2003 年 AMC 8 真题（P版）

1 答案 E

一个立方体有 \(12\) 条棱、\(8\) 个顶点和 \(6\) 个面。相加得到 \[ 12 + 8 + 6 = 26\text{。}\]

所以正确答案是 E。

English

A cube has \(12\) edges, \(8\) corners, and \(6\) faces. Adding these together yields \[ 12 + 8 + 6 = 26. \]

Thus, E is the correct answer.

2 答案 C

最小的质数是 \(2\)。因此任何偶数都会是答案。

所以正确答案是 C。

English

Note that the smallest prime number is \(2.\) This means that any even number would be our answer.

Thus, C is the correct answer.

3 答案 D

不是填充物的部分为 \(120 - 30 = 90\) 克。因此百分比为 \[ 100 \cdot \dfrac{90}{120} = 100 \cdot \dfrac{3}{4} = 75 \%\text{。}\]

所以正确答案是 D。

English

We get that \(120 - 30 = 90\) grams are not filler. The percentage is therefore \[ 100 \cdot \dfrac{90}{120} = 100 \cdot \dfrac{3}{4} = 75 \%. \]

Thus, D is the correct answer.

4 答案 C

设 \(b\) 为自行车数量，\(t\) 为三轮车数量。可列方程组：\[ \begin{gather*} b + t = 7, \\ 2b + 3t = 19\text{。} \end{gather*} \] 将第一个方程乘以 \(2\)，再从第二个方程中减去所得等式，得到 \(t = 5\)。

所以正确答案是 C。

English

Let \(b\) be the number of bicycles and \(t\) be the number of tricycles. Then we can set up the following system of equations: \[ \begin{gather*} b + t = 7, \\ 2b + 3t = 19. \end{gather*} \] Multiplying the first equation by \(2\) and subtracting from the second equation, we get \(t = 5.\)

Thus, C is the correct answer.

5 答案 B

因为这个数的 \(20\%\) 是 \(12\)，所以这个数的 \(10\%\) 是 \(6\)。

因此同一个数的 \(30\%\) 是 \(3\cdot6=18\)。

所以正确答案是 B。

English

Since \(20\%\) of the number is \(12,\) \(10\%\) of the number is \(6.\)

Therefore \(30\%\) of the same number is \(3\cdot6=18.\)

Thus, B is the correct answer.

6 答案 B

三个正方形的边长分别为 \[ \sqrt{169} = 13\text{、}\]\[ \sqrt{144} = 12 \] 和 \[ \sqrt{25} = 5\text{。}\]这些长度构成一个勾股数组。

因此内部三角形是直角三角形，面积为 \[ \dfrac{1}{2} \cdot 5 \cdot 12 = 30\text{。}\]

所以正确答案是 B。

English

The side lengths of the squares are \[ \sqrt{169} = 13,\]\[ \sqrt{144} = 12, \] and \[ \sqrt{25} = 5. \] These lengths form a Pythagorean triple.

Therefore, the interior triangle is right. Its area is \[ \dfrac{1}{2} \cdot 5 \cdot 12 = 30. \]

Thus, B is the correct answer.

7 答案 A

两人四次测试的总分差为 \[ 10 - 10 + 20 \cdot 2 = 40\text{，}\]所以平均分之差为 \(40 \div 4 = 10\)。

所以正确答案是 A。

English

The total point difference between the two is \[ 10 - 10 + 20 \cdot 2 = 40. \] The average of this difference is \(40 \div 4 = 10.\)

Thus, A is the correct answer.

8 答案 A

因为所有饼干厚度相同且使用相同数量的面团，所以面积最大的饼干形状会产生最少的饼干。

Art 的梯形面积为 \(\frac12(3+5)\cdot3=12\)。Roger 的长方形面积为 \(4\cdot2=8\)。Paul 的平行四边形面积为 \(3\cdot2=6\)，Trisha 的三角形面积为 \(\frac12\cdot3\cdot4=6\)。

Art 的饼干面积最大，所以 Art 用一批面团得到的饼干最少。

所以正确答案是 A。

English

Since the cookies all have the same thickness and use the same amount of dough, the largest cookie shape produces the fewest cookies.

