2004 年 AMC 8 真题（P版）

1 答案 B

\(1\) 厘米表示 \(72 \div 12 = 6\) 千米。因此 \(17\) 厘米表示 \(6 \cdot 17 = 102\) 千米。

所以正确答案是 B。

English

Note that \(1\) cm represents \(72 \div 12 = 6\) kilometers. This means that \(17\) cm represents \(6 \cdot 17 = 102\) kilometers.

Thus, B is the correct answer.

2 答案 B

有 \(2\) 个非零数字可以作为千位数字。

选定后，需要安排另外 \(3\) 个数字。另一个非零数字有 \(3\) 个位置可放。

因此共有 \(2 \cdot 3 = 6\) 个可能的数。

所以正确答案是 B。

English

Note that there are \(2\) non-zero digits that could be the thousands digit.

After choosing that, we need to arrange the other \(3\) digits. There are \(3\) spots for the other non-zero digit.

This gives us \(2 \cdot 3 = 6\) possible numbers.

Thus, B is the correct answer.

3 答案 A

\(12\) 份餐能喂饱 \(18\) 个人，所以一份餐能喂饱 \(\dfrac{18}{12} = \dfrac{3}{2}\) 个人。

因此所需餐数为 \(12 \div \dfrac{3}{2} = 8\)，正好够 \(12\) 个人吃。

所以正确答案是 A。

English

Note that \(12\) meals feed \(18\) people. This means that one meal feeds \(\dfrac{18}{12} = \dfrac{3}{2}\) people.

This means that they need \(12 \div \dfrac{3}{2} = 8\) meals for \(12\) people.

Thus, A is the correct answer.

4 答案 B

若有 \(3\) 名首发，则有一人不首发。选择不首发的人就确定了首发。

不首发的人有 \(4\) 种选择。

所以正确答案是 B。

English

If there are \(3\) starters, then one person must not be starting. Choosing the person who doesn’t start determines the starters.

There are \(4\) choices for the person who doesn’t start.

Thus, B is the correct answer.

5 答案 D

每场比赛后都会淘汰一支队伍。要只剩下一支队伍，必须淘汰 \(15\) 支队伍。

因此必须打 \(15\) 场比赛。

所以正确答案是 D。

English

Note that after every game, one team gets eliminated. For there to be one team remaining, \(15\) teams must have been eliminated.

This means that \(15\) games had to have been played.

Thus, D is the correct answer.

6 答案 C

萨莉在最初的 \(20\) 次投篮中命中了 \(20 \times 0.55 = 11\) 次。于是 \[ \dfrac{11 + x}{25} = 0.56\text{，}\]所以 \[ 11 + x = 14 \] 且 \(x = 3\)。

所以正确答案是 C。

English

Sally made \(20 \times 0.55 = 11\) of her first \(20\) shots. Then we get that \[ \dfrac{11 + x}{25} = 0.56, \] which tells us that \[ 11 + x = 14 \] and \(x = 3.\)

Thus, C is the correct answer.

7 答案 B

这名运动员的最大心率为 \(220 - 26 = 194\)。目标心率为 \(194 \times 0.8 \approx 155\)。

所以正确答案是 B。

English

The maximum heart rate for this athlete would be \(220 - 26 = 194.\) Then the target heart rate would be \(194 \times 0.8 \approx 155.\)

Thus, B is the correct answer.

8 答案 B

十位数字可以从 \(1\) 到 \(7\)，而这个数字会确定个位数字。

因此共有 \(7\) 个数。

所以正确答案是 B。

English

Note that the tens digit can range from \(1\) to \(7,\) and this digit determines the units digit.

Therefore, there are \(7\) numbers.

Thus, B is the correct answer.

9 答案 D

\(5\) 个数的总和是 \(54 \cdot 5 = 270\)。前 \(2\) 个数的总和是 \(48 \cdot 2 = 96\)。

后 \(3\) 个数的总和是 \(270 - 96 = 174\)，所以平均数是 \(174 \div 3 = 58\)。

所以正确答案是 D。

English

The sum of all \(5\) numbers is \(54 \cdot 5 = 270.\) The sum of the first \(2\) numbers is \(48 \cdot 2 = 96.\)

The sum of the last \(3\) numbers is \(270 - 96 = 174.\) The average is therefore \(174 \div 3 = 58.\)

Thus, D is the correct answer.

10 答案 E

亚伦星期一工作 \(75\) 分钟，星期二工作 \(50\) 分钟，星期三工作 \(145\) 分钟，星期五工作 \(30\) 分钟。

总时间为 \(75+50+145+30=300\) 分钟，即 \(5\) 小时。

按每小时 \(\$3\) 计算，他赚了 \(5\cdot3=15\) 美元。

所以正确答案是 E。

English

Aaron worked \(75\) minutes on Monday, \(50\) minutes on Tuesday, \(145\) minutes on Wednesday, and \(30\) minutes on Friday.

