1999 年 AMC 8 真题（P版）

1 答案 A

我们有 \[ 4 - (2 - 1) = 4 - 1 = 3\text{。}\]

因此 \[ (6 \ ? \ 3) + 3 = 5 \] \[ 6 \ ? \ 3 = 2\text{。}\]

唯一可行的选择是 \(\div\)。

所以正确答案是 A。

English

We have that \[ 4 - (2 - 1) = 4 - 1 = 3. \]

Then \[ (6 \ ? \ 3) + 3 = 5 \]\[ 6 \ ? \ 3 = 2. \]

The only choice that works is \(\div.\)

Thus, A is the correct answer.

2 答案 C

\(10\) 点整时，时针指向 \(10\)，分针指向 \(12\)。

两根指针相隔整个圆的 \(\frac{1}{6}\)。所以角度为 \[ 360^{\circ} \cdot \dfrac{1}{6} = 60^{\circ}\text{。}\]

所以正确答案是 C。

English

At \(10\) o’clock, we have that the hour hand is at \(10\) and the minute hand is at \(12.\)

This means that the hands are a \(\frac{1}{6}\) of the entire circle apart. This is equal to \[ 360^{\circ} \cdot \dfrac{1}{6} = 60^{\circ}. \]

Thus, C is the correct answer.

3 答案 D

A 等于 \[ \dfrac{3}{6} + \dfrac{2}{6} + \dfrac{1}{6} = 1\text{。}\]

B 化简为 \[ 2 - 2 + 1 = 1\text{。}\]

C 的和是 \[ 0.1 + 0.3 + 0.6 = 1\text{。}\]

但 D 的和为 \[ 1.1 - 2.1 + 1.0 = 0\text{。}\]

为确认起见，E 的和为 \[ -\dfrac{8}{2} + 5 = 5 - 4 = 1\text{。}\]

所以正确答案是 D。

English

We have that A is the same as \[ \dfrac{3}{6} + \dfrac{2}{6} + \dfrac{1}{6} = 1. \]

B reduces to \[ 2 - 2 + 1 = 1. \]

C adds up to \[ 0.1 + 0.3 + 0.6 = 1. \]

D however adds to \[ 1.1 - 2.1 + 1.0 = 0. \]

To make sure, we check that E has a sum of \[ -\dfrac{8}{2} + 5 = 5 - 4 = 1. \]

Thus, D is the correct answer.

4 答案 A

从图中可见，\(4\) 小时后阿尔韦托骑了 \(60\) 英里，比约恩骑了 \(45\) 英里。

两者的距离差是 \(60 - 45 = 15\) 英里。

所以正确答案是 A。

English

Looking at the graph, we have that Alberto has biked \(60\) miles and Bjorn \(45\) miles after \(4\) hours.

The difference between the two distances is \( 60 - 45 = 15 \) miles.

Thus, A is the correct answer.

5 答案 D

当前栅栏的周长为 \[ 2(50 + 10) = 2 \cdot 60 = 120 \] 英尺。如果四条边都相等，那么每条边长为 \(120 \div 4 = 30\) 英尺。因此正方形面积为 \(30^2 = 900\) 平方英尺。原花园面积为 \(50 \cdot 10 = 500\) 平方英尺。

因此面积差是 \(900 - 500 = 400\) 平方英尺。

所以正确答案是 D。

English

The current length of the fence is \[ 2(50 + 10) = 2 \cdot 60 = 120 \] feet. If all the sides become the same, then each side length is \( 120 \div 4 = 30 \) feet. As such, the area of the square is then \( 30^2 = 900 \) square feet. Note that the original area of the garden is \( 50 \cdot 10 = 500 \) square feet.

Therefore, the difference is \( 900 - 500 = 400 \) square feet.

Thus, D is the correct answer.

6 答案 E

设 \(B, C, F, J\) 和 \(M\) 分别表示博、科、弗洛、乔和莫的钱数。

那么 \[ J \lt F,\quad B \lt F\text{。}\]

还知道 \[ B \gt M,\quad C \gt M\text{。}\]

最后，题目给出 \[ M \lt J \lt B\text{。}\]

由第一组和第二组不等式可知 \(F \gt M\)。

因此每个人的钱都比莫多。

所以正确答案是 E。

English

Let \(B, C, F, J,\) and \(M\) represent the amount of money that Bo, Coe, Flo, Jo, and Moe have respectively.