Art’s trapezoid has area \(\frac12(3+5)\cdot3=12.\) Roger’s rectangle has area \(4\cdot2=8.\) Paul’s parallelogram has area \(3\cdot2=6,\) and Trisha’s triangle has area \(\frac12\cdot3\cdot4=6.\)

Art has the largest cookie area, so Art gets the fewest cookies from one batch.

Thus, A is the correct answer.

9 答案 C

Art 做 \(12\) 块饼干，每块卖 \(60\) 分，所以一批收入 \(12\cdot60=720\) 分。

这批面团面积为 \(12\cdot12=144\) 平方英寸，每块 Roger 饼干面积为 \(4\cdot2=8\)。因此 Roger 能做 \(144\div8=18\) 块饼干。

一批需要赚到 \(720\) 分，而 Roger 有 \(18\) 块饼干，因此每块应卖 \(720\div18=40\) 分。

所以正确答案是 C。

English

Art makes \(12\) cookies that sell for \(60\) cents each, so one batch earns \(12\cdot60=720\) cents.

The batch has \(12\cdot12=144\) square inches of dough area, and each Roger cookie has area \(4\cdot2=8.\) Thus Roger makes \(144\div8=18\) cookies.

To earn \(720\) cents from \(18\) cookies, each Roger cookie should cost \(720\div18=40\) cents.

Thus, C is the correct answer.

10 答案 E

Art 的一块饼干面积为 \(12\) 平方英寸，所以整批面团面积为 \(12\cdot12=144\) 平方英寸。

Trisha 的三角形饼干面积为 \(\frac12\cdot3\cdot4=6\) 平方英寸。

因此 Trisha 每批可以做 \(144\div6=24\) 块饼干。

所以正确答案是 E。

English

Art’s cookie area is \(12\) square inches, so the whole batch has area \(12\cdot12=144\) square inches.

Trisha’s triangular cookie has area \(\frac12\cdot3\cdot4=6\) square inches.

Therefore Trisha can make \(144\div6=24\) cookies per batch.

Thus, E is the correct answer.

11 答案 B

星期五鞋价提高 \(10\%\)，所以标价变为 \(40\cdot1.10=44\) 美元。

星期一再打 \(10\%\) 的折扣，从 \(\$44\) 降到 \(44\cdot0.90=39.60\) 美元。

所以正确答案是 B。

English

On Friday, the shoes are marked up by \(10\%,\) so the listed price becomes \(40\cdot1.10=44\) dollars.

On Monday, the \(10\%\) discount is taken from \(\$44,\) giving \(44\cdot0.90=39.60\) dollars.

Thus, B is the correct answer.

12 答案 E

如果数字 \(6\) 的面可见，那么可见数字的乘积能被 \(6\) 整除。

如果数字 \(6\) 在底面，那么可见面仍包含 \(2\) 和 \(3\)，所以它们的乘积仍能被 \(6\) 整除。

每种可能的掷骰结果都满足条件，所以概率为 \(1\)。

所以正确答案是 E。

English

If the face numbered \(6\) is visible, then the visible product is divisible by \(6.\)

If the face numbered \(6\) is on the bottom, then the visible faces include both \(2\) and \(3,\) so their product is still divisible by \(6.\)

Every possible toss works, so the probability is \(1.\)

Thus, E is the correct answer.

13 答案 B

一个小立方体恰好有四个面被涂色，当且仅当它恰好与两个其他立方体相连。

\(4\) 个顶部立方体只接触一个其他立方体，所以它们有 \(5\) 个涂色面。\(4\) 个底部角落立方体接触三个其他立方体，所以它们有 \(3\) 个涂色面。

剩下的 \(14-4-4=6\) 个立方体各恰好接触两个其他立方体，所以它们恰好有四个涂色面。

所以正确答案是 B。

English

A cube has exactly four painted faces exactly when it is attached to exactly two other cubes.

The \(4\) top cubes touch only one other cube, so they have \(5\) painted faces. The \(4\) bottom corner cubes touch three other cubes, so they have \(3\) painted faces.

The remaining \(14-4-4=6\) cubes each touch exactly two other cubes, so they have exactly four painted faces.

Thus, B is the correct answer.

