2007 年 AMC 8 真题（P版）

1 答案 D

前 \(5\) 周 Theresa 总共工作 \[8+11+7+12+10=48\] 小时。

她承诺总共工作 \(10\cdot 6=60\) 小时。

因此她最后一周必须工作 \(60-48=12\) 小时才能得到门票。

所以正确答案是 D。

English

During the first \(5\) weeks, Theresa works for a total of \[8+11+7+12+10=48\] hours.

She promised to work \(10\cdot 6=60\) hours in all.

This means that she has to work \(60-48=12\) hours during the final week to earn the tickets.

Thus, D is the correct answer.

2 答案 E

喜欢意大利面条的学生有 \(250\) 人，喜欢意式夹馅卷的学生有 \(100\) 人。

比值为 \[ \dfrac{250}{100} = \dfrac{5}{2}\text{。}\]

所以正确答案是 E。

English

There are \(250\) students who preferred spaghetti and \(100\) that preferred manicotti.

The ratio is therefore \[ \dfrac{250}{100} = \dfrac{5}{2}. \]

Thus, E is the correct answer.

3 答案 C

对 \(250\) 分解质因数：\[ 250 = 2 \cdot 5^3\text{。}\]因此 \(250\) 只有两个质因数：\(2\) 和 \(5\)。它们的和为 \(7\)。

所以正确答案是 C。

English

We can prime factorize \(250\) to get \[ 250 = 2 \cdot 5^3. \] From this, we can see that \(250\) only has two prime factors: \(2\) and \(5.\) The sum of these is \(7.\)

Thus, C is the correct answer.

4 答案 D

Georgie 有 \(6\) 种选择进入的窗户。但离开时只能选择不同的窗户，所以有 \(5\) 种选择。

路径总数为 \(6 \cdot 5 = 30\)。

所以正确答案是 D。

English

Georgie has \(6\) options for which window he enters through. He, however, only has \(5\) options for the exit since it must be different from the entrance.

The total number of paths is therefore \(6 \cdot 5 = 30.\)

Thus, D is the correct answer.

5 答案 B

Chandler 的生日钱总额为 \(50+35+15=100\) 美元。因此他还需要 \(500-100=400\) 美元。

送报需要 \(400\div 16=25\) 周赚到这笔钱。

所以正确答案是 B。

English

The total amount of birthday money Chandler gets is \(50+35+15=100\) dollars. Therefore, he still needs \(500-100=400\) dollars.

This can be earned in \(400\div 16=25\) weeks through his paper route.

Thus, B is the correct answer.

6 答案 E

费用下降了 \(41-7=34\) 美分每分钟。

百分比下降为 \(100\cdot\dfrac{34}{41}\%\)，略大于 \(80\%\)，所以最接近的答案是 \(80\)。

所以正确答案是 E。

English

The decrease in cost is \(41-7=34\) cents per minute.

The percent decrease is \(100\cdot\dfrac{34}{41}\%\), which is a little more than \(80\%\), so the closest answer is \(80\).

Thus, E is the correct answer.

7 答案 D

起初，房间里所有人的总年龄为 \(5 \cdot 30 = 150\) 岁。

这个人离开后，总年龄为 \(150 - 18 = 132\)。剩下 \(4\) 人，平均年龄为 \(\dfrac{132}{4} = 33\)。

所以正确答案是 D。

English

Initially, the total age of everyone in the room is \(5 \cdot 30 = 150\) years.

After the person leaves, the total age is \(150 - 18 = 132.\) With \(4\) people remaining, the average age becomes \(\dfrac{132}{4} = 33.\)

Thus, D is the correct answer.

8 答案 B

我们有 \[ EC = DC - DE\] \[ = 6 - 3 \] \[= 3\text{。}\]还知道 \[ BE = AD = 3\text{。}\]因此 \(\triangle BEC\) 的面积为 \[ \dfrac{1}{2} \cdot 3 \cdot 3 = \dfrac{9}{2}\text{。}\]

所以正确答案是 B。

English

We know that \[ EC = DC - DE\]\[ = 6 - 3 \]\[= 3. \] We also know that \[ BE = AD = 3. \] Therefore, the area of \(\triangle BEC\) is \[ \dfrac{1}{2} \cdot 3 \cdot 3 = \dfrac{9}{2}. \]

Thus, B is the correct answer.