The total is \(75+50+145+30=300\) minutes, or \(5\) hours.

At \(\$3\) per hour, he earned \(5\cdot3=15\) dollars.

Thus, E is the correct answer.

11 答案 C

最大数、最小数和中位数都不能在第一位或最后一位。

因此第一位和最后一位是 \(4\) 和 \(9\)，顺序不限。它们的平均数为 \[ (4 + 9) \div 2 = 13 \div 2 = 6.5\text{。}\]

所以正确答案是 C。

English

Note that the largest, smallest, and median numbers cannot be the first or last number.

This means that the first and last numbers are \(4\) and \(9\) in some order. Their average is \[ (4 + 9) \div 2 = 13 \div 2 = 6.5. \]

Thus, C is the correct answer.

12 答案 B

不使用时，手机每小时消耗 \(\frac{1}{24}\) 的电量。使用时，每小时消耗 \(\frac{1}{3}\) 的电量。

妮基的手机开机 \(9\) 小时，其中 \(8\) 小时待机，\(1\) 小时通话。

因此手机已消耗 \(\frac{2}{3}\) 的电量。剩余 \(\frac{1}{3}\) 电量在不使用时还能持续 \(8\) 小时。

所以正确答案是 B。

English

When not in use, her cell phone uses up \(\frac{1}{24}\) of its battery per hour. When it is in use, it uses up \(\frac{1}{3}\) of its battery per hour.

Niki’s phone has been on for \(9\) hours, with \(8\) of those hours being idle and \(1\) hour being used to talk on the phone.

This means that the phone has used up \(\frac{2}{3}\) of its battery. In order to drain the remaining \(\frac{1}{3}\) of the battery, the phone can last for \(8\) more hours without being used.

Thus, B is the correct answer.

13 答案 E

如果比尔最大，那么陈述 \(\mathrm{I}\) 和 \(\mathrm{II}\) 都为真，所以比尔不是最大的。

如果赛琳最大，那么陈述 \(\mathrm{II}\) 和 \(\mathrm{III}\) 都为真，所以赛琳也不是最大的。

因此艾米最大。陈述 \(\mathrm{I}\) 和 \(\mathrm{II}\) 为假，所以陈述 \(\mathrm{III}\) 必须是唯一为真的陈述。因此赛琳不是最小，比尔最小。

顺序是艾米、赛琳、比尔。

所以正确答案是 E。

English

If Bill were oldest, then statements \(\mathrm{I}\) and \(\mathrm{II}\) would both be true, so Bill is not oldest.

If Celine were oldest, then statements \(\mathrm{II}\) and \(\mathrm{III}\) would both be true, so Celine is not oldest.

Therefore Amy is oldest. Statements \(\mathrm{I}\) and \(\mathrm{II}\) are false, so statement \(\mathrm{III}\) must be the single true statement. Thus Celine is not youngest, leaving Bill youngest.

The order is Amy, Celine, Bill.

Thus, E is the correct answer.

14 答案 C

将四边形放入外接的 \(10\times10\) 正方形中，其面积为 \(100\)。

五个外侧部分的面积分别为 \(15\)、\(\frac12\cdot6\cdot7=21\)、\(\frac12\cdot1\cdot3=\frac32\)、\(\frac12\cdot4\cdot5=10\)、\(\frac12\cdot6\cdot10=30\)。

外侧面积为 \(15+21+\frac32+10+30=77\frac12\)，所以四边形面积为 \(100-77\frac12=22\frac12\)。

所以正确答案是 C。

English

Place the quadrilateral inside the surrounding \(10\times10\) square, whose area is \(100.\)

The five outside pieces have areas \(15,\) \(\frac12\cdot6\cdot7=21,\) \(\frac12\cdot1\cdot3=\frac32,\) \(\frac12\cdot4\cdot5=10,\) and \(\frac12\cdot6\cdot10=30.\)

The outside area is \(15+21+\frac32+10+30=77\frac12,\) so the quadrilateral area is \(100-77\frac12=22\frac12.\)

Thus, C is the correct answer.

15 答案 C

原图有 \(13\) 块阴影瓷砖和 \(6\) 块未阴影瓷砖。

新边框是围绕图形的下一圈六边形环，有 \(18\) 块未阴影瓷砖。

新图形有 \(6+18=24\) 块未阴影瓷砖和 \(13\) 块阴影瓷砖，所以差为 \(24-13=11\)。

所以正确答案是 C。

English

The original figure has \(13\) shaded tiles and \(6\) unshaded tiles.

The new border is the next hexagonal ring around the figure, which has \(18\) unshaded tiles.

The new figure has \(6+18=24\) unshaded tiles and \(13\) shaded tiles, so the difference is \(24-13=11.\)

Thus, C is the correct answer.