14 答案 D

因为两个 \(T\) 都是 \(7\)，所以 \(O\) 是 \(4\) 或 \(5\)。由于 \(O\) 是偶数，得到 \(O = 4\)。

于是 \(R = 4 + 4 = 8\)。还知道 \(W + W\) 不会进位，否则 \(O\) 会是 \(5\)。

因此 \(W\) 小于 \(5\)，且不能是 \(4\) 或 \(1\)。如果 \(W = 0\)，则 \(U = 0\)，两个字母会代表同一个数字。如果 \(W = 2\)，则 \(U = 4\)，这也不允许。

所以 \(W = 3\)。

所以正确答案是 D。

English

Since both \(T\)’s are \(7,\) we get that \(O\) is either \(4\) or \(5.\) Since \(O\) is even, we get that \(O = 4.\)

Then, we get that \(R = 4 + 4 = 8.\) We also know that \(W + W\) doesn’t carry over, since otherwise \(O\) would be \(5.\)

Therefore, \(W\) is less than \(5\) and cannot be \(4\) or \(1.\) If \(W = 0,\) then \(U = 0,\) which gives two letters the same digit. If \(W = 2,\) then \(U = 4,\) which is also not allowed.

This makes \(W = 3.\)

Thus, D is the correct answer.

15 答案 B

正视图需要一个有三个可见位置的 L 形，所以至少需要三个立方体。

假设恰好有三个立方体。正视图中上下相叠的两个立方体必须共享一个面，因此它们处于同一深度。第三个立方体在正视图中位于下方立方体的旁边。若要和前两个立方体中的一个共享面，它也必须和下方立方体处于同一深度；否则它与这两个立方体都不相连。因此三个立方体都在同一个深度列中，侧视图便只有一列，而不是题目要求的 L 形。

图中的四立方体构造具有两个所需视图，所以最小值是 \(4\)。

所以正确答案是 B。

English

The front view requires an L-shape with three visible positions, so at least three cubes are needed.

Suppose there were exactly three cubes. The two cubes that appear one above the other must share a face, so they have the same depth. The third cube appears beside the lower one. To share a face with either of the first two cubes, it must share the lower cube’s depth as well; otherwise it would be disconnected from both. Thus all three cubes would occupy one depth column, giving a one-column side view instead of the required L-shape.

The shown four-cube construction has both required views, so the minimum is \(4.\)

Thus, B is the correct answer.

16 答案 D

驾驶座有 \(2\) 种选择。另一个前排座位有 \(3\) 种选择，第一个后排座位有 \(2\) 种选择。

最后一个人必须坐最后一个座位，所以共有 \[ 2 \cdot 3 \cdot 2 = 12 \] 种座位安排。

所以正确答案是 D。

English

There are \(2\) options for who sits in the driver’s seat. There are \(3\) options for the other front seat, and \(2\) options for the first back seat.

The last person has to sit in the last seat, for a total of \[ 2 \cdot 3 \cdot 2 = 12 \] possible seating arrangements.

Thus, D is the correct answer.

17 答案 E

Nadeen、Austin 和 Sue 是唯一与 Jim 共享某个特征的 \(3\) 个人。需要找出其中哪一个与其他人完全不同。

Austin 和 Sue 都有蓝色眼睛，所以 Nadeen 是不属于这一组的人。因此 Austin 和 Sue 是 Jim 的兄弟姐妹。

所以正确答案是 E。

English

Note that Nadeen, Austin, and Sue are the only individuals who share a characteristic with Jim. We need to find which of the \(3\) are completely different from the others.

Austin and Sue both have blue eyes, which makes Nadeen the odd one out. Therefore, Austin and Sue are Jim’s siblings.

Thus, E is the correct answer.

18 答案 D

Sarah 邀请她的朋友，以及至少和她某个朋友是朋友的同学。用图论语言说，她邀请距离 Sarah 为 \(1\) 或 \(2\) 条线段的点。

从图中可见，有 \(4\) 个点与 Sarah 所在连通分量断开，另外还有 \(2\) 个点在 Sarah 所在连通分量中但距离为 \(3\) 条线段。

因此不会被邀请的同学有 \(4+2=6\) 位。

所以正确答案是 D。

English

Sarah invites her friends and the classmates who are friends with at least one of her friends. In graph terms, she invites dots that are \(1\) or \(2\) line segments away from Sarah.

