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AMC8 1990

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AMC8 · 1990

Q1
What is the smallest sum of two 3-digit numbers that can be obtained by placing each of the six digits 4, 5, 6, 7, 8, 9 in one of the six boxes in this addition problem?
通过将六个数字4、5、6、7、8、9各放置在该加法题的六个方框中，能得到的最小两位3位数之和是多少？
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Correct Answer: C
In the smallest such sum, the two smallest digits are in the hundred’s places, the next two digits in the ten’s places and the two largest digits are in the one’s places. One example is $468 + 579 = 1047$.
在这样的最小和中，两位最小的数字位于百位处，接下来两位的数字位于十位处，两位最大的数字位于个位处。一个例子是 $468 + 579 = 1047$。
Q2
Which digit of .12345, when changed to 9, gives the largest number?
将小数 .12345 的哪一位数字改为9，能得到最大的数？
Correct Answer: A
An increase in the tenth’s place gives a larger value than an increase in any of the other decimal places. Since 1 is in the tenth’s place of .12345, (A) is correct.
十分位处的增加比其他任何小数位处的增加给出的值都大。由于 .12345 的十分位处是1，因此 (A) 是正确的。
Q3
What fraction of the square is shaded?
正方形中有多少分数被涂黑？
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Correct Answer: E
Each shaded piece above or below the diagonal is matched by an identical unshaded piece meaning $\frac{1}{2}$ of the total area is shaded.
对角线上方或下方的每个涂黑部分都有一个相同的未涂黑部分与之匹配，这意味着总面积的 $\frac{1}{2}$ 被涂黑。
Q4
Which of the following could not be the unit's digit of the square of a whole number?
下列哪一个不可能是某个整数平方的个位数字？
Correct Answer: E
The unit’s digit of the square of a whole number is determined by the square of the unit’s digit of that whole number. The possible units digits are $0,1,4,5,6,9$. Note that $2,3,7,$ or $8$ will never occur.
整数平方的个位数字由该整数个位数字的平方决定。可能的个位数字是 $0,1,4,5,6,9$。注意 $2,3,7$ 或 $8$ 永远不会出现。
Q5
Which of the following is the closest to the product $(.48017)(.48017)(.48017)$?
下列哪一个最接近于乘积 $(.48017)(.48017)(.48017)$？
Correct Answer: B
The desired product is about $\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{8} = 0.125$.
所需的乘积大约是 $\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{8} = 0.125$。
Q6
Which of these five numbers is the largest?
这五个数中哪个最大？
Correct Answer: D
All of the choices, except (C) and (D), are near 13579. In (D), the result is the product $(13579)(2468)$ while in (C) the result is much less than 13579.
除了 (C) 和 (D) 外的所有选项都接近 13579。在 (D) 中，结果是 $(13579)(2468)$ 的乘积，而在 (C) 中，结果远小于 13579。
Q7
When three different numbers from the set $\{-3, -2, -1, 4, 5\}$ are multiplied, the largest possible product is
从集合 $\{-3, -2, -1, 4, 5\}$ 中选取三个不同的数相乘，最大可能积是
Correct Answer: C
For the product of three numbers to be positive, either all three must be positive or two negative and one positive. Since there are only two positives, the max is $(-3)(-2)(5)=30$.
三个数的乘积为正，要么全正，要么两个负一个正。由于只有两个正数，最大值为 $(-3)(-2)(5)=30$。
Q8
A dress originally priced at $80 was put on sale at 25% off. If 10% tax was added to the sale price, then the total selling price of the dress was
一件原价 80 美元的连衣裙打 25% 折销售。销售价上再加 10% 税，总售价是
Correct Answer: D
The sale price was $\frac{3}{4}(\$80) = \$60$. Thus the tax was $\$$6 and the total was $\$$66.
销售价为 $\frac{3}{4}(\$80) = \$60$。税为 $\$$6，总价为 $\$$66。
Q9
The fifteen scores in Mr. Freeman's class were: 89, 72, 54, 97, 77, 92, 85, 74, 75, 63, 84, 78, 71, 80, 90. In Mr. Freeman's class, what percent of the students received a grade of C?
弗里曼先生班上的 15 个成绩是：89, 72, 54, 97, 77, 92, 85, 74, 75, 63, 84, 78, 71, 80, 90。在弗里曼先生班上，有百分之多少的学生获得 C 等级？
stem
Correct Answer: D
Five of the fifteen scores [77, 75, 84, 78, 80] are in the “C range”, so $5/15 = 1/3 = 33\frac{1}{3}\%$.