Then \[ J \lt F,\quad B \lt F. \]

We also have \[ B \gt M,\quad C \gt M. \]

Finally, we are given \[ M \lt J \lt B. \]

From the first and second inequalities, we can get that \(F \gt M.\)

This shows that everyone has more money than Moe.

Thus, E is the correct answer.

7 答案 E

第三个和第十个出口之间相差 \(160 - 40 = 120\) 英里。

这意味着全程的 \(\frac{3}{4}\) 处位于 \[ 40 + \dfrac{3}{4} \cdot 120 = 130 \] 英里标处。

所以正确答案是 E。

English

There are \(160 - 40 = 120\) miles between the third and tenth exits.

This means \(\frac{3}{4}\) of the way is at the \[ 40 + \dfrac{3}{4} \cdot 120 = 130 \] milepost.

Thus, E is the correct answer.

8 答案 A

设立方体的黄色面朝上。

这时白色、绿色和橙色面可以向下折。

因为蓝色面连在绿色面上，所以它会向后折。

因此蓝色面最终会与白色面相对。

所以正确答案是 A。

English

Consider the cube with the yellow face facing upwards.

Then we can fold the white, green, and orange faces down.

Since the blue face is attached to the green face, it will end up folding backwards.

This means that the blue face will end up facing opposite the white face.

Thus, A is the correct answer.

9 答案 C

有 \(50 + 100 = 150\) 株植物在两个花坛中，且没有植物同时在三个花坛中。

所以植物总数为 \[ 500 + 450 + 350 - 150 = 1150\text{。}\]减去重叠部分是为了去掉被计算两次的植物。

所以正确答案是 C。

English

Note there are \(50 + 100 = 150\) plants that are in two beds and there are no plants in all three beds.

The total number of plants is then \[ 500 + 450 + 350 - 150 = 1150. \] We subtract to get rid of the plants that we counted twice.

Thus, C is the correct answer.

10 答案 E

在一个周期中，交通灯不是绿灯的时间为 \(30 + 5 = 35\) 秒。

所以不是绿灯的概率为 \[ \dfrac{35}{60} = \dfrac{7}{12}\text{。}\]

所以正确答案是 E。

English

During a given cycle, the light is not green for \(30 + 5 = 35\) seconds.

Then the probability that it is not green is \[ \dfrac{35}{60} = \dfrac{7}{12}. \]

Thus, E is the correct answer.

11 答案 D

设中心方格中的数为 \(x\)。横排之和加上竖列之和时，其他数各计算一次，而中心数计算两次。

所以公共和的两倍是 \(1+4+7+10+13+x\) \(=35+x\)。当 \(x=13\) 时，这个值最大。

因此公共和至多为 \(\dfrac{35+13}{2}=24\)，而且可以达到：把 \(13\) 放在中心，把 \(1\) 和 \(10\) 放在一排的两端，把 \(4\) 和 \(7\) 放在另一排的两端。两个和都是 \(24\)。

所以正确答案是 D。

English

Let \(x\) be the number in the center. The horizontal sum plus the vertical sum counts every number once, except the center number is counted twice.

So twice the common sum is \(1+4+7+10+13+x\) \(=35+x\). This is largest when \(x=13\).

The largest common sum is therefore at most \(\dfrac{35+13}{2}=24\), and it is attainable: put \(13\) in the center, \(1\) and \(10\) at the ends of one row, and \(4\) and \(7\) at the ends of the other. Both sums are \(24\).

Thus, D is the correct answer.

12 答案 B

可以把这支队看成每 \(15\) 场比赛中赢 \(11\) 场、输 \(4\) 场。

输掉的百分比为 \(\dfrac4{15}\cdot100\%\approx 27\%\)。

所以正确答案是 B。

English

The team can be viewed as winning \(11\) games and losing \(4\) games in each batch of \(15\) games.

The percent lost is \(\dfrac4{15}\cdot100\%\approx 27\%\).

Thus, B is the correct answer.

13 答案 C

全营所有人的年龄总和为 \(40 \cdot 17 = 680\)。

女孩年龄总和为 \(20 \cdot 15 = 300\)，男孩年龄总和为 \(15 \cdot 16 = 240\)。

因此成年人的年龄总和为 \(680 - 300 - 240 = 140\)。

所以成年人的平均年龄为 \[ 140 \div 5 = 28\text{。}\]

所以正确答案是 C。

English

The sum of the ages of everybody at the camp is \( 40 \cdot 17 = 680. \)

The sum of the ages of the girls is \( 20 \cdot 15 = 300 \) and of the boys is \( 15 \cdot 16 = 240. \)

As such, we know that the ages of the adults must be \( 680 - 300 - 240 = 140. \)

And therefore, the average age of the adults is then \[ 140 \div 5 = 28. \]

Thus, C is the correct answer.

