1997 年 AMC 8 真题（P版）

1 答案 C

把所有分数化成小数，得到 \[ \begin{aligned} 0.1 + 0.09 + 0.009 &+ 0.0007 \\ &= 0.1997 \end{aligned}\text{。}\]

所以正确答案是 C。

English

Converting all the fractions to decimals, we get \[ \begin{aligned} 0.1 + 0.09 + 0.009 &+ 0.0007 \\ &= 0.1997. \end{aligned} \]

Thus, C is the correct answer.

2 答案 D

要得到最大结果，应从 \(200\) 中减去尽可能小的数。

最小的两位数是 \(10\)。让安选择这个数，就会得到 \[ 2 (200 - 10) = 2 \cdot 190 = 380 \] 作为最终结果。

所以正确答案是 D。

English

To get the largest number, we would want to subtract the smallest number possible from \(200.\)

The smallest two-digit number is \(10.\) Having Ahn choose this number will give us \[ 2 (200 - 10) = 2 \cdot 190 = 380 \] as the final result.

Thus, D is the correct answer.

3 答案 B

所有数的十分位数字相同，所以比较百分位数字。

最大的百分位数字是 \(7\)，因此只需看百分位为七的选项。

其中最大的千分位数字是 \(9\)，对应 \(0.979\)。

所以正确答案是 B。

English

We have that all the tenths place digits are the same, so we then look at the hundredths digit.

The largest hundredths digit is \(7,\) so we can limit the answer choices to the ones with this value.

The largest thousandths digit is \(9,\) which is achieved by \(0.979.\)

Thus, B is the correct answer.

4 答案 E

半小时是 \(30\) 分钟，朱莉可以说 \[ 150 \cdot 30 = 4500 \] 个词。四分之三小时是 \(45\) 分钟，她可以说 \[ 150 \cdot 45 = 6750 \] 个词。选项中唯一介于这两个数之间的是 \(5650\)。

所以正确答案是 E。

English

One-half hour is \(30\) minutes, in which Julie can speak \[ 150 \cdot 30 = 4500 \] words. Three-quarters of an hour is \(45\) minutes, in which Julie can speak \[ 150 \cdot 45 = 6750 \] words. The only answer choice in between these two values is \(5650.\)

Thus, E is the correct answer.

5 答案 A

\(7\) 的两位数倍数为 \[ 14, 21, 28, 35, 42, 49, 56, 63, 70\text{、}\]\[ 77, 84, 91, 98\text{。}\]其中数位和为 \(10\) 的是 \(28\) 和 \(91\)。

它们的和是 \(28 + 91 = 119\)。

所以正确答案是 A。

English

Listing out all the two-digit multiples of \(7,\) we get \[ 14, 21, 28, 35, 42, 49, 56, 63, 70,\] \[ 77, 84, 91, 98. \] We have that \(28\) and \(91\) are the two multiples whose digits add to \(10.\)

The sum of these numbers is \(28 + 91 = 119.\)

Thus, A is the correct answer.

6 答案 C

每个数字所在位的位值都是它右边一位的十倍。

数字 \(9\) 所在位比数字 \(3\) 所在位向左 \(5\) 位。

因此它的位值是 \[ 10^5 = 100,000 \] 倍。

所以正确答案是 C。

English

Note that each digit has ten times the value of the digit to its right.

The place occupied by \(9\) is \(5\) spaces to the left of the place occupied by \(3.\)

This means that it is \[ 10^5 = 100,000 \] times as great.

Thus, C is the correct answer.

7 答案 D

可以把圆内切在正方形中，使它与正方形四边的中点相切。

因此正方形边长是圆半径的两倍，即 \(2 \cdot 4 = 8\)。

正方形面积为 \[ 8^2 = 64\text{。}\]

所以正确答案是 D。

English

We can inscribe the circle inside the square so that it is tangent to the midpoints of each side of the square.