15 个成绩中有 5 个 [77, 75, 84, 78, 80] 在“C 范围”内，所以 $5/15 = 1/3 = 33\frac{1}{3}\%$。
Q10
On this monthly calendar, the date behind one of the letters is added to the date behind C. If this sum equals the sum of the dates behind A and B, then the letter is
在这个月历上，其中一个字母后面的日期加上 C 后面的日期，其和等于 A 和 B 后面日期的和，则该字母是
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Correct Answer: A
Since the date behind C is one less than that behind A, the date behind the desired letter must be one more than that behind B. This date is behind P.
由于 C 后面的日期比 A 后面的少 1，因此所求字母后面的日期必须比 B 后面的多 1。这个日期在 P 后面。
Q11
The numbers on the faces of this cube are consecutive whole numbers. The sums of the two numbers on each of the three pairs of opposite faces are equal. The sum of the six numbers on this cube is
这个立方体的各个面上的数字是连续的整数。三对相对面的两个数字之和相等。这个立方体上六个数字的总和是
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Correct Answer: E
11,12,13,14,15 on five faces, remaining 10 or 16. 10 opp 15 impossible, so 16 opp 11,12 opp 15,14 opp 13, each pair 27, total $3\times27=81$.
五个面标11,12,13,14,15，剩余面为10或16。10相对15不可能，因此16相对11，12相对15，14相对13，每对和为27，总和$3\times27=81$。
Q12
There are twenty-four 4-digit whole numbers that use each of the four digits 2, 4, 5, and 7 exactly once. Listed in numerical order from smallest to largest, the number in the 17th position in the list is
有二十四个4位数的整数，它们恰好各使用一次数字2、4、5和7。从最小到最大按数值顺序排列，列表中第17个位置的数是
Correct Answer: B
1/4 of 24=6 begin with each digit. 1-6:2xxx,7-12:4xxx,13-18:5xxx. Fifth with 5: 5247,5274,5427,5472,5724.
24的1/4=6，以每个数字开头各有6个。1-6:2xxx，7-12:4xxx，13-18:5xxx。以5开头的第五个：5247,5274,5427,5472,5724。
Q13
One proposal for new postage rates for a letter was 30¢ for the first ounce and 22¢ for each additional ounce (or fraction of an ounce). The postage for a letter weighing 4.5 ounces was
一封信的新邮资方案是首盎司30美分，每额外盎司（或零头）22美分。重4.5盎司的信的邮资是
Correct Answer: C
First ounce $0.30$. Additional 3.5 oz costs $4*0.22= 0.88$. Total \$1.18.
首盎司$0.30$。额外3.5盎司需 $4*0.22= 0.88$。总计\$1.18。
Q14
A bag contains only blue balls and green balls. There are 6 blue balls. If the probability of drawing a blue ball at random from this bag is $\frac{1}{4}$, then the number of green balls in the bag is
一个袋子里只有蓝球和绿球。有6个蓝球。从袋中随机抽取一个蓝球的概率是$\frac{1}{4}$，则袋子里的绿球数是
Correct Answer: B
Blue $\frac{1}{4}$, 6 blue, total 24 balls, green 18. Or green 3 times blue: $3\times6=18$.
蓝球占$\frac{1}{4}$，6个蓝球，总数24个球，绿球18个。或者绿球是蓝球的3倍：$3\times6=18$。
Q15
The area of this figure is 100 cm². Its perimeter is
这个图形的面积是100 cm²。其周长是
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Correct Answer: E
Four squares total 100 cm², each 25 cm², side 5 cm, perimeter 10×5=50 cm.
四个正方形总面积100 cm²，每个25 cm²，边长5 cm，周长10×5=50 cm。
Q16
1990 - 1980 + 1970 - 1960 + \dots - 20 + 10 =
1990 - 1980 + 1970 - 1960 + \dots - 20 + 10 =
Correct Answer: D
By grouping as shown below, there are $\frac{199+1}{2} = 100$ groups of 10, for a sum of 1000: \[[1990 - 1980] + [1970 - 1960] + \cdots + [30 - 20] + 10.\]
By grouping as shown below, there are $\frac{199+1}{2} = 100$ groups of 10, for a sum of 1000: \[[1990 - 1980] + [1970 - 1960] + \cdots + [30 - 20] + 10.\]
Q17
A straight concrete sidewalk is to be 3 feet wide, 60 feet long and 3 inches thick. How many cubic yards of concrete must a contractor order for the sidewalk if concrete must be ordered in a whole number of cubic yards?