14 答案 D

设从 \(B\) 到 \(\overline{AD}\) 的高交 \(\overline{AD}\) 于 \(H\)。

于是我们有 \[ AH = \dfrac{16 - 8}{2} = 4\text{，}\]因为 \(AB = CD\)。

于是 \[ AB = \sqrt{4^2 + 3^2} = \sqrt{25} = 5\text{。}\]

周长为 \[ 8 + 16 + 2 \cdot 5 = 34\text{。}\]

所以正确答案是 D。

English

Let \(H\) be where the altitude from \(B\) to \(\overline{AD}\) intersects \(\overline{AD}.\)

Then we have that \[ AH = \dfrac{16 - 8}{2} = 4, \] since \(AB = CD.\)

We then have that \[ AB = \sqrt{4^2 + 3^2} = \sqrt{25} = 5. \]

Then the perimeter is \[ 8 + 16 + 2 \cdot 5 = 34. \]

Thus, D is the correct answer.

15 答案 D

目前一共可以制作 \[5 \cdot 3 \cdot 4 = 60\] 个牌照。

如果两个字母都加入第一个集合，那么可以制作 \[ 7 \cdot 3 \cdot 4 = 84 \] 个牌照。

如果两个字母都加入第二个集合，那么有 \[ 5 \cdot 5 \cdot 4 = 100 \] 个牌照。

如果两个字母都加入第三个集合，那么有 \[ 5 \cdot 3 \cdot 6 = 90 \] 种选择。

如果一个字母加入第一个集合，另一个加入第二个集合，那么有 \[ 6 \cdot 4 \cdot 4 = 96 \] 个牌照。

如果另一个字母改为加入第三个集合，那么可以得到 \[ 6 \cdot 3 \cdot 5 = 90 \] 个牌照。

最后，如果两个字母分别加入第二个和第三个集合，那么有 \[ 5 \cdot 4 \cdot 5 = 100 \] 个牌照。

最大可以达到 \(100\) 个牌照，因此额外增加 \(100 - 60 = 40\) 个。

所以正确答案是 D。

English

There are currently \[5 \cdot 3 \cdot 4 = 60\] license plates that can be made.

If both letters are added to the first set, then there are \[ 7 \cdot 3 \cdot 4 = 84 \] possible plates.

If they are both added to the second, there are \[ 5 \cdot 5 \cdot 4 = 100 \] plates.

If they are added to the third, there are \[ 5 \cdot 3 \cdot 6 = 90 \] choices.

If one is added to the first set and the other to the second set, there are \[ 6 \cdot 4 \cdot 4 = 96 \] plates.

If the other is added to the third set, we get \[ 6 \cdot 3 \cdot 5 = 90 \] possible plates.

Finally, if the letters are added to the second and third sets, there are \[ 5 \cdot 4 \cdot 5 = 100 \] plates.

We see that \(100\) is the greatest number of plates that we can achieve. This is an additional \(100 - 60 = 40\) plates.

Thus, D is the correct answer.

16 答案 B

她答对了 \(10 \cdot 0.7 = 7\) 道算术题，答对了 \(30 \cdot 0.4 = 12\) 道代数题，还答对了 \(35 \cdot 0.6 = 21\) 道几何题。

因此她总共答对 \(7 + 12 + 21 = 40\) 道题。

要达到 \(60\%\)，托里必须答对 \[ 75 \cdot 0.6 = 45 \] 道题。所以她还需要多答对 \(45 - 40 = 5\) 道题。

所以正确答案是 B。

English

She answered \( 10 \cdot 0.7 = 7 \) arithmetic questions correctly. She answered \( 30 \cdot 0.4 = 12 \) algebra ones correctly. She also got \(35 \cdot 0.6 = 21 \) geometry questions correctly.

This means she got a total of \( 7 + 12 + 21 = 40 \) questions correct.

As such, in order to get a \(60\%,\) Tori must have answered \[ 75 \cdot 0.6 = 45 \] questions correctly. This means that she would have needed to answer an additional \(45 - 40 = 5\) questions correctly.

Thus, B is the correct answer.