This means that the side length of the square is two times the radius of the circle, making it \(2 \cdot 4 = 8.\)

Then the area of the square is \[ 8^2 = 64. \]

Thus, D is the correct answer.

8 答案 B

从沃尔特上校车到回到家之间经过 \(8.5\) 小时。

换算成分钟，共有 \[ 60 \cdot 8.5 = 510 \] 分钟。

他在学校的总时间是 \[ 6 \cdot 50 + 30 + 2 \cdot 60 =\] \[ 300 + 30 + 120 = 450 \] 分钟。

因此他在校车上的时间是 \[ 510 - 450 = 60 \] 分钟。

所以正确答案是 B。

English

There are \(8.5\) hours between the time Walter catches the school bus and arrives at home.

This is a total of \[ 60 \cdot 8.5 = 510 \] minutes.

The total time Walter spends at school is \[ 6 \cdot 50 + 30 + 2 \cdot 60 =\]\[ 300 + 30 + 120 = 450 \] minutes.

This means that Walter spends \[ 510 - 450 = 60 \] minutes on the bus.

Thus, B is the correct answer.

9 答案 C

队首有 \(3\) 种选择，中间有 \(2\) 种选择。

最后剩 \(1\) 种选择，所以共有 \(3 \cdot 2 \cdot 1 = 6\) 种排列。

其中只有一种按字母顺序排列，所以概率为 \(\dfrac{1}{6}\)。

所以正确答案是 C。

English

There are \(3\) options for the person in front. Then, there are \(2\) options for the person in the middle.

This leaves \(1\) choice for the person at the end. There are \(3 \cdot 2 \cdot 1 = 6\) ways for the people to line up.

Only one of these lines is correct, and the probability it occurs is \(\dfrac{1}{6}.\)

Thus, C is the correct answer.

10 答案 C

可以把大正方形分成 \(6^2 = 36\) 个小正方形。涂色小正方形数为 \[ 3 + 7 + 11 = 21\text{。}\]

涂色比例为 \[ \dfrac{21}{36} = \dfrac{7}{12}\text{。}\]

所以正确答案是 C。

English

We can split the square into \(6^2 = 36\) unit squares. The number of shaded small squares is \[ 3 + 7 + 11 = 21. \]

The fraction of the square that is shaded is \[ \dfrac{21}{36} = \dfrac{7}{12}. \]

Thus, C is the correct answer.

11 答案 A

\(11\) 是质数，所以它只有 \(2\) 个正因数。

\(20\) 的质因数分解是 \[ 2^2 \cdot 5\text{。}\]一个数的因数个数等于其质因数分解中各指数加一后的乘积。

这里得到 \[ (2 + 1) (1 + 1) = 3 \cdot 2 = 6\text{。}\]

因此里面的乘积是 \(2 \cdot 6 = 12\)。而 \(12\) 的质因数分解是 \[ 2^2 \cdot 3\text{，}\]所以它也有 \[ (2 + 1) (1 + 1) = 3 \cdot 2 = 6 \] 个因数。

所以正确答案是 A。

English

We know that \(11\) is prime, which means that it only has \(2\) divisors.

The prime factorization of \(20\) is \[ 2^2 \cdot 5. \] Recall that the number of divisors a number has is the product of all the exponents plus one in the prime factorization.

Here, that product would be \[ (2 + 1) (1 + 1) = 3 \cdot 2 = 6. \]

Then \(2 \cdot 6 = 12.\) We have the prime factorization of \(12\) is \[ 2^2 \cdot 3. \] This also has \[ (2 + 1) (1 + 1) = 3 \cdot 2 = 6 \] divisors.

Thus, A is the correct answer.

