2026 年 AMC 8 真题（P版）

1

下面这个表达式的值是多少？\[ 1 + 2 - 3 + 4 + 5 - 6 + {} \] \[ 7 + 8 - 9 + 10 + 11 - 12 \]

A\(18\) B\(21\) C\(24\) D\(27\) E\(30\)

English

What is the value of the following expression? \[ 1 + 2 - 3 + 4 + 5 - 6 + {} \] \[ 7 + 8 - 9 + 10 + 11 - 12 \]

(A) \(18\)(B) \(21\)(C) \(24\)(D) \(27\)(E) \(30\)

2

在下图数组中，三个 \(3\) 被一圈 \(2\) 包围，而这一圈又被一圈 \(1\) 包围。数组中所有数的和是多少？

\[ \begin{array}{ccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 & 2 & 2 & 1 \\ 1 & 2 & 3 & 3 & 3 & 2 & 1 \\ 1 & 2 & 2 & 2 & 2 & 2 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \end{array} \]

A\(49\) B\(51\) C\(53\) D\(55\) E\(57\)

English

In the array shown below, three \(3\)s are surrounded by \(2\)s, which are in turn surrounded by a border of \(1\)s. What is the sum of the numbers in the array?

\[ \begin{array}{ccccccc} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 & 2 & 2 & 1 \\ 1 & 2 & 3 & 3 & 3 & 2 & 1 \\ 1 & 2 & 2 & 2 & 2 & 2 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \end{array} \]

(A) \(49\)(B) \(51\)(C) \(53\)(D) \(55\)(E) \(57\)

3

Haruki 有一根长 \(24\) 厘米的铁丝。他想把它分别弯成下面各图形。

边长为 \(5\) 厘米的正六边形。

面积为 \(36\) 平方厘米的正方形。

两条直角边分别为 \(6\) 和 \(8\) 厘米的直角三角形。

Haruki 能弯成哪些图形？

A只有三角形 B只有六边形和正方形 C只有六边形和三角形 D只有正方形和三角形 E三种都可以

English

Haruki has a piece of wire that is \(24\) centimeters long. He wants to bend it to form each of the following shapes, one at a time.

A regular hexagon with side length \(5\) cm.

A square of area \(36\) cm².

A right triangle whose legs are \(6\) and \(8\) cm long.

Which of the shapes can Haruki make?

(A) Triangle only(B) Hexagon and square only(C) Hexagon and triangle only(D) Square and triangle only(E) Hexagon, triangle, and square

4

Brynn 的存款在七月减少了 \(20\%\)，然后在八月增加了 \(50\%\)。现在她的存款是原来的百分之多少？

A\(80\) B\(90\) C\(100\) D\(110\) E\(120\)

English

Brynn’s savings decreased by \(20\%\) in July, then increased by \(50\%\) in August. Brynn’s savings are now what percent of the original amount?

(A) \(80\)(B) \(90\)(C) \(100\)(D) \(110\)(E) \(120\)

5

Casey 开车旅行了 \(100\) 英里，途中只停下来吃了一次午饭。整趟旅行总共用了 \(3\) 小时，开车时的平均速度是每小时 \(40\) 英里。她的午饭休息用了多少分钟？

A\(15\) B\(30\) C\(40\) D\(45\) E\(60\)

English

Casey went on a road trip that covered \(100\) miles, stopping only for a lunch break along the way. The trip took \(3\) hours in total and her average speed while driving was \(40\) miles per hour. In minutes, how long was the lunch break?

(A) \(15\)(B) \(30\)(C) \(40\)(D) \(45\)(E) \(60\)

6

Peter 住在一片长满黑莓灌木的长方形田地附近。田地长 \(10\) 米、宽 \(8\) 米，Peter 能摘到距离田地边界 \(1\) 米以内的所有黑莓，如图中阴影部分所示。他能摘到黑莓的区域占整片田地面积的几分之几？

A\(\dfrac{1}{6}\) B\(\dfrac{1}{4}\) C\(\dfrac{1}{3}\) D\(\dfrac{3}{8}\) E\(\dfrac{2}{5}\)

English

Peter lives near a rectangular field that is filled with blackberry bushes. The field is \(10\) meters long and \(8\) meters wide, and Peter can reach any blackberries that are within \(1\) meter of an edge of the field. The portion of the field he can reach is shaded in the figure below. What fraction of the area of the field can Peter reach?