一条直线混凝土人行道宽 3 英尺，长 60 英尺，厚 3 英寸。承包商需要订购多少立方码混凝土，如果混凝土必须以整数量的立方码订购？
Correct Answer: A
Answer (A): The number of cubic feet is $3 \times 60 \times \frac{1}{4} = 45$. Since there are 27 cubic feet in 1 cubic yard, there are $\frac{45}{27} = 1\tfrac{2}{3}$ cubic yards of concrete required. Thus 2 cubic yards must be ordered.
答案（A）：立方英尺的数量为 $3 \times 60 \times \frac{1}{4} = 45$。 由于 1 立方码等于 27 立方英尺，因此需要的混凝土体积为 $\frac{45}{27} = 1\tfrac{2}{3}$ 立方码。 因此必须订购 2 立方码的混凝土。
Q18
Each corner of a rectangular prism is cut off. Two (of the eight) cuts are shown. How many edges does the new figure have?
一个长方体每个角都被切掉。图中展示了八个切口中的两个。新图形有多少条边？
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Correct Answer: C
The original prism had 12 edges. Each "cut-off" corner yields 3 additional edges, so the new figure has a total of 12 + 8 x 3 = 36 edges.
原来有 12 条边。每个切角增加 3 条边，8×3=24，总共 12+24=36。
Q19
There are 120 seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone?
一排有 120 个座位。必须占用多少个座位，才能使下一个坐下的人必须坐在某人旁边？（最少）
Correct Answer: B
Answer (B): In order for the fewest number of seats to be occupied, there must be someone in every third seat, beginning with \#2 and ending with \#119. There are a total of $\frac{120}{3} = 40$ occupied seats. OR Consider some simpler cases and make a table: Number of seats in the row: \[3 \quad 6 \quad 9 \quad 12 \] Number of occupied seats in the row: \[1 \quad 2 \quad 3 \quad 4\] In each case, the middle seat in every group of three seats must be occupied, so the desired number of occupied seats in a row of 120 seats is $\frac{120}{3} = 40$.
答案（B）：为了使被占用的座位数最少，必须每隔两个座位坐一个人，也就是从 \#2 开始到 \#119 结束。被占用的座位总数为 $\frac{120}{3} = 40$。 或者 考虑一些更简单的情况，并列出表格： 一排座位数：\[3 \quad 6 \quad 9 \quad 12 \] 被占用的座位数：\[1 \quad 2 \quad 3 \quad 4\] 在每种情况下，每三个座位为一组时，中间的座位必须被占用，因此一排 120 个座位时，被占用的座位数为 $\frac{120}{3} = 40$。
Q20
The annual incomes of 1,000 families range from \$8200 to \$98,000. In error, the largest income was entered on the computer as $980,000. The difference between the mean of the incorrect data and the mean of the actual data is
1000 个家庭的年收入从 8200 美元到 98,000 美元。由于输入错误，最高收入在计算机上被输入为 980,000 美元。不正确数据平均数与实际数据平均数的差值为
Correct Answer: A
Answer (A): The difference between the incorrect sum and the actual sum is $\$980{,}000 - \$98{,}000 = \$882{,}000$. Since this difference is equally shared by all 1000 families, the difference between the means is $\frac{882000}{1000} = \$882$.
答案（A）：错误总和与实际总和之间的差为 $\$980{,}000 - \$98{,}000 = \$882{,}000$。 由于这笔差额由 1000 个家庭平均分担，因此两个平均数之间的差为 $\frac{882000}{1000} = \$882$。
Q21
A list of 8 numbers is formed by beginning with two given numbers. Each new number in the list is the product of the two previous numbers. Find the first number if the last three are shown: \[ ? , \underline{\hspace{1cm}} , \underline{\hspace{1cm}} , \underline{\hspace{1cm}} , \underline{\hspace{1cm}} , \underline{\hspace{1cm}} , 16 , 64 , 1024 \] what is the first number?