12 答案 D

我们有 \[ \angle 1 = 180^{\circ} - 70^{\circ} - 40^{\circ} = 70^{\circ}\text{，}\]这里用到了三角形内角和为 \(180^{\circ}\) 这一事实。

由此可得 \[ \angle 2 = 180^{\circ} - \angle 1 = 110^{\circ}\text{，}\]因为 \(\angle 1\) 和 \(\angle 2\) 互补。

最后，\[ \angle 3 + \angle 4 + 110^{\circ} = 180^{\circ} \Rightarrow\] \[ 2 \angle 4 = 70^{\circ} \Rightarrow \angle 4 = 35^{\circ}\text{。}\]

所以正确答案是 D。

English

We have that \[ \angle 1 = 180^{\circ} - 70^{\circ} - 40^{\circ} = 70^{\circ} \] using the fact that the interior angles of a triangle add to \(180^{\circ}.\)

This tells us that \[ \angle 2 = 180^{\circ} - \angle 1 = 110^{\circ} \] since \(\angle 1\) and \(\angle 2\) are supplementary.

Finally, \[ \angle 3 + \angle 4 + 110^{\circ} = 180^{\circ} \Rightarrow\]\[ 2 \angle 4 = 70^{\circ} \Rightarrow \angle 4 = 35^{\circ}. \]

Thus, D is the correct answer.

13 答案 A

第一袋中有 \[ 26 \cdot 0.5 = 26 \div 2 = 13 \] 颗黄色软糖，第二袋中有 \[ 28 \cdot 0.25 = 28 \div 4 = 7 \] 颗黄色软糖，第三袋中有 \[ 30 \cdot 0.2 = 30 \div 5 = 6 \] 颗黄色软糖。

黄色软糖总数为 \[ 13 + 7 + 6 = 26\text{，}\]软糖总数为 \[ 26 + 28 + 30 = 84\text{。}\]黄色软糖所占的百分比为 \[ \dfrac{26}{84} \cdot 100 \% = \dfrac{13}{42} \cdot 100 \%\] \[ \approx 30.9 \%\text{。}\]

所以正确答案是 A。

English

There are \[ 26 \cdot 0.5 = 26 \div 2 = 13 \] yellow jelly beans in the first bag, \[ 28 \cdot 0.25 = 28 \div 4 = 7 \] yellow jelly beans in the second bag, and \[ 30 \cdot 0.2 = 30 \div 5 = 6 \] yellow jelly beans in the third bag.

The total number of yellow jelly beans is \[ 13 + 7 + 6 = 26 \] and the total number of jelly beans is \[ 26 + 28 + 30 = 84. \] The ratio of yellow jelly beans to all the beans is \[ \dfrac{26}{84} \cdot 100 \% = \dfrac{13}{42} \cdot 100 \%\]\[ \approx 30.9 \%. \]

Thus, A is the correct answer.

14 答案 D

五个数的和是 \(5 \cdot 5 = 25\)。

按从小到大排列，中间的数是 \(5\)。因为 \(8\) 是唯一的众数，最后两个数都必须是 \(8\)；但不能有三个 \(8\)，否则中位数也会是 \(8\)。

这三个已知数的和是 \(5 + 8 + 8 = 21\)，所以两个较小的数之和必须是 \(25 - 21 = 4\)。它们必须是不同的正整数，否则另一个数也会与 \(8\) 并列为众数。因此它们是 \(1\) 和 \(3\)。

因此所求的差为 \[ 8 - 1 = 7\text{。}\]

所以正确答案是 D。

English

The sum of all the numbers in the list is \(5 \cdot 5 = 25.\)

In increasing order, the middle number is \(5\). Because \(8\) is the only mode, the last two numbers must both be \(8\); there cannot be three \(8\)s, since then the median would be \(8\).

These three known numbers sum to \(5 + 8 + 8 = 21,\) so the two smaller numbers must add to \(25 - 21 = 4.\) They must be distinct positive integers; otherwise a second value would tie \(8\) as a mode. Thus they are \(1\) and \(3\).

The desired difference is then \[ 8 - 1 = 7. \]

Thus, D is the correct answer.

