 \[(\text{风寒}) = (\text{空气温度}) - 0.7 \times (\text{风速}),\] 其中温度以华氏度 $(^{\circ}\text{F})$ 测量，风速以英里每小时 (mph) 测量。假设空气温度为 $36^{\circ}\text{F}$，风速为 $18$ mph。下列哪个是最接近于近似风寒的数值？
Q4
The numbers from $1$ to $49$ are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number $7.$ How many of these four numbers are prime?
数字从 $1$ 到 $49$ 在一个正方形网格上以螺旋图案排列，从中心开始。下面网格中已填入了前几个数字。考虑将出现在阴影方块中的四个数字，这些方块与数字 $7$ 在同一条对角线上。其中有多少个是质数？
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Q5
A lake contains $250$ trout, along with a variety of other fish. When a marine biologist catches and releases a sample of $180$ fish from the lake, $30$ are identified as trout. Assume that the ratio of trout to the total number of fish is the same in both the sample and the lake. How many fish are there in the lake?
一个湖中有 $250$ 条鳟鱼，还有各种其他鱼类。当海洋生物学家从湖中捕获并放回 $180$ 条鱼的样本时，其中 $30$ 条被识别为鳟鱼。假设样本与湖中鳟鱼与总鱼类的比例相同。湖中有多少条鱼？
Q6
The digits $2,0,2,$ and $3$ are placed in the expression below, one digit per box. What is the maximum possible value of the expression?
数字 $2,0,2,$ 和 $3$ 被放置在下面的表达式中，每个方框一个数字。表达式的最大可能值是多少？
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Q7
A rectangle, with sides parallel to the $x$-axis and $y$-axis, has opposite vertices located at $(15, 3)$ and $(16, 5)$. A line is drawn through points $A(0, 0)$ and $B(3, 1)$. Another line is drawn through points $C(0, 10)$ and $D(2, 9)$. How many points on the rectangle lie on at least one of the two lines?
一个矩形，其边平行于 $x$ 轴和 $y$ 轴，对角顶点位于 $(15, 3)$ 和 $(16, 5)$。一条直线通过点 $A(0, 0)$ 和 $B(3, 1)$。另一条直线通过点 $C(0, 10)$ 和 $D(2, 9)$。矩形上有多少点位于至少一条直线上？
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Q8
Lola, Lolo, Tiya, and Tiyo participated in a ping pong tournament. Each player competed against each of the other three players exactly twice. Shown below are the win-loss records for the players. The numbers $1$ and $0$ represent a win or loss, respectively. For example, Lola won five matches and lost the fourth match. What was Tiyo’s win-loss record?
Lola、Lolo、Tiya 和 Tiyo 参加了一场乒乓球锦标赛。每位选手与每位其他三位选手各比赛两次。下面显示了选手们的胜负记录。数字 $1$ 和 $0$ 分别代表胜利或失败。例如，Lola 赢了五场比赛，输了第四场比赛。Tiyo 的胜负记录是什么？
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Q9
Malaika is skiing on a mountain. The graph below shows her elevation, in meters, above the base of the mountain as she skis along a trail. In total, how many seconds does she spend at an elevation between $4$ and $7$ meters?
Malaika 在山上滑雪。下面图表显示了她滑雪沿小径时相对于山底的海拔高度，单位为米。总共，她在海拔 $4$ 到 $7$ 米之间花费了多少秒？
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Q10
Harold made a plum pie to take on a picnic. He was able to eat only $\frac{1}{4}$ of the pie, and he left the rest for his friends. A moose came by and ate $\frac{1}{3}$ of what Harold left behind. After that, a porcupine ate $\frac{1}{3}$ of what the moose left behind. How much of the original pie still remained after the porcupine left?
Harold 做了一个李子派带去野餐。他只吃了派的三分之一，将剩下的留给了朋友。一只驼鹿过来吃了 Harold 留下的三分之一。在此之后，一只豪猪吃了驼鹿留下三分之一的三分之一。豪猪离开后，原派还剩下多少？
Q11
NASA’s Perseverance Rover was launched on July $30,$ $2020.$ After traveling $292{,}526{,}838$ miles, it landed on Mars in Jezero Crater about $6.5$ months later. Which of the following is closest to the Rover’s average interplanetary speed in miles per hour?
NASA的Perseverance漫游车于2020年7月$30,$日发射。在飞行$292{,}526{,}838$英里后，它在大约$6.5$个月后登陆火星的Jezero陨石坑。以下哪项最接近漫游车的平均行星际速度（英里/小时）？
Q12
The figure below shows a large white circle with a number of smaller white and shaded circles in its interior. What fraction of the interior of the large white circle is shaded?
下图显示一个大的白色圆圈，内部有多个较小的白色和阴影圆圈。大的白色圆圈内部的阴影部分占几分之几？
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Q13
Along the route of a bicycle race, $7$ water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also $2$ repair stations evenly spaced between the start and finish lines. The $3$rd water station is located $2$ miles after the $1$st repair station. How long is the race in miles?
在自行车赛的路线上，起点和终点之间有$7$个均匀分布的水站，如图所示。起点和终点之间还有$2$个均匀分布的维修站。第$3$个水站位于第$1$个维修站后$2$英里处。比赛总长多少英里？
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Q14
Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of $5$-cent, $10$-cent, and $25$-cent stamps, with exactly $20$ of each type. What is the greatest number of stamps Nicolas can use to make exactly $\$7.10$ in postage? (Note: The amount $\$7.10$ corresponds to $7$ dollars and $10$ cents. One dollar is worth $100$ cents.)