From the graph, \(4\) dots are disconnected from Sarah’s component, and \(2\) more dots in Sarah’s component are \(3\) segments away.

Those \(4+2=6\) classmates will not be invited.

Thus, D is the correct answer.

19 答案 C

如果一个数 \(x\) 以这三个数为因数，那么它们的最小公倍数也必须整除 \(x\)。

这些数的质因数分解为 \[ 15 = 3 \cdot 5\text{，}\]\[ 20 = 2^2 \cdot 5\text{，}\]\[ 25 = 5^2\text{。}\]

因此最小公倍数为 \[ 2^2 \cdot 3 \cdot 5^2 = 300\text{。}\]

\(300\) 的倍数中，位于 \(1000\) 和 \(2000\) 之间的是 \(1200\)、\(1500\) 和 \(1800\)。

所以正确答案是 C。

English

If a number \(x\) has these three numbers as factors, then their least common multiple must also divide \(x.\)

These numbers have the following prime factorizations: \[ 15 = 3 \cdot 5,\]\[ 20 = 2^2 \cdot 5,\]\[ 25 = 5^2. \]

From these values, we get that the least common multiple is \[ 2^2 \cdot 3 \cdot 5^2 = 300. \]

Therefore, the multiples of \(300\) between \(1000\) and \(2000\) are \(1200,\) \(1500,\) and \(1800.\)

Thus, C is the correct answer.

20 答案 D

在 \(4{:}20\)，时针走过了相邻小时刻度间距的 \(\frac{1}{3}\)，因而位于 \(4\) 和 \(5\) 之间。

每个小时刻度代表 \(360 \div 12 = 30^{\circ}\)。所以时针又移动了 \(30 \div 3 = 10^{\circ}\)，超过 \(4\) 的位置。

\(20\) 分钟时，分针指向 \(4\)，所以两根指针形成的锐角为 \(10^{\circ}\)。

所以正确答案是 D。

English

At \(4{:}20,\) the hour hand will be \(\frac{1}{3}\) of the way between \(4\) and \(5.\)

Each hour represents \(360 \div 12 = 30^{\circ}.\) This means the hour hand will be \(30 \div 3 = 10^{\circ}\) past \(4.\)

At \(20\) minutes, the minute hand points to \(4,\) so the acute angle between the hands is \(10^{\circ}.\)

Thus, D is the correct answer.

21 答案 B

从 \(B\) 和 \(C\) 向 \(\overline{AD}\) 作垂线，垂足分别为 \(E\) 和 \(F\)。

在左右两个直角三角形中，\(AE=\sqrt{10^2-8^2}=6\)，\(FD=\sqrt{17^2-8^2}=15\)。

两侧三角形面积分别为 \(\frac12\cdot6\cdot8=24\) 和 \(\frac12\cdot15\cdot8=60\)。中间长方形面积为 \(8\cdot BC\)。

因此 \(164=24+60+8BC\)，所以 \(8BC=80\)，\(BC=10\)。

所以正确答案是 B。

English

Drop perpendiculars from \(B\) and \(C\) to \(\overline{AD},\) meeting it at \(E\) and \(F.\)

In the left and right right triangles, \(AE=\sqrt{10^2-8^2}=6\) and \(FD=\sqrt{17^2-8^2}=15.\)

The two side triangles have areas \(\frac12\cdot6\cdot8=24\) and \(\frac12\cdot15\cdot8=60.\) The middle rectangle has area \(8\cdot BC.\)

Thus \(164=24+60+8BC,\) so \(8BC=80\) and \(BC=10.\)

Thus, B is the correct answer.