9 答案 B

第二行最后一个格子必须是 \(1\)，因为该行已有 \(2\) 和 \(3\)，最后一列已有 \(4\)。

接着右上角格子必须是 \(3\)，因为第一行已有 \(1\) 和 \(2\)，最后一列已有 \(1\) 和 \(4\)。

这迫使 \(2\) 位于右下角方格。

所以正确答案是 B。

English

The last square in the second row must be \(1\), since that row already contains \(2\) and \(3\), and the last column already contains \(4\).

Then the top-right square must be \(3\), since the top row already contains \(1\) and \(2\), and the last column already contains \(1\) and \(4\).

This forces the \(2\) to be in the bottom-right square.

Thus, B is the correct answer.

10 答案 D

首先，由于 \(11\) 是质数，\[ \boxed{11} = 1 + 11 = 12\text{。}\]接着求 \(\boxed{12}\)。\(12\) 的因数为 \[ 1, 2, 3, 4, 6, 12\text{。}\]相加得到 \(\boxed{12} = 28\)。

所以正确答案是 D。

English

First, we find that \[ \boxed{11} = 1 + 11 = 12 \] since \(11\) is prime. Then we need to find \(\boxed{12}.\) The factors of \(12\) are \[ 1, 2, 3, 4, 6, 12. \] Adding these yields \(\boxed{12} = 28.\)

Thus, D is the correct answer.

11 答案 D

只有瓷片 \(III\) 的边上有 \(0\)，所以瓷片 \(III\) 必须放在 \(C\) 或 \(D\)，使这个边可以在外侧。

瓷片 \(III\) 的右边还有 \(5\)，且没有其他瓷片有 \(5\)。因此瓷片 \(III\) 必须放在 \(D\)，让 \(5\) 在外侧。

\(1\) 标在瓷片 \(III\) 的左边，唯一能与之匹配的是瓷片 \(IV\)，所以瓷片 \(IV\) 放在 \(C\)。

所以正确答案是 D。

English

Only Tile \(III\) has a \(0\) on an edge, so Tile \(III\) must be placed in \(C\) or \(D\), where that edge can be on the outside.

Tile \(III\) also has a \(5\) on its right edge, and no other tile has a \(5\). Therefore Tile \(III\) must be placed in \(D\), with its \(5\) on the outside.

The only tile that can match the \(1\) on the left edge of Tile \(III\) is Tile \(IV\), so Tile \(IV\) is placed in \(C\).

Thus, D is the correct answer.

12 答案 A

可以把六边形分成 \(6\) 个全等等边三角形，如下图。

因为每个内部三角形都与一个外侧三角形共边，所以所有这些三角形全等。因此面积比为 \(1:1\)。

所以正确答案是 A。

English

Note that we can split the hexagon into \(6\) congruent equilateral triangles as follows.

Since each of them share an edge with an exterior triangle, all the triangles are congruent. Therefore, the ratio of areas is \(1:1.\)

Thus, A is the correct answer.

13 答案 C

设 \(x\) 为集合 \(A\) 和 \(B\) 各自的元素个数，并设 \(y\) 为它们交集的元素个数。

条件给出 \[ 2x - y = 2007 \] 且 \[ y = 1001\text{。}\]将 \(y\) 代入第一式得到 \[ \begin{align*} 2x - 1001 &= 2007 \\ 2x &= 3008 \\ x &= 1504 \end{align*}\text{。}\]

所以正确答案是 C。

English

Let \(x\) be the number of elements in each \(A\) and \(B.\) Also let \(y\) be the number of elements in their intersection.

The conditions give us that \[ 2x - y = 2007 \] and \[ y = 1001. \] Plugging \(y\) into the first equation yields \[ \begin{align*} 2x - 1001 &= 2007 \\ 2x &= 3008 \\ x &= 1504. \end{align*} \]

Thus, C is the correct answer.