17 答案 C

因为学生平均每人吃 \(2\) 个饼干，他们总共会吃 \(108 \cdot 2 = 216\) 个饼干。

每份食谱做 \(15\) 个饼干，所以需要 \[ \left\lceil\dfrac{216}{15}\right\rceil = 15 \] 份完整食谱，才能做够。

每盘需要 \(2\) 个鸡蛋，所以需要 \(15 \cdot 2 = 30\) 个鸡蛋。半打是 \(6\) 个鸡蛋，因此需要 \(30 \div 6 = 5\) 个半打。

所以正确答案是 C。

English

Since the students eat an average of \(2\) cookies each, they will eat a total of \(108 \cdot 2 = 216\) cookies.

Each recipe makes \(15\) cookies, which means we need \[ \left\lceil\dfrac{216}{15}\right\rceil = 15 \] full recipes to make enough cookies.

Each pan requires \(2\) eggs, which means we need \(15 \cdot 2 = 30\) eggs. There are \(6\) eggs in a half-dozen, so we need \(30 \div 6 = 5\) half-dozens.

Thus, C is the correct answer.

18 答案 E

现在聚会上只有 \[ 108 \cdot \left(1 - \dfrac{1}{4}\right) = 108 \cdot \dfrac{3}{4} = 81 \] 人。这意味着需要 \[ \left\lceil \dfrac{81 \cdot 2}{15} \right\rceil = \left\lceil \dfrac{162}{15} \right\rceil = 11 \] 份食谱。

所以正确答案是 E。

English

There will now only be \[ 108 \cdot \left(1 - \dfrac{1}{4}\right) = 108 \cdot \dfrac{3}{4} = 81 \] people at the party. This means we need \[ \left\lceil \dfrac{81 \cdot 2}{15} \right\rceil = \left\lceil \dfrac{162}{15} \right\rceil = 11 \] recipes.

Thus, E is the correct answer.

19 答案 B

要做 \(216\) 个饼干，他们必须做 \[ \left\lceil \dfrac{216}{15} \right\rceil = 15 \] 盘。每盘需要 \(3\) 汤匙黄油，所以所有饼干共需要 \(15 \cdot 3 = 45\) 汤匙。

因此他们需要 \[ \left\lceil \dfrac{45}{8} \right\rceil = 6 \] 条黄油。

所以正确答案是 B。

English

To make \(216\) cookies, they have to make \[ \left\lceil \dfrac{216}{15} \right\rceil = 15 \] pans. Since each pan requires \(3\) tablespoons of butter, all the pans will need \(15 \cdot 3 = 45\) tablespoons.

They will then need \[ \left\lceil \dfrac{45}{8} \right\rceil = 6 \] sticks of butter.

Thus, B is the correct answer.

20 答案 B

注意某一列在正面视图中的高度，是前后两个位置高度的最大值。

左列中，后面的立方体堆较高，高度为 \(2\)。

中列中，前面的立方体堆较高，高度为 \(3\)。

右列的高度为 \(4\)。

所以正确答案是 B。

English

Note that the height of the column is the maximum height of the front and back columns.

In the left column, we have that the back column is taller with height \(2.\)

In the middle column, we have that the front column is taller with height \(3.\)

Finally, the right column has height \(4.\)

Thus, B is the correct answer.

21 答案 B

如下标记顶点。

由补角关系，\[ \angle ABC = 180^{\circ} - 100^{\circ} = 80^{\circ}\text{。}\]同样由补角关系，\[ \angle CED = 180^{\circ} - 110^{\circ} = 70^{\circ}\text{。}\]利用三角形内角和为 \(180^{\circ}\)，得到 \[ \begin{aligned} \angle ECD &= 180^{\circ} - 70^{\circ} - 40^{\circ} \\ &= 70^{\circ} \end{aligned}\text{。}\]再由对顶角相等，\[ \angle ACB = \angle ECD = 70^{\circ}\text{。}\]最后在三角形中使用内角和，得到 \[ \angle A = 180^{\circ} - 80^{\circ} - 70^{\circ} = 30^{\circ}\text{。}\]

所以正确答案是 B。

English

Label the vertices as below.