Nicolas计划寄包裹给他的朋友Anton，后者是集邮爱好者。为了支付邮费，Nicolas想用大量邮票覆盖包裹。假设他有5美分、10美分和25美分邮票各正好20张。Nicolas能用多少张邮票来正好凑成$\$7.10$的邮费？ （注：$\$7.10$对应7美元10美分。1美元=100美分。）
Q15
Viswam walks half a mile to get to school each day. His route consists of $10$ city blocks of equal length and he takes $1$ minute to walk each block. Today, after walking $5$ blocks, Viswam discovers he has to make a detour, walking $3$ blocks of equal length instead of $1$ block to reach the next corner. From the time he starts his detour, at what speed, in mph, must he walk, in order to get to school at his usual time? Here’s a hint: if you aren’t correct, think about using conversions, maybe that’s why you’re wrong! -RyanZ4552
Viswam每天走半英里上学。他的路线包括10个等长城市街区，每块街区走1分钟。今天，走5个街区后，Viswam发现必须绕道，走3个等长街区代替1个街区到达下一个拐角。从开始绕道时起，他必须以多少mph的速度走，才能按平常时间到校？ 提示：如果不对，想想单位换算，也许那是错的原因！-RyanZ4552
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Q16
The letters $\text{P}, \text{Q},$ and $\text{R}$ are entered into a $20\times20$ table according to the pattern shown below. How many $\text{P}$s, $\text{Q}$s, and $\text{R}$s will appear in the completed table?
字母 $\text{P}, \text{Q},$ 和 $\text{R}$ 按照下面所示的模式填入一个 $20\times20$ 表格中。完成的表格中会出现多少个 $\text{P}$、$\text{Q}$ 和 $\text{R}$？
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Q17
A regular octahedron has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the right of $Q$?
一个正八面体有八个等边三角形面，每个顶点处有四个面相交。Jun 将通过折叠左边所示的纸张来制作右边所示的正八面体。哪个编号的面最终会位于 $Q$ 的右侧？
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Q18
Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump $5$ pads to the right or $3$ pads to the left. What is the fewest number of jumps Greta must make to reach the lily pad located $2023$ pads to the right of her starting position?
绿草hopper Greta 坐在池塘中一长排睡莲上。从任何睡莲上，Greta 可以向右跳 $5$ 个睡莲或向左跳 $3$ 个睡莲。Greta 到达起始位置右侧 $2023$ 个睡莲的位置需要的最少跳跃次数是多少？
Q19
An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is $\frac23$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?
一个等边三角形被放置在大等边三角形内部，使得它们之间的区域可以分成三个全等的梯形，如下图所示。内三角形的边长是大三角形的 $\frac23$。一个梯形的面积与内三角形面积的比是多少？
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Q20
Two integers are inserted into the list $3, 3, 8, 11, 28$ to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers?
将两个整数插入列表 $3, 3, 8, 11, 28$ 中，使其范围加倍。众数和中位数保持不变。两个附加数字的最大可能和是多少？
Q21
Alina writes the numbers $1, 2, \dots , 9$ on separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the numbers in each group will be the same. In how many ways can this be done?
Alina 将数字 $1, 2, \dots , 9$ 分别写在单独的卡片上，每张卡片一个数字。她希望将这些卡片分成 $3$ 组，每组 $3$ 张卡片，使得每组数字之和相同。有多少种方法可以做到这一点？
Q22
In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term is $4000$. What is the first term?
在一个正整数序列中，从第三个项开始，每一项都是前两项的乘积。第六项是 $4000$。第一项是多少？
Q23
Each square in a $3 \times 3$ grid is randomly filled with one of the $4$ gray and white tiles shown below on the right. What is the probability that the tiling will contain a large gray diamond in one of the smaller $2 \times 2$ grids? Below is an example of such tiling.
一个 $3 \times 3$ 网格中的每个方格随机填充右边所示的 $4$ 种灰白瓦片之一。\n\n求该铺砖包含至少一个较小的 $2 \times 2$ 网格中有一个大灰色菱形的概率？下面是一个这样的铺砖示例。
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Q24
Isosceles $\triangle ABC$ has equal side lengths $AB$ and $BC$. In the figure below, segments are drawn parallel to $\overline{AC}$ so that the shaded portions of $\triangle ABC$ have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height of $h$ of $\triangle ABC$? (Diagram not drawn to scale.)
等腰 $\triangle ABC$ 有相等的边长 $AB$ 和 $BC$。在下面的图中，画了与 $\overline{AC}$ 平行的线段，使得 $\triangle ABC$ 的阴影部分面积相同。两个非阴影部分的的高度分别为 $11$ 和 $5$ 个单位。求 $\triangle ABC$ 的高度 $h$？（图未按比例绘制。）
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Q25
Fifteen integers $a_1, a_2, a_3, \dots, a_{15}$ are arranged in order on a number line. The integers are equally spaced and have the property that \[1 \le a_1 \le 10, 13 \le a_2 \le 20, \text{ and } 241 \le a_{15}\le 250.\] What is the sum of digits of $a_{14}?$
十五个整数 $a_1, a_2, a_3, \dots, a_{15}$ 按顺序排列在数线上。这些整数等间距排列，且满足 \[1 \le a_1 \le 10, 13 \le a_2 \le 20, \text{ and } 241 \le a_{15}\le 250.\] $a_{14}$ 的数字和是多少？
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