15 答案 B

设大正方形边长为 \(3x\)。于是内接正方形的边长可以由 \[ \sqrt{(2x)^2 + x^2} = \sqrt{5x^2} = x\sqrt{5} \] 根据勾股定理求出。

大正方形面积为 \[ (3x)^2 = 9x^2\text{，}\]内接正方形面积为 \[ (x\sqrt{5})^2 = 5x^2\text{。}\]

因此面积比为 \(\dfrac{5}{9}\)。

所以正确答案是 B。

English

Let \(3x\) be the side length of the large square. Then we can find the side length of the inner square via \[ \sqrt{(2x)^2 + x^2} = \sqrt{5x^2} = x\sqrt{5} \] from the Pythagorean Theorem.

The area of the larger square is \[ (3x)^2 = 9x^2 \] and that of the inner square is \[ (x\sqrt{5})^2 = 5x^2. \]

The ratio of the areas is then \(\dfrac{5}{9}.\)

Thus, B is the correct answer.

16 答案 E

第一年后，AA 的价值变为 \[ \$ 100 \cdot 1.2 = \$ 120\text{，}\]BB 的价值变为 \[ \$ 100 \cdot 0.75 = \$ 75\text{。}\]

第二年后，AA 的价值为 \[ \$ 120 \cdot 0.8 = \$ 96\text{，}\]BB 的价值为 \[ \$ 75 \cdot 1.25 = \$ 93.75\text{。}\]

CC 价值仍不变，所以最终顺序是 \[ B \lt A \lt C\text{。}\]

所以正确答案是 E。

English

After the first year, AA’s stock’s worth goes up to \[ \$ 100 \cdot 1.2 = \$ 120 \] and BB’s goes down to \[ \$ 100 \cdot 0.75 = \$ 75. \]

After the second year, AA’s stock is worth \[ \$ 120 \cdot 0.8 = \$ 96 \] and BB’s is worth \[ \$ 75 \cdot 1.25 = \$ 93.75. \]

CC’s stock worth remains the same, so the ordering of the stock worths is now \[ B \lt A \lt C. \]

Thus, E is the correct answer.

17 答案 E

每个面有两条对角线，连接这个面上两对相对顶点。

另外，对于每个顶点，都有一个与它在立方体内部相对的顶点。

这样空间对角线共有 \(8 \div 2 = 4\) 条。

总对角线数为 \[ 6 \cdot 2 + 4 = 12 + 4 = 16\text{。}\]

所以正确答案是 E。

English

Each face has two diagonals connecting each of the two pairs of opposite vertices.

Also, for each vertex, there is one corresponding vertex that lies opposite it on the cube.

There are then \(8 \div 2 = 4\) interior space diagonals in the cube.

The total number of diagonals is then \[ 6 \cdot 2 + 4 = 12 + 4 = 16. \]

Thus, E is the correct answer.

18 答案 B

原来每盒价格为 \[ \$ 5 \div 4 = \$ 1.25\text{。}\]

现在每盒价格为 \[ \$ 4 \div 5 = \$ 0.8\text{。}\]

下降百分比为 \[ \begin{gathered} \dfrac{1.25 - 0.8}{1.25} \cdot 100 \% \\ = \dfrac{45}{125} \cdot 100 \% \\ = \dfrac{9}{25} \cdot 100 \% \\ = 36 \% \end{gathered}\text{。}\]

所以正确答案是 B。

English

Originally, each box is worth \[ \$ 5 \div 4 = \$ 1.25. \]

Now, each box is worth \[ \$ 4 \div 5 = \$ 0.8. \]

The percent decrease is then \[ \begin{gathered} \dfrac{1.25 - 0.8}{1.25} \cdot 100 \% \\ = \dfrac{45}{125} \cdot 100 \% \\ = \dfrac{9}{25} \cdot 100 \% \\ = 36 \%. \end{gathered} \]

Thus, B is the correct answer.

19 答案 D

每个分数的分子都会与右边分数的分母约掉。

全部约分后得到 \[ \dfrac{a}{2} = 9 \Rightarrow a = 18 \Rightarrow b = 17\text{。}\]

所求和为 \[ 18 + 17 = 35\text{。}\]

所以正确答案是 D。

English

Note that the numerator of each fraction cancels with the denominator of the fraction to its right.