14 答案 C

作 \(\overline{BD}\) 为从 \(B\) 到 \(\overline{AC}\) 的高。

则 \[ 60 = \dfrac{1}{2} \cdot BD \cdot 24\text{，}\]得到 \(BD = 5\)。

在 \(\triangle ABD\) 中使用勾股定理：\[ AB^2 = 5^2 + 12^2 = 169 = 13^2\text{。}\]因此 \(AB = 13\)。

所以正确答案是 C。

English

Construct \(\overline{BD}\) as the altitude from \(B\) to \(\overline{AC}.\)

Then \[ 60 = \dfrac{1}{2} \cdot BD \cdot 24, \] which gives us that \(BD = 5.\)

From this, we apply the Pythagorean Theorem on \(\triangle ABD:\) \[ AB^2 = 5^2 + 12^2 = 169 = 13^2. \] This gives us that \(AB = 13.\)

Thus, C is the correct answer.

15 答案 A

已知 \(b \lt c\) 且 \(0 \lt a\)。把这两个不等式相加得到 \[ b \lt c + a\text{。}\]这说明 A 不可能，因此就是正确答案。

为确认这一点，可以给出其他选项可行的例子。

B 和 C：取 \(a = 1\)、\(b = 2\)、\(c = 4\)。

D：取 \(a = \dfrac{1}{3}\)、\(b = \dfrac{1}{2}\)、\(c = 1\)。

E：取 \(a = \dfrac{1}{2}\)、\(b = 1\)、\(c = 2\)。

所以正确答案是 A。

English

We know that \(b \lt c\) and \(0 \lt a.\) Adding these two inequalities together yields \[ b \lt c + a. \] This shows that A is impossible, and therefore the right answer.

To ensure that this is correct, we can show that the other options are possible.

B and C : \(a = 1,\) \(b = 2,\) and \(c = 4\)

D : \(a = \dfrac{1}{3},\) \(b = \dfrac{1}{2},\) and \(c = 1\)

E : \(a = \dfrac{1}{2},\) \(b = 1,\) and \(c = 2\)

Thus, A is the correct answer.

16 答案 A

半径 \(1\) 到 \(5\) 的圆的周长分别为 \(2\pi\)、\(4\pi\)、\(6\pi\)、\(8\pi\) 和 \(10\pi\)，对应面积分别为 \(\pi\)、\(4\pi\)、\(9\pi\)、\(16\pi\) 和 \(25\pi\)。

当 \(C\) 等量增加时，\(A\) 增加得越来越快，所以这些点形成递增的二次形状。只有图 A 有这种形状。

所以正确答案是 A。

English

The circumferences of circles with radii \(1\) through \(5\) are \(2\pi,\) \(4\pi,\) \(6\pi,\) \(8\pi,\) and \(10\pi\). Their respective areas are \(\pi,\) \(4\pi,\) \(9\pi,\) \(16\pi,\) and \(25\pi\).

As \(C\) increases by equal amounts, \(A\) increases more and more quickly, so the points form an increasing quadratic pattern. Only graph A has that shape.

Thus, A is the correct answer.

17 答案 C

原混合物中黄色颜料为 \(0.3 \times 30 = 9\) 升。加入 \(5\) 升后，黄色颜料共有 \(14\) 升。

新混合物总量为 \(35\) 升，所以黄色颜料百分比为 \[ 100 \cdot \dfrac{14}{35} = 100 \cdot \dfrac{2}{5} = 40 \%\text{。}\]

所以正确答案是 C。

English

The amount of yellow tint in the original mixture is \(0.3 \times 30 = 9\) liters. Adding \(5\) liters results in a total of \(14\) liters of yellow tint.

The new mixture has a total of \(35\) liters, so the percent of yellow tint is \[ 100 \cdot \dfrac{14}{35} = 100 \cdot \dfrac{2}{5} = 40 \%. \]

Thus, C is the correct answer.

18 答案 D

只有最后四位会影响乘积的千位和个位。这两个数分别以 \(0303\) 和 \(0505\) 结尾，所以只需计算 \(303\cdot 505\)。

因为 \(303\cdot 505=153015\)，完整乘积的最后四位是 \(3015\)。

因此 \(A=3\)，\(B=5\)，所以 \(A+B=8\)。

所以正确答案是 D。

English

Only the last four digits can affect the thousands digit and units digit of the product. The two numbers end in \(0303\) and \(0505\), so it is enough to compute \(303\cdot 505\).