We then have that \[ \angle ABC = 180^{\circ} - 100^{\circ} = 80^{\circ} \] by supplementary angles. Then we have \[ \angle CED = 180^{\circ} - 110^{\circ} = 70^{\circ} \] again by supplementary angles. Using the sum of the interior angles of a triangle is \(180^{\circ},\) we get \[ \begin{aligned} \angle ECD &= 180^{\circ} - 70^{\circ} - 40^{\circ} \\ &= 70^{\circ}. \end{aligned} \] Then, using vertical angles, we have \[ \angle ACB = \angle ECD = 70^{\circ}. \] Finally, using the sum of the interior angles of a triangle, we get \[ \angle A = 180^{\circ} - 80^{\circ} - 70^{\circ} = 30^{\circ}. \]

Thus, B is the correct answer.

22 答案 D

三条鱼值两个面包，所以一条鱼值 \(\dfrac23\) 个面包。

一条面包值 \(4\) 袋米，所以一条鱼值 \(\dfrac23\cdot4=\dfrac83=2\dfrac23\) 袋米。

所以正确答案是 D。

English

Three fish are worth two loaves of bread, so one fish is worth \(\dfrac23\) of a loaf.

One loaf is worth \(4\) bags of rice, so one fish is worth \(\dfrac23\cdot4=\dfrac83=2\dfrac23\) bags of rice.

Thus, D is the correct answer.

23 答案 C

正方形面积为 \(3^2 = 9\)，所以每个区域的面积是 \(9 \div 3 = 3\)。

因此 \(\triangle BMC\) 的面积为 \(3\)，所以 \[ \dfrac{1}{2} \cdot 3 \cdot BM = 3 \] \[ BM = 2\text{。}\]

因为 \(\triangle BMC\) 是直角三角形，\[ CM = \sqrt{2^2 + 3^2} = \sqrt{13}\text{。}\]

所以正确答案是 C。

English

The area of the square is \(3^2 = 9,\) which means that the area of one region is \(9 \div 3 = 3.\)

This means the area of \(\triangle BMC\) is \(3,\) which means that \[ \dfrac{1}{2} \cdot 3 \cdot BM = 3 \]\[ BM = 2. \]

Since \(\triangle BMC\) is right, we have \[ CM = \sqrt{2^2 + 3^2} = \sqrt{13}. \]

Thus, C is the correct answer.

24 答案 D

要求除以 \(5\) 的余数，只需要关注个位数。

因此只需观察个位数 \(9\) 的幂的规律。

\(9\) 的幂的个位数在 \(9\) 和 \(1\) 之间交替：\[ 9, 81, 729, \cdots\text{。}\]因为指数是偶数，所以 \(1999^{2000}\) 的个位数为 \(1\)。

除以 \(5\) 的余数就是 \(1\)，因为它比 \(10\) 的倍数多 \(1\)。

所以正确答案是 D。

English

Note that to find the remainder when divided by \(5,\) we only care about the units digit.

This means we only have to observe how the powers of the units digit work, namely the powers of \(9.\)

Then, looking at powers of \(9,\) we see that the units digit alternates between \(9\) and \(1:\) \[ 9, 81, 729, \cdots. \] This means that \(1999^{2000}\) ends in a \(1\) since the power is even.

The remainder when divided by \(5\) is then \(1,\) since it is \(1\) more than a multiple of \(10.\)

Thus, D is the correct answer.

25 答案 A

第一个阴影三角形的两条直角边为 \(3\) 和 \(3\)，所以面积为 \(\dfrac12\cdot3\cdot3=\dfrac92\)。

之后每一步，相关三角形的边长变为原来的一半，因此阴影面积乘以 \(\left(\dfrac12\right)^2=\dfrac14\)。

许多步后的阴影总面积非常接近等比和 \[ \begin{gathered} \frac92\left(1+\frac14+\frac1{16}+\cdots\right) \\ {}= \frac92\cdot\frac1{1-\frac{1}{4}} \\ {}= 6 \end{gathered}\text{。}\]

\(100\) 步后被省略的尾项极小，所以总面积最接近 \(6\)。

所以正确答案是 A。

English

The first shaded triangle has legs \(3\) and \(3\), so its area is \(\dfrac12\cdot3\cdot3=\dfrac92\).

At each later step, the relevant triangle has half the leg length, so the shaded area is multiplied by \(\left(\dfrac12\right)^2=\dfrac14\).

The total shaded area after many steps is therefore very close to the geometric sum \[ \begin{gathered} \frac92\left(1+\frac14+\frac1{16}+\cdots\right) \\ {}= \frac92\cdot\frac1{1-\frac{1}{4}} \\ {}= 6. \end{gathered} \]

After \(100\) steps the omitted tail is tiny, so the total area is nearest \(6\).

Thus, A is the correct answer.