16 答案 C

第一个水壶含有 \(600 \cdot \dfrac{1}{3} = 200\) 毫升橙汁。第二个含有 \(600 \cdot \dfrac{2}{5} = 240\) 毫升。

大容器中共有 \(200 + 240 = 440\) 毫升橙汁。混合液总量为 \(2 \cdot 600 = 1200\) 毫升。

因此橙汁所占比例为 \(\dfrac{440}{1200} = \dfrac{11}{30}\)。

所以正确答案是 C。

English

The first pitcher contains \(600 \cdot \dfrac{1}{3} = 200\) mL of orange juice. The second one has \(600 \cdot \dfrac{2}{5} = 240\) mL.

The large container then has \(200 + 240 = 440\) mL of orange juice. The total amount of mixture is \(2 \cdot 600 = 1200\) mL.

Then the fraction of orange juice is \(\dfrac{440}{1200} = \dfrac{11}{30}.\)

Thus, C is the correct answer.

17 答案 D

先给每个朋友一支铅笔。还剩 \(3\) 支铅笔要分给 \(3\) 个朋友。

如果一个朋友得到全部剩下的 \(3\) 支，有 \(3\) 种方式。如果剩下的铅笔按 \(2\) 和 \(1\) 分，有 \(3\cdot2=6\) 种方式。如果每个朋友再各得一支，有 \(1\) 种方式。

总方式数为 \(3+6+1=10\)。

所以正确答案是 D。

English

First give each friend one pencil. Then \(3\) pencils remain to distribute among the \(3\) friends.

If one friend gets all \(3\) remaining pencils, there are \(3\) ways. If the remaining pencils split as \(2\) and \(1,\) there are \(3\cdot2=6\) ways. If each friend gets one more, there is \(1\) way.

The total number of ways is \(3+6+1=10.\)

Thus, D is the correct answer.

18 答案 A

本得到 \(4\) 分的唯一方式是 \(1\) 和 \(3\)，因为他不能两次击中 \(2\)。

辛迪的得分可能来自 \[ 1 + 6,\quad 2 + 5,\quad 3 + 4\text{。}\]本已经击中了 \(1\) 和 \(3\)，所以辛迪必须击中 \(2\) 和 \(5\)。

同理，戴夫必须击中 \(4\) 和 \(7\)。最后，因为 \(7\) 已经被使用，爱丽丝只能击中 \(6\) 和 \(10\)，埃伦击中 \(8\) 和 \(9\)。

所以正确答案是 A。

English

The only way to get Ben’s score of \(4\) is with a \(1\) and \(3\) since he can’t hit \(2\) twice.

Cindy can achieve her score with \[ 1 + 6,\quad 2 + 5,\quad 3 + 4. \] Ben already hit \(1\) and \(3,\) so Cindy must have hit \(2\) and \(5.\)

Similarly, Dave must have hit \(4\) and \(7.\) Finally, since \(7\) is already used, Alice is forced to have hit \(6\) and \(10\) with Ellen hitting \(8\) and \(9.\)

Thus, A is the correct answer.

19 答案 B

设这个数为 \(x\)。那么 \(x - 2\) 能被 \(3\)、\(4\)、\(5\) 和 \(6\) 整除。

这些数的最小公倍数是 \(60\)，所以 \(x\) 为 \(62\)。

所以正确答案是 B。

English

Let \(x\) be the number. Then \(x - 2\) is divisible by \(3,\) \(4,\) \(5,\) and \(6.\)

The least common multiple of these numbers is \(60,\) which makes \(x\) \(62.\)

Thus, B is the correct answer.

20 答案 D

因为 \(\frac{3}{4}\) 的椅子被坐了，所以 \(6\) 把空椅子是所有椅子的 \(\frac{1}{4}\)。因此共有 \(24\) 把椅子。

坐着的人数是 \(\frac{3}{4}\) 乘 \(24\)，即 \(18\)。

这 \(18\) 个坐着的人是所有人的 \(\frac{2}{3}\)，所以总人数为 \(18\div\frac23=27\)。

所以正确答案是 D。

English

Since \(\frac{3}{4}\) of the chairs are occupied, the \(6\) empty chairs are \(\frac{1}{4}\) of all the chairs. Thus there are \(24\) chairs.

The number of seated people is \(\frac{3}{4}\) of \(24,\) which is \(18.\)

Those \(18\) seated people are \(\frac{2}{3}\) of all the people, so the total number of people is \(18\div\frac23=27.\)

Thus, D is the correct answer.

21 答案 D

要使乘积为偶数，至少一个转盘必须落在偶数上。

使用补集计数，计算两个转盘都落在奇数上的概率。

这个概率为 \[ \dfrac{1}{2} \cdot \dfrac{2}{3} = \dfrac{1}{3}\text{。}\]

因此至少一个为偶数的概率为 \[1 - \dfrac{1}{3} = \dfrac{2}{3}\text{。}\]

所以正确答案是 D。

English

For the product to be even, at least one of the spinners must land on an even number.