22 答案 C

在图形 \(A\) 中，阴影面积是一个 \(2\) 乘 \(2\) 正方形面积减去半径为 \(1\) 的圆面积，即 \(4-\pi\)。

图形 \(B\) 由四个边长为图形 \(A\) 一半的副本组成，所以每个副本的面积是原图的四分之一，总阴影面积也为 \(4-\pi\)。

在图形 \(C\) 中，圆的半径为 \(1\)，内接正方形的对角线为 \(2\)，所以其面积为 \(2\)。阴影面积为 \(\pi-2\)。

因为 \(\pi>3\)，所以 \(\pi-2>4-\pi\)，因此图形 \(C\) 的阴影面积最大。

所以正确答案是 C。

English

In figure \(A,\) the shaded area is the area of a \(2\) by \(2\) square minus a circle of radius \(1,\) so it is \(4-\pi.\)

Figure \(B\) is made of four copies of figure \(A\) with half the side length, so each copy has one-fourth the area and the total shaded area is also \(4-\pi.\)

In figure \(C,\) the circle has radius \(1,\) and the inscribed square has diagonal \(2,\) so its area is \(2.\) The shaded area is \(\pi-2.\)

Since \(\pi>3,\) we have \(\pi-2>4-\pi,\) so figure \(C\) has the largest shaded area.

Thus, C is the correct answer.

23 答案 A

猫的位置每 \(4\) 次移动重复一次，老鼠的位置每 \(8\) 次移动重复一次。

因为 \(247\equiv3\pmod{4}\)，猫的位置与第 \(3\) 次移动后相同：在右下方正方形。

因为 \(247\equiv7\pmod{8}\)，老鼠的位置与第 \(7\) 次移动后相同：在左下方正方形的左边。

所以正确答案是 A。

English

The cat’s position repeats every \(4\) moves, and the mouse’s position repeats every \(8\) moves.

Since \(247\equiv3\pmod{4},\) the cat is in the same position as after the \(3\)rd move: the lower right square.

Since \(247\equiv7\pmod{8},\) the mouse is in the same position as after the \(7\)th move: the left side of the lower left square.

Thus, A is the correct answer.

24 答案 B

从 \(A\) 到 \(B\) 的半圆路径上每一点到圆心 \(X\) 的距离都相同，所以图像一开始是水平的。

在从 \(B\) 到 \(C\) 的直线路径上，船先靠近 \(X\)，然后远离 \(X\)。

只有图 \(B\) 先保持水平，然后下降再上升。

所以正确答案是 B。

English

Every point on the semicircular path from \(A\) to \(B\) is the same distance from the center \(X,\) so the graph starts horizontally.

On the straight path from \(B\) to \(C,\) the ship first gets closer to \(X\) and then farther away from \(X.\)

Only graph \(B\) starts flat and then decreases before increasing.

Thus, B is the correct answer.

25 答案 C

正方形 \(WXYZ\) 的边长为 \(5\) cm，因为 \(\sqrt{25} = 5\)。

\(\overline{WZ}\) 到 \(\overline{BC}\) 的距离是 \(2\)，因为它等于 \(2\) 个单位正方形的边长之和。

最后，\(A\) 到 \(\overline{BC}\) 的距离等于 \(\overline{BC}\) 到 \(O\) 的距离，即 \[ 2 + \dfrac{5}{2} = \dfrac{9}{2} \text{ cm}\text{。}\]

现在可求 \(BC\)：\[ WZ - 2 = 5 - 2 = 3 \text{ cm}\text{。}\]

因此 \(\triangle ABC\) 的面积为 \[ \dfrac{1}{2} \cdot 3 \cdot \dfrac{9}{2} = \dfrac{27}{4} \text{ cm}^2\text{。}\]

所以正确答案是 C。

English

The side length of \(WXYZ\) is \(5\) cm, since \(\sqrt{25} = 5.\)

We also know that the distance from \(\overline{WZ}\) to \(\overline{BC}\) is \(2\) since it is the sum of the side lengths of \(2\) unit squares.

Finally, the distance from \(A\) to \(\overline{BC}\) is the same as the distance from \(\overline{BC}\) to \(O,\) which is \[ 2 + \dfrac{5}{2} = \dfrac{9}{2} \text{ cm}. \]

Now, we can find \(BC,\) which is \[ WZ - 2 = 5 - 2 = 3 \text{ cm}. \]

Therefore, the area of \(\triangle ABC\) is \[ \dfrac{1}{2} \cdot 3 \cdot \dfrac{9}{2} = \dfrac{27}{4} \text{ cm}^2. \]

Thus, C is the correct answer.