(A) \(\dfrac{1}{6}\)(B) \(\dfrac{1}{4}\)(C) \(\dfrac{1}{3}\)(D) \(\dfrac{3}{8}\)(E) \(\dfrac{2}{5}\)

7

Mika 想估计她的电动自行车充满电能骑多远。两次骑行一共骑了 \(40\) 英里。第一次用了 \(\frac{1}{2}\) 的电量，第二次用了 \(\frac{3}{10}\) 的电量。充满电大约可以骑多少英里？

A\(45\) B\(48\) C\(50\) D\(52\) E\(55\)

English

Mika would like to estimate how far she can ride a new model of electric bike on a fully charged battery. She completed two trips totaling \(40\) miles. The first trip used \(\frac{1}{2}\) of the total battery power, while the second trip used \(\frac{3}{10}\) of the total battery power. How many miles can this electric bike go on a fully charged battery?

(A) \(45\)(B) \(48\)(C) \(50\)(D) \(52\)(E) \(55\)

8

一项调查询问人们是否喜欢解数学题。正好有 \(74\%\) 的人回答“是”。参加调查的人数最少可能是多少？

A\(10\) B\(20\) C\(25\) D\(50\) E\(100\)

English

A poll asked a number of people if they liked solving mathematics problems. Exactly \(74\%\) answered “yes.” What is the fewest possible number of people who could have been asked the question?

(A) \(10\)(B) \(20\)(C) \(25\)(D) \(50\)(E) \(100\)

9

下面这个表达式的值是多少？\[ \frac{\sqrt{16\sqrt{81}}}{\sqrt{81\sqrt{16}}} \]

A\(\dfrac{4}{9}\) B\(\dfrac{2}{3}\) C\(1\) D\(\dfrac{3}{2}\) E\(\dfrac{9}{4}\)

English

What is the value of this expression? \[ \frac{\sqrt{16\sqrt{81}}}{\sqrt{81\sqrt{16}}} \]

(A) \(\dfrac{4}{9}\)(B) \(\dfrac{2}{3}\)(C) \(1\)(D) \(\dfrac{3}{2}\)(E) \(\dfrac{9}{4}\)

10

五名跑步者 Luke、Melina、Nico、Olympia 和 Pedro 完成了艰苦的 Xmarathon 马拉松赛。

• Nico 比 Pedro 晚 \(11\) 分钟到达终点。

• Olympia 比 Melina 早 \(2\) 分钟到达，但比 Pedro 晚 \(3\) 分钟到达。

• Olympia 比 Luke 早 \(6\) 分钟到达。

哪名跑步者第四个到达终点？

ALuke BMelina CNico DOlympia EPedro

English

Five runners completed the grueling Xmarathon: Luke, Melina, Nico, Olympia, and Pedro.

• Nico finished \(11\) minutes behind Pedro.

• Olympia finished \(2\) minutes ahead of Melina, but \(3\) minutes behind Pedro.

• Olympia finished \(6\) minutes ahead of Luke.

Which runner finished fourth?

(A) Luke(B) Melina(C) Nico(D) Olympia(E) Pedro

11

边长为 \(1\)、\(1\)、\(2\)、\(3\) 和 \(5\) 的正方形拼成了图中的长方形。曲线由每个正方形中的四分之一圆弧组成，圆弧按正方形从小到大的顺序连接。曲线的长度是多少？

A\(4\pi\) B\(6\pi\) C\(\dfrac{13}{2}\pi\) D\(8\pi\) E\(13\pi\)

English

Squares of side length \(1,\) \(1,\) \(2,\) \(3,\) and \(5\) are arranged to form the rectangle shown below. A curve is drawn by inscribing a quarter circle in each square and joining the quarter circles in order, from shortest to longest. What is the length of the curve?