We can use complementary counting and calculate the probability of both spinners landing on odds.

This happens with a probability of \[ \dfrac{1}{2} \cdot \dfrac{2}{3} = \dfrac{1}{3}. \]

Then the probability of landing on at least one even is \[1 - \dfrac{1}{3} = \dfrac{2}{3}.\]

Thus, D is the correct answer.

22 答案 B

为了便于按比例计算，设房间里有 \(5\) 名女性。那么单身女性有 \(5 \cdot \dfrac{2}{5} = 2\) 名。

因此已婚女性有 \(5 - 2 = 3\) 名，也就是已婚男性的人数。

房间里共有 \(5 + 3 = 8\) 人。已婚男性所占比例为 \(\dfrac{3}{8}\)。

所以正确答案是 B。

English

For a convenient ratio model, let there be \(5\) women in the room. Then there are \(5 \cdot \dfrac{2}{5} = 2\) single women.

This means that there are \(5 - 2 = 3\) married women, which is also the number of married men.

There are a total of \(5 + 3 = 8\) people in the room. The fraction of married men is \(\dfrac{3}{8}.\)

Thus, B is the correct answer.

23 答案 D

特丝从 \(J\) 出发跑向 \(K\) 时，她到家的距离不断增加。

从 \(K\) 到 \(L\)，距离继续增加，但变化方式不同，并在对角 \(L\) 达到最大。

从 \(L\) 到 \(M\)，再从 \(M\) 回到 \(J\)，距离逐渐减小到 \(0\)，并在 \(M\) 处再次改变变化方式。

只有图 \(D\) 先以一种方式增加，再改变增加方式；随后改变减少方式，最后减小到 \(0\)。

所以正确答案是 D。

English

As Tess runs from \(J\) to \(K,\) her distance from home increases.

From \(K\) to \(L,\) her distance continues to increase, but with a different shape, reaching its maximum at the opposite corner \(L.\)

From \(L\) to \(M\) and then from \(M\) back to \(J,\) her distance decreases back to \(0,\) again changing behavior at \(M.\)

Graph \(D\) is the only graph with this increase, changed-rate increase, changed-rate decrease, and final decrease to \(0.\)

Thus, D is the correct answer.

24 答案 C

长方形边长为 \(10\) 和 \(8\)，所以面积为 \(80\)。

四个角上三角形的总面积为 \[ \begin{aligned} &2\cdot\frac12\cdot3\cdot4 \\ &\quad{}+2\cdot\frac12\cdot6\cdot5 \\ &=12+30=42 \end{aligned}\text{。}\]因此平行四边形 \(EFGH\) 的面积为 \(80-42=38\)。

线段 \(HE\) 长为 \(5\)，这可由 \(3-4-5\) 直角三角形得到。因为平行四边形面积为 \(HE\cdot d\)，所以 \(5d=38\)，从而 \(d=7.6\)。

所以正确答案是 C。

English

The rectangle has side lengths \(10\) and \(8,\) so its area is \(80.\)

The four corner triangles have total area \[ \begin{aligned} &2\cdot\frac12\cdot3\cdot4 \\ &\quad{}+2\cdot\frac12\cdot6\cdot5 \\ &=12+30=42. \end{aligned} \] Thus parallelogram \(EFGH\) has area \(80-42=38.\)

Segment \(HE\) has length \(5\) by the \(3-4-5\) right triangle. Since the parallelogram area is \(HE\cdot d,\) we have \(5d=38,\) so \(d=7.6.\)

Thus, C is the correct answer.

25 答案 D

两个 \(4\times4\) 正方形总面积为 \(32\)，但它们的重叠部分是一个 \(2\times2\)、面积为 \(4\) 的正方形。因此两个正方形并集的面积为 \(32-4=28\)。

圆的直径是这个 \(2\times2\) 重叠正方形的对角线，所以直径为 \(2\sqrt2\)，半径为 \(\sqrt2\)。

圆面积为 \(\pi(\sqrt2)^2=2\pi\)，所以阴影面积为 \(28-2\pi\)。

所以正确答案是 D。

English

The two \(4\times4\) squares have total area \(32,\) but their overlap is a \(2\times2\) square with area \(4.\) Thus the area covered by the union of the two squares is \(32-4=28.\)

The circle’s diameter is the diagonal of that \(2\times2\) overlap square, so the diameter is \(2\sqrt2\) and the radius is \(\sqrt2.\)

The circle area is \(\pi(\sqrt2)^2=2\pi,\) so the shaded area is \(28-2\pi.\)

Thus, D is the correct answer.