Since \(303\cdot 505=153015\), the last four digits of the full product are \(3015\).

This gives \(A=3\) and \(B=5\), so \(A+B=8\).

Thus, D is the correct answer.

19 答案 C

设连续正整数为 \(x\) 和 \(x+1\)。它们的和为 \(2x+1\)，小于 \(100\)。

它们的平方差为 \[ \begin{gathered} (x+1)^2-x^2 \\ = (x+1+x)(x+1-x) \\ = 2x+1 \end{gathered}\text{。}\]

所以这个差必须是小于 \(100\) 的奇数。选项中只有 \(79\) 可能，它由 \(39\) 和 \(40\) 得到。

所以正确答案是 C。

English

Let the consecutive positive integers be \(x\) and \(x+1\). Their sum is \(2x+1\), which is less than \(100\).

The difference of their squares is \[ \begin{gathered} (x+1)^2-x^2 \\ = (x+1+x)(x+1-x) \\ = 2x+1. \end{gathered} \]

So the difference must be an odd number less than \(100\). Among the choices, \(79\) is the only possibility, and it occurs for \(39\) and \(40\).

Thus, C is the correct answer.

20 答案 A

设 Unicorns 分区赛前打了 \(x\) 场比赛。那么分区赛前赢了 \(0.45x\) 场，分区赛后共赢了 \(0.45x+6\) 场。

他们总共打了 \(x+8\) 场。题意给出 \[0.45x+6=\dfrac{x+8}{2}\text{。}\]

解得 \(0.45x+6=0.5x+4\)，所以 \(2=0.05x\)，\(x=40\)。因此 Unicorns 总共打了 \(40+8=48\) 场。

所以正确答案是 A。

English

Let \(x\) be the number of games the Unicorns had played before district play. Then they had won \(0.45x\) games before district play and \(0.45x+6\) games after district play.

In total, they played \(x+8\) games. The problem statement tells us that \[0.45x+6=\dfrac{x+8}{2}.\]

Solving gives \(0.45x+6=0.5x+4\), so \(2=0.05x\), and \(x=40\). Therefore, the Unicorns played \(40+8=48\) games in all.

Thus, A is the correct answer.

21 答案 D

抽出第一张牌后，还剩 \(7\) 张。其中 \(3\) 张同色，\(1\) 张同字母。因此会与第一张牌形成获胜牌对的有 \(4\) 张，它们都在剩余的 \(7\) 张牌中。

概率为 \(\dfrac{4}{7}\)。

所以正确答案是 D。

English

After drawing the first card, there are \(7\) left. \(3\) of them have the same color, and \(1\) of them has the same letter. Therefore, there are \(4\) out of \(7\) possibilities that result in a winning pair.

The probability of drawing a pair is then \(\dfrac{4}{7}.\)

Thus, D is the correct answer.

22 答案 C

根据题目给出的长度，旅鼠停下后仍在正方形内。

因为旅鼠仍在正方形内，它到两条水平边的距离和为 \(10\) 米，到两条竖直边的距离和也一样。因此 \(4\) 个距离之和为 \(20\) 米，平均数为 \(20 \div 4 = 5\) 米。

所以正确答案是 C。

English

Based on the lengths given in the problem, the lemming is still in the square after it stops.

Since the lemming is still in the square, the sum of the distances to the horizontal sides is \(10\) meters and the same for the vertical sides. Therefore, the \(4\) distances sum to \(20\) meters, making the average \(20 \div 4 = 5\) meters.

Thus, C is the correct answer.

23 答案 B

可以用总面积减去未阴影面积来求阴影面积。

有 \(4\) 个边长为 \(1\) 的正方形，每个面积为 \(1\)，对未阴影面积贡献共 \(4\)。

还有 \(4\) 个底为 \(3\)、高为 \(\dfrac{5}{2}\) 的三角形，总面积为 \[ 4 \cdot \dfrac{1}{2} \cdot 3 \cdot \dfrac{5}{2} = 15\text{。}\]

未阴影总面积为 \(4 + 15 = 19\)。

整个方格面积为 \(5^2 = 25\)，所以阴影面积为 \(25 - 19 = 6\)。

所以正确答案是 B。

English

We can find the area of the shaded part by subtracting the area of the unshaded part from the whole area.