一个由8个数组成的列表，从给定的两个数开始形成。列表中的每个新数是前两个数的乘积。如果最后三个数已知： \[? , \underline{\hspace{1cm}} , \underline{\hspace{1cm}} , \underline{\hspace{1cm}} , \underline{\hspace{1cm}} , \underline{\hspace{1cm}} , 16 , 64 , 1024\] 求第一个数。
Correct Answer: B
Answer (B): Working backward from $1024$, divide each number (e.g. $1024$) by the preceding number (e.g. $64$) to get the previous number (e.g. $16$) in the list. Thus $64 \div 16 = 4$, $4 \div 4 = 1$, and so on: $\frac{1}{4},\ 4,\ 1,\ 4,\ 4,\ 16,\ 64,\ 1024$
答案（B）：从 $1024$ 开始向前推，用当前数（例如 $1024$）除以前一个数（例如 $64$），即可得到再前一个数（例如 $16$）。因此 $64 \div 16 = 4$，$4 \div 4 = 1$，依此类推： $\frac{1}{4},\ 4,\ 1,\ 4,\ 4,\ 16,\ 64,\ 1024$
Q22
Several students are seated at a large circular table. They pass around a bag containing 100 pieces of candy. Each person receives the bag, takes one piece of candy and then passes the bag to the next person. If Chris takes the first and the last piece of candy, then the number of students at the table could be
几个学生围坐在一张大圆桌旁。他们传着一个装有100块糖果的袋子。每人拿到袋子时，取一块糖果，然后传给下一个。克里斯拿了第一块和最后一块糖果，则桌旁学生人数可能是
Correct Answer: B
Answer (B): Since Chris takes the last piece of candy, each person receives the same number of the other $99$ pieces of candy. Thus the number of students at the table must be a factor of $99$. Only (B) fulfills this condition.
答案（B）：由于 Chris 拿走了最后一块糖，其余的 $99$ 块糖必须平均分给每个人。因此，桌旁学生的人数必须是 $99$ 的因数。只有选项（B）满足这一条件。
Q23
The graph relates the distance traveled [in miles] to the time elapsed [in hours] on a trip taken by an experimental airplane. During which hour was the average speed of this airplane the largest?
该图表示实验飞机一次飞行中行驶距离[英里]与经过时间[小时]的关系。在哪一小时该飞机的平均速度最大？
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Correct Answer: B
Answer (B): The rate will be the largest when the graph is the “steepest”. During the second hour the distance traveled is about $500$ miles, so the average speed during that hour is about $500$ mph. For the other hours, the speeds are less than $350$ mph.
答案（B）：当图像最“陡”时，速度最大。在第二个小时内，行驶的距离约为 $500$ 英里，因此该小时的平均速度约为 $500$ 英里/小时。其余时间段的速度都小于 $350$ 英里/小时。
Q24
Three $\triangle$ and a $\diamond$ will balance nine $\bullet$'s. One $\triangle$ will balance a $\diamond$ and a $\bullet$. How many $\bullet$'s will balance the two $\diamond$'s in this balance?
三个 $\triangle$ 和一个 $\diamond$ 可以平衡九个 $\bullet$。一个 $\triangle$ 可以平衡一个 $\diamond$ 和一个 $\bullet$。在这个天平中，两个 $\diamond$ 可以平衡多少个 $\bullet$？
stem stem stem
Correct Answer: C
Answer (C): The second balance shows each $\triangle$ balances $\diamond\bullet$. Replace each $\triangle$ on the first balance with $\diamond\bullet$. Then after removing three $\bullet$'s from each side, the balance has $\diamond\diamond\diamond\diamond\diamond$ on the left and $\bullet\bullet\bullet\bullet\bullet$ on the right. Thus $\diamond\diamond$ will be balanced by $\bullet\bullet\bullet$.
答案（C）：第二个天平表明每个 $\triangle$ 与 $\diamond\bullet$ 平衡。将第一个天平上的每个 $\triangle$ 都替换为 $\diamond\bullet$。然后从天平两边各去掉三个 $\bullet$，天平左边剩下 $\diamond\diamond\diamond\diamond\diamond$，右边剩下 $\bullet\bullet\bullet\bullet\bullet$。因此，$\diamond\diamond$ 与 $\bullet\bullet\bullet$ 平衡。
Q25
How many difference patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different.
在九个方格中恰好涂黑两个，能形成多少种不同的图案？可以通过翻转和/或旋转匹配的图案不视为不同。
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Correct Answer: C
Answer (C): There are $8$. Be systematic. You can begin with the five cases that have one corner square. Then consider the other three cases that do not have a corner square.
答案（C）：共有 $8$ 种。要有条理地分类。可以先从包含一个角落方格的 5 种情况开始，再考虑另外 3 种不包含角落方格的情况。
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