(A) \(4\pi\)(B) \(6\pi\)(C) \(\dfrac{13}{2}\pi\)(D) \(8\pi\)(E) \(13\pi\)

12

图中六个圆圈要填入数字 \(1\) 到 \(6\)。每个数字恰好用一次。相邻圆圈之间方框里的数表示这两个圆圈中数字的和。最上面的圆圈中应该填什么数字？

A\(2\) B\(3\) C\(4\) D\(5\) E不可能

English

In the figure below, each circle will be filled with a digit from \(1\) to \(6.\) Each digit must appear exactly once. The sum of the digits in neighboring circles is shown in the box between them. What digit must be placed in the top circle?

(A) \(2\)(B) \(3\)(C) \(4\)(D) \(5\)(E) it is impossible to fill the circles

13

图中由 \(1 \times 1\) 的单位正方形铺成，相邻两行之间水平错开半个单位。阴影正方形的每个顶点都是某个单位正方形的顶点。阴影正方形的面积是多少？

A\(10\) B\(\dfrac{21}{2}\) C\(\dfrac{32}{3}\) D\(11\) E\(\dfrac{34}{3}\)

English

The figure below shows a tiling of \(1 \times 1\) unit squares. Each row of unit squares is shifted horizontally by half a unit relative to the row above it. A shaded square is drawn on top of the tiling. Each vertex of the shaded square is a vertex of one of the unit squares. In square units, what is the area of the shaded square?

(A) \(10\)(B) \(\dfrac{21}{2}\)(C) \(\dfrac{32}{3}\)(D) \(11\)(E) \(\dfrac{34}{3}\)

14

Jami 选了三个等间隔的整数。第一个数和第二个数的和是 \(40\)，第二个数和第三个数的和是 \(60\)。这三个数的和是多少？

A\(70\) B\(75\) C\(80\) D\(85\) E\(90\)

English

Jami picked three equally spaced integer numbers on the number line. The sum of the first and the second numbers is \(40,\) while the sum of the second and third numbers is \(60.\) What is the sum of all three numbers?

(A) \(70\)(B) \(75\)(C) \(80\)(D) \(85\)(E) \(90\)

15

Elijah 有一些立方体，每个立方体有 \(4\) 个面未涂色，另外 \(2\) 个相邻的面涂了阴影。他把这些立方体面贴面地粘在一起。图中显示 \(2\) 个这样的立方体粘在一起后，仍有 \(3\) 个阴影面可见。为了保证无论怎样旋转整个图形，都没有阴影面可见，至少需要多少个立方体？

A\(4\) B\(6\) C\(8\) D\(9\) E\(27\)

English

Elijah has a large collection of identical wooden cubes which are plain on \(4\) faces and shaded on \(2\) faces that share an edge. He glues some cubes together face-to-face. The figure below shows \(2\) cubes being glued together, leaving \(3\) shaded faces visible. What is the fewest number of cubes that he could glue together to ensure that no shaded faces are visible, no matter how he rotates the figure?

(A) \(4\)(B) \(6\)(C) \(8\)(D) \(9\)(E) \(27\)

16

考虑所有只由偶数数字组成的正四位整数。其中有多少比例能被 \(4\) 整除？

A\(\dfrac{1}{4}\) B\(\dfrac{2}{5}\) C\(\dfrac{1}{2}\) D\(\dfrac{3}{5}\) E\(\dfrac{3}{4}\)

English

Consider all positive four-digit integers consisting of only even digits. What fraction of these integers are divisible by \(4?\)

(A) \(\dfrac{1}{4}\)(B) \(\dfrac{2}{5}\)(C) \(\dfrac{1}{2}\)(D) \(\dfrac{3}{5}\)(E) \(\dfrac{3}{4}\)

17

四名学生坐成一排。他们可以和坐在自己旁边的人聊天。现在重新安排座位，使得没有任何一对原来相邻的学生仍然相邻。这样的重新安排有多少种？

A\(2\) B\(4\) C\(9\) D\(12\) E\(24\)

English

Four students are seated in a row. They chat with the people sitting next to them, then rearrange themselves so that they are no longer seated next to any of the same people. How many rearrangements are possible?