We can then cancel out all these terms to get a final equation of \[ \dfrac{a}{2} = 9 \Rightarrow a = 18 \Rightarrow b = 17. \]

The desired sum is then \[ 18 + 17 = 35. \]

Thus, D is the correct answer.

20 答案 A

按第一颗骰子的点数分类。如果点数为 \(1\)、\(2\)、\(3\) 或 \(4\)，乘积不可能大于 \(36\)。

如果点数是 \(5\)，另一颗必须掷出 \(8\)，否则乘积小于 \(36\)。

如果第一颗是 \(6\)，另一颗必须是 \(7\) 或 \(8\)，有 \(2\) 种可能。如果第一颗是 \(7\)，另一颗必须是 \(6\)、\(7\) 或 \(8\)，有 \(3\) 种可能。

最后，如果第一颗掷出 \(8\)，另一颗可以是 \(5\)、\(6\)、\(7\) 或 \(8\)，又有 \(4\) 种可能。

有利结果共有 \[ 1 + 2 + 3 + 4 = 10 \] 种，总结果数是 \(8^2 = 64\)，所以概率为 \[ \dfrac{10}{64} = \dfrac{5}{32}\text{。}\]

所以正确答案是 A。

English

We can case on the value of the first die. If its value is \(1,\) \(2,\) \(3,\) or \(4,\) then it is impossible for the product to be greater than \(36.\)

If it is a \(5,\) then the other die has to roll an \(8,\) otherwise the product is less than \(36.\)

If the first die is a \(6,\) then the other die must be \(7\) or \(8,\) giving \(2\) possibilities. If the first die is a \(7,\) then the other die must be \(6,\) \(7,\) or \(8,\) giving \(3\) possibilities.

Finally, if the first roll is an \(8,\) the other die can be \(5,\) \(6,\) \(7,\) or \(8,\) giving \(4\) more possibilities.

The total number of working pairs is \[ 1 + 2 + 3 + 4 = 10, \] and the total number of pairs is \(8^2 = 64.\) The desired probability is then \[ \dfrac{10}{64} = \dfrac{5}{32}. \]

Thus, A is the correct answer.

21 答案 D

任意两个角立方体之间都有一个单位立方体隔开，所以去掉一个角不会影响其他角。

去掉一个角时，失去三个单位正方形表面，但也露出三个新的单位正方形表面。

因此去掉角立方体不改变表面积。原立方体表面积为 \[ 6 \cdot 3^2 = 6 \cdot 9 = 54 \] 平方厘米。

所以正确答案是 D。

English

Note that there is one unit cube between any pair of corner cubes, so the removal of each does not affect the others.

When we remove a corner, we are losing three unit squares. We, however, gain these back from the three faces that get uncovered.

This means that removing a corner cube does not change the surface area. The surface area of the original cube is \[ 6 \cdot 3^2 = 6 \cdot 9 = 54 \] square centimeters.

Thus, D is the correct answer.

22 答案 E

两英寸立方体由 \(2^3 = 8\) 个单位立方体组成。每个单位立方体价值 \[ 200 \div 8 = 25 \] 美元。三英寸立方体需要 \(3^3 = 27\) 个单位立方体，所以价值 \[ 25 \cdot 27 = 675 \] 美元。

所以正确答案是 E。

English

The two-inch cube consists of \(2^3 = 8\) unit cubes. Each of these unit cubes is worth \[ 200 \div 8 = 25 \] dollars. To form a three-inch cube, you need \(3^3 = 27\) unit cubes. This means that it is worth \[ 25 \cdot 27 = 675 \] dollars.

Thus, E is the correct answer.

