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AMC8 1996

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AMC8 · 1996

Q1
How many positive factors of 36 are also multiples of 4?
36 有多少个正因数同时也是 4 的倍数？
Q2
José, Thuy, and Kareem each start with the number 10. José subtracts 1 from the number 10, doubles his answer, and then adds 2. Thuy doubles the number 10, subtracts 1 from her answer, then adds 2. Kareem subtracts 1 from the number 10, adds 2 to his answer, and then doubles the result. Who gets the largest final answer?
José、Thuy 和 Kareem 都从数字 10 开始。José 从 10 中减去 1，将结果加倍，然后加 2。Thuy 将 10 加倍，从结果中减去 1，然后加 2。Kareem 从 10 中减去 1，将结果加 2，然后加倍。谁得到最大的最终结果？
Q3
The 64 whole numbers from 1 through 64 are written, one per square, on a checkerboard (an 8 by 8 array of 64 squares). The first 8 numbers are written in order across the first row, the next 8 across the second row, and so on. After all 64 numbers are written, the sum of the numbers in the four corners will be
从 1 到 64 的 64 个整数，每个方格一个，写在一个棋盘上（8×8 的 64 个方格）。前 8 个数字按顺序写在第一行，接下来的 8 个写在第二行，依此类推。写完所有 64 个数字后，四角数字的和是
Q4
\(\frac{2+4+6+\cdots+34}{3+6+9+\cdots+51} =\)
\(\frac{2+4+6+\cdots+34}{3+6+9+\cdots+51} =\)
stem
Q5
The letters P, Q, R, S, and T represent numbers located on the number line as shown. Which of the following expressions represents a negative number?
字母 P、Q、R、S 和 T 表示数轴上所示的位置。以下哪个表达式表示一个负数？
Q6
What is the smallest result that can be obtained by the following process? • Choose three different numbers from the set $\{3, 5, 7, 11, 13, 17\}$. • Add two of these numbers. • Multiply their sum by the third number.
通过以下过程能得到的最小结果是多少？ • 从集合 $\{3, 5, 7, 11, 13, 17\}$ 中选择三个不同的数。 • 将其中两个数相加。 • 将它们的和乘以第三个数。
Q7
Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has 4 goldfish at the same time that Gretel has 128 goldfish, then in how many months from that time will they have the same number of goldfish?
Brent的金鱼每月增加到四倍（数量变为原来的四倍），Gretel的金鱼每月增加到两倍。如果Brent有4条金鱼时Gretel有128条金鱼，那么从那时起多少个月后它们金鱼数量相同？
Q8
Points A and B are 10 units apart. Points B and C are 4 units apart. Points C and D are 3 units apart. If A and D are as close as possible, then the number of units between them is
点A和B相距10个单位。点B和C相距4个单位。点C和D相距3个单位。如果A和D尽可能接近，那么它们之间的单位数是
Q9
If 5 times a number is 2, then 100 times the reciprocal of the number is
如果一个数的5倍是2，那么该数的倒数的100倍是
Q10
When Walter drove up to the gasoline pump, he noticed that his gasoline tank was $\frac{1}{8}$ full. He purchased 7.5 gallons of gasoline for \$10. With this additional gasoline, his gasoline tank was then $\frac{5}{8}$ full. The number of gallons of gasoline his tank holds when it is full is
当Walter开车到加油泵时，他注意到他的油箱有 $\frac{1}{8}$ 满。他买了7.5加仑汽油，花了\$10。加了这部分汽油后，他的油箱有 $\frac{5}{8}$ 满。他的油箱满时容纳的加仑数是
Q11
Let $x$ be the number $0.\underbrace{0000\dots0000}_{1996\ zeros}1$, where there are 1996 zeros after the decimal point. Which of the following expressions represents the largest number?
设 $x$ 为小数 $0.\underbrace{0000\dots0000}_{1996\ zeros}1$，其中小数点后有 1996 个零。以下哪个表达式表示最大的数？
Q12
The number should be removed from the list 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 so that the average of the remaining numbers is 6.1?