(A) \(2\)(B) \(4\)(C) \(9\)(D) \(12\)(E) \(24\)

18

有多少种方法可以把 \(60\) 写成两个或更多个连续正奇数的和，并且这些奇数按递增顺序排列？

A\(1\) B\(2\) C\(3\) D\(4\) E\(5\)

English

In how many ways can \(60\) be written as the sum of two or more consecutive odd positive integers that are arranged in increasing order?

(A) \(1\)(B) \(2\)(C) \(3\)(D) \(4\)(E) \(5\)

19

Miguel 带着狗 Luna 散步。在公园入口处，他把球直直地向前扔到一棵树旁，然后自己继续以稳定速度向前走。Luna 跑到停在树旁的球那里，再跑回 Miguel 身边。Luna 跑步的速度是 Miguel 走路速度的 \(5\) 倍。当 Luna 把球带回 Miguel 身边时，Miguel 已经走过了入口到树之间距离的几分之几？

A\(\dfrac{1}{6}\) B\(\dfrac{1}{5}\) C\(\dfrac{1}{4}\) D\(\dfrac{1}{3}\) E\(\dfrac{2}{5}\)

English

Miguel is walking with his dog, Luna. When they reach the entrance to a park, Miguel throws a ball straight ahead and continues to walk at a steady pace. Luna sprints toward the ball, which stops by a tree. As soon as the dog reaches the ball, she brings it back to Miguel. Luna runs \(5\) times faster than Miguel walks. What fraction of the distance between the entrance and the tree does Miguel cover by the time Luna brings him the ball?

(A) \(\dfrac{1}{6}\)(B) \(\dfrac{1}{5}\)(C) \(\dfrac{1}{4}\)(D) \(\dfrac{1}{3}\)(E) \(\dfrac{2}{5}\)

20

Catania 有两种硬币：金币厚 \(1\) 毫米，银币厚 \(3\) 毫米。用这些硬币堆成高度为 \(8\) 毫米的一摞共有多少种方法？硬币的上下顺序不同算不同方法。

A\(3\) B\(7\) C\(10\) D\(13\) E\(16\)

English

The land of Catania uses gold coins and silver coins. Gold coins are \(1\) mm thick and silver coins are \(3\) mm thick. In how many ways can Taylor make a stack of coins that is \(8\) mm tall using any arrangement of gold and silver coins, assuming order matters?

(A) \(3\)(B) \(7\)(C) \(10\)(D) \(13\)(E) \(16\)

21

蜘蛛 Charlotte 在图中形状像 \(5\) 角星的蛛网上随机行走。蛛网有 \(5\) 个外顶点和 \(5\) 个内顶点。每次到达一个顶点时，她都会随机选择一个相邻顶点并走过去。她从一个外顶点出发，走 \(3\) 步后在外顶点的概率是多少？

A\(\dfrac{1}{5}\) B\(\dfrac{1}{4}\) C\(\dfrac{2}{5}\) D\(\dfrac{1}{2}\) E\(\dfrac{3}{5}\)

English

Charlotte the spider is walking along a web shaped like a \(5\)-pointed star, shown in the figure below. The web has \(5\) outer points and \(5\) inner points. Each time Charlotte reaches a point, she randomly chooses a neighboring point and moves to that point. Charlotte starts at one of the outer points and makes \(3\) moves (re-visiting points is allowed). What is the probability she is now at one of the outer points of the star?