There are \(4\) squares with side length \(1,\) contributing \(1\) each to the unshaded area, for a total of \(4.\)

There are also \(4\) triangles with base \(3\) and height \(\dfrac{5}{2}.\) This contributes a total area of \[ 4 \cdot \dfrac{1}{2} \cdot 3 \cdot \dfrac{5}{2} = 15. \]

The total unshaded area is therefore \(4 + 15 = 19.\)

The total area is \(5^2 = 25,\) and subtracting the unshaded area yields \(25 - 19 = 6\) as the area of the shaded region.

Thus, B is the correct answer.

24 答案 C

一个数能被 \(3\) 整除，当且仅当它的数字和能被 \(3\) 整除。

从 \(1,2,3,4\) 中取三个不同数字，数字和能被 \(3\) 整除的只有 \((1,2,3)\) 和 \((2,3,4)\)。

因此所组成的数是 \(3\) 的倍数，当且仅当袋中剩下的是 \(1\) 或 \(4\)。

每种事件概率都是 \(\dfrac{1}{4}\)，总概率为 \(2\cdot\dfrac{1}{4}=\dfrac{1}{2}\)。

所以正确答案是 C。

English

Recall that a number is divisible by \(3\) if the sum of its digits is divisible by \(3\).

The only triples of distinct digits from \(1,2,3,4\) whose sum is divisible by \(3\) are \((1,2,3)\) and \((2,3,4)\).

This means the constructed number is a multiple of \(3\) exactly when either \(1\) or \(4\) is left in the bag.

Each of these events has probability \(\dfrac{1}{4}\), for a total probability of \(2\cdot\dfrac{1}{4}=\dfrac{1}{2}\).

Thus, C is the correct answer.

25 答案 B

外圆面积为 \(6^2\pi = 36\pi\)，内圆面积为 \(3^2\pi = 9\pi\)。因此外环面积为 \(36\pi - 9\pi = 27\pi\)。

所以击中一个外环扇区的概率为 \(\dfrac{9\pi}{36\pi} = \dfrac{1}{4}\)。类似地，击中一个内圆扇区的概率为 \(\dfrac{3\pi}{36\pi} = \dfrac{1}{12}\)。

汇总所有可能，击中 \(1\) 分区域的概率为 \[ \dfrac{1}{12} + 2 \cdot \dfrac{1}{4} = \dfrac{7}{12}\text{。}\]击中 \(2\) 分区域的概率为 \[ 2 \cdot \dfrac{1}{12} + \dfrac{1}{4} = \dfrac{5}{12}\text{。}\]

得到奇数得分的唯一方式是一支击中 \(1\) 分，另一支击中 \(2\) 分。

两种顺序的总概率为 \[ 2 \cdot \dfrac{5}{12} \cdot \dfrac{7}{12} = \dfrac{35}{72}\text{。}\]

所以正确答案是 B。

English

The area of the outer circle is \(6^2\pi = 36\pi,\) and the area of the inner circle is \(3^2\pi = 9\pi.\) Therefore, the area of the outer ring is \(36\pi - 9\pi = 27\pi.\)

This means that the probability of hitting an outer segment is \(\dfrac{9\pi}{36\pi} = \dfrac{1}{4}.\) Similarly, the probability of hitting an inner segment is \(\dfrac{3\pi}{36\pi} = \dfrac{1}{12}.\)

Summing over all possibilities, the probability of hitting a \(1\) is \[ \dfrac{1}{12} + 2 \cdot \dfrac{1}{4} = \dfrac{7}{12}. \] Similarly, the probability of hitting a \(2\) is \[ 2 \cdot \dfrac{1}{12} + \dfrac{1}{4} = \dfrac{5}{12}. \]

The only way to get an odd score is to hit one \(1\) and one \(2.\)

The probability of hitting these numbers in either order is \[ 2 \cdot \dfrac{5}{12} \cdot \dfrac{7}{12} = \dfrac{35}{72}. \]

Thus, B is the correct answer.