23 答案 C

如果这个数有五位，最小的递增正数字会使平方和为 \(1^2+2^2+3^2+4^2+5^2=55\)，已经太大。因此最大的有效数最多有四位。

对于四位数 \(abcd\)，其中 \(0<a<b<c<d\)，末位不能是 \(7\) 或更大，因为即使是 \(1^2+2^2+3^2+7^2=63>50\)。

尝试 \(d=6\)，其余数字的平方和必须是 \(50-36=14\)，而 \(1^2+2^2+3^2=14\)。这给出有效数 \(1236\)。

任何以 \(5\) 或更小数字结尾的数都小于 \(1236\)，所以最大的有效整数是 \(1236\)。

它的各位数字乘积为 \(1\cdot2\cdot3\cdot6=36\)。

所以正确答案是 C。

English

If the number had five digits, the smallest possible increasing positive digits would give square-sum \(1^2+2^2+3^2+4^2+5^2=55,\) already too large. So the largest valid number has at most four digits.

For a four-digit number \(abcd\) with \(0<a<b<c<d\), the last digit cannot be \(7\) or larger, since even \(1^2+2^2+3^2+7^2=63>50\).

Trying \(d=6,\) the remaining squares must sum to \(50-36=14,\) and \(1^2+2^2+3^2=14.\) This gives the valid number \(1236.\)

Any number ending with \(5\) or less is smaller than \(1236,\) so the largest valid integer is \(1236.\)

The product of its digits is \(1\cdot2\cdot3\cdot6=36.\)

Thus, C is the correct answer.

24 答案 C

取 \(AE=10\)，使 \(AC=4\)、\(CE=6\)。大半圆半径为 \(5\)，面积为 \(\frac{25\pi}{2}\)。

以 \(AC\) 为直径的半圆半径为 \(2\)，面积为 \(2\pi\)；以 \(CE\) 为直径的半圆半径为 \(3\)，面积为 \(\frac{9\pi}{2}\)。

上方阴影区域面积为 \[\frac{25\pi}{2}-2\pi+\frac{9\pi}{2}=15\pi\text{。}\]

整个圆面积为 \(25\pi\)，所以下方区域面积为 \(25\pi-15\pi=10\pi\)。

所求比为 \(15\pi:10\pi=3:2\)。

所以正确答案是 C。

English

Choose \(AE=10\) so that \(AC=4\) and \(CE=6.\) The large semicircle has radius \(5,\) so its area is \(\frac{25\pi}{2}.\)

The semicircle on \(AC\) has radius \(2\) and area \(2\pi,\) while the semicircle on \(CE\) has radius \(3\) and area \(\frac{9\pi}{2}.\)

The upper shaded region has area \[\frac{25\pi}{2}-2\pi+\frac{9\pi}{2}=15\pi.\]

The whole circle has area \(25\pi,\) so the lower region has area \(25\pi-15\pi=10\pi.\)

The desired ratio is \(15\pi:10\pi=3:2.\)

Thus, C is the correct answer.

25 答案 D

我们只关心个位数字，所以十位数字不重要。

因数的个位可以分成 \(10\) 组，每组个位乘积为 \[ 2 \cdot 4 \cdot 6 \cdot 8 = 384\text{。}\]

每组的个位数字是 \(4\)。因此需要求 \(4^{10}\) 的个位数字。

\(4^2 = 16\) 的个位是 \(6\)，所以只需求 \(6^5\) 的个位。

\(6\) 的任何正整数次幂都以 \(6\) 结尾，例如 \(6\)、\(36\)、\(216\)、\(\ldots\)。

所以正确答案是 D。

English

We only care about the units digit, which means that the tens digits don’t matter.

Then we have \(10\) groups of \[ 2 \cdot 4 \cdot 6 \cdot 8 = 384. \]

The units digit of each group is \(4.\) We now need to find the units digit of \(4^{10}.\)

The units digit of \(4^2 = 16\) is \(6.\) This means we only need to find the units digit of \(6^5.\)

Note that every power of \(6\) always ends in a \(6\) (e.g. \(6,\) \(36,\) \(216,\) \(\ldots\)).

Thus, D is the correct answer.