从列表 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 中移除哪个数，使得剩余数字的平均数为 6.1？
Q13
In the fall of 1996, a total of 800 students participated in an annual school clean-up day. The organizers of the event expect that in each of the years 1997, 1998, and 1999, participation will increase by 50% over the previous year. The number of participants the organizers expect in the fall of 1999 is
1996 年秋季共有 800 名学生参加年度学校清洁日。组织者预计 1997、1998 和 1999 年每年的参与人数将比前一年增加 50%。组织者预计 1999 年秋季的参与人数为
Q14
Six different digits from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ are placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12. The sum of the six digits used is
从集合 $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ 中选取六个不同的数字放置在图示的方格中，使得垂直列的数字和为 23，水平行的数字和为 12。所用六个数字的和为
stem
Q15
The remainder when the product $1492 \cdot 1776 \cdot 1812 \cdot 1996$ is divided by 5 is
$1492 \cdot 1776 \cdot 1812 \cdot 1996$ 的乘积除以 5 的余数是
Q16
$1 - 2 - 3 + 4 + 5 - 6 - 7 + 8 + 9 - 10 - 11 + 12 + 13 - \dots + 1992 + 1993 - 1994 - 1995 + 1996 = \dots$
$1 - 2 - 3 + 4 + 5 - 6 - 7 + 8 + 9 - 10 - 11 + 12 + 13 - \dots + 1992 + 1993 - 1994 - 1995 + 1996 = \dots$
Q17
Figure $OPQR$ is a square. Point $O$ is the origin, and point $Q$ has coordinates $(2, 2)$. What are the coordinates for $T$ so that the area of triangle $PQT$ equals the area of square $OPQR$?
图 $OPQR$ 是一个正方形。点 $O$ 是原点，点 $Q$ 的坐标为 $(2, 2)$。求点 $T$ 的坐标，使得三角形 $PQT$ 的面积等于正方形 $OPQR$ 的面积。
stem
Q18
Ana's monthly salary was $2000 in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in June and July, Ana's monthly salary was
Ana 五月份的月薪是 2000 美元。六月份她涨薪 20%。七月份她减薪 20%。经过六月和七月的两次变化后，Ana 的月薪是
Q19
The pie charts at the right indicate the percent of students who prefer golf, bowling, or tennis at East Junior High School and West Middle School. The total number of students at East is 2000 and at West, 2500. In the two schools combined, the percent of students who prefer tennis is
右边的饼图显示了东初中和西中学的学生偏好高尔夫、保龄球或网球的百分比。东校总学生数 2000 人，西校 2500 人。两校合计，偏好网球的学生百分比是
stem
Q20
Suppose there is a special key on a calculator that replaces the number $x$ currently displayed with the number given by the formula $\frac{1}{1-x}$. For example, if the calculator is displaying 2 and the special key is pressed, then the calculator will display $-1$ since $\frac{1}{1-2} = -1$. Now suppose that the calculator is displaying 5. After the special key is pressed 100 times in a row, the calculator will display
假设计算器上有一个特殊按键，将当前显示的数字 $x$ 替换为公式 $\frac{1}{1-x}$ 给出的数字。例如，如果计算器显示 2，按下特殊键后，将显示 $-1$，因为 $\frac{1}{1-2} = -1$。现在假设计算器显示 5。连续按下特殊键 100 次后，计算器将显示
Q21
How many subsets containing three different numbers can be selected from the set $\{89, 95, 99, 132, 166, 173\}$ so that the sum of the three numbers is even?
从集合 \{89, 95, 99, 132, 166, 173\} 中能选出多少个包含三个不同数字的子集，使得三个数的和为偶数？
Q22
The horizontal and vertical distances between adjacent points equal 1 unit. The area of triangle $ABC$ is
相邻点之间的水平和垂直距离均为1单位。三角形 $ABC$ 的面积是
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Q23
The manager of a company planned to distribute a \$50 bonus to each employee from the company fund, but the fund contained \$5 less than what was needed. Instead the manager gave each employee a \$45 bonus and kept the remaining \$95 in the company fund. The amount of money in the company fund before any bonuses were paid was
公司经理计划从公司基金中向每位员工发放50美元奖金，但基金比所需少5美元。于是经理改发给每位员工45美元奖金，并将剩余的95美元留在公司基金中。在发放任何奖金前，公司基金中的金额是
Q24
The measure of angle $ABC$ is $50^\circ$, $\overline{AD}$ bisects angle $BAC$, and $\overline{DC}$ bisects angle $BCA$. The measure of angle $ADC$ is
角 $ABC$ 的度数为 $50^\circ$，$\overline{AD}$ 平分角 $BAC$，$\overline{DC}$ 平分角 $BCA$。角 $ADC$ 的度数是
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Q25
A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region?
从圆形区域内随机选择一点。该点比到区域边界的距离更靠近区域中心的概率是多少？
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