(A) \(\dfrac{1}{5}\)(B) \(\dfrac{1}{4}\)(C) \(\dfrac{2}{5}\)(D) \(\dfrac{1}{2}\)(E) \(\dfrac{3}{5}\)

22

将整数 \(1\) 到 \(25\) 任意分成五组，每组 \(5\) 个数。找出每组的中位数，再在这五个中位数中取中位数，记为 \(M\)。\(M\) 的最小可能值是多少？

A\(9\) B\(10\) C\(12\) D\(13\) E\(14\)

English

The integers from \(1\) through \(25\) are arbitrarily separated into five groups of \(5\) numbers each. The median of each group is identified. Let \(M\) equal the median of the five medians. What is the least possible value of \(M?\)

(A) \(9\)(B) \(10\)(C) \(12\)(D) \(13\)(E) \(14\)

23

Lakshmi 有 \(5\) 枚圆形硬币，每枚直径为 \(4\) 厘米。她把硬币按图中 \(2\) 行摆在桌上，并用一根弹性带紧紧围住它们。弹性带的长度是多少厘米？

A\(2\pi + 20\) B\(\dfrac{5}{2}\pi + 20\) C\(4\pi + 20\) D\(\dfrac{9}{2}\pi + 20\) E\(5\pi + 20\)

English

Lakshmi has \(5\) round coins of diameter \(4\) centimeters. She arranges the coins in \(2\) rows on a table top, as shown below, and wraps an elastic band tightly around them. In centimeters, what will be the length of the band?

(A) \(2\pi + 20\)(B) \(\dfrac{5}{2}\pi + 20\)(C) \(4\pi + 20\)(D) \(\dfrac{9}{2}\pi + 20\)(E) \(5\pi + 20\)

24

记号 \(n!\)（读作“\(n\) 的阶乘”）定义为前 \(n\) 个正整数的乘积。（例如，\(3! = 1 \cdot 2 \cdot 3 = 6\)。）定义正整数的超级阶乘 \(n^!\) 为前 \(n\) 个整数的阶乘的乘积。（例如，\(3^! = 1! \cdot 2! \cdot 3! = 12\)。）\(51^!\)（也就是 \(51\) 的超级阶乘）的质因数分解中含有多少个因数 \(7\)？

A\(147\) B\(150\) C\(156\) D\(168\) E\(171\)

English

The notation \(n!\) (read “\(n\) factorial”) is defined as the product of the first \(n\) positive integers. (For example, \(3! = 1 \cdot 2 \cdot 3 = 6.\)) Define the superfactorial of a positive integer, denoted by \(n^!,\) to be the product of the factorials of the first \(n\) integers. (For example, \(3^! = 1! \cdot 2! \cdot 3! = 12.\)) How many factors of \(7\) appear in the prime factorization of \(51^!,\) the superfactorial of \(51?\)

(A) \(147\)(B) \(150\)(C) \(156\)(D) \(168\)(E) \(171\)

25

一个等角六边形的所有内角都是 \(120^\circ\)。图中例子边长依次为 \(2\)、\(3\)、\(1\)、\(3\)、\(2\) 和 \(2\)，并且这个六边形内接于等边三角形 \(\triangle ABC\)。考虑所有边长为正整数、内接于三角形 \(ABC\) 的等角六边形，六个顶点都在三角形边上。按旋转和反射视为相同，一共有多少种？

A\(4\) B\(5\) C\(6\) D\(7\) E\(8\)

English

In an equiangular hexagon, all interior angles measure \(120^\circ.\) An example of such a hexagon with side lengths of \(2,\) \(3,\) \(1,\) \(3,\) \(2,\) and \(2\) is shown below, inscribed in equilateral triangle \(ABC.\) Consider all equiangular hexagons with positive integer side lengths that can be inscribed in \(\triangle ABC,\) with all six vertices on the sides of the triangle. What is the total number of such hexagons? Hexagons that differ only by a rotation or a reflection are considered the same.

(A) \(4\)(B) \(5\)(C) \(6\)(D) \(7\)(E) \(8\)