1996 AMC 8 考试题目

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1.

How many positive factors of $36$ are also multiples of $4?$

$2$

$3$

$4$

$5$

$6$

Answer: B

Concepts:factor multiple

Difficulty rating: 560

Solution:

The positive factors of $36$ are $1, 2, 3, 4, 6, 9, 12, 18, 36$ . Of these, only $4, 12,$ and $36$ are multiples of $4$ .

Thus, the correct answer is B .

2.

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, and 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

Kareem

José and Thuy

Thuy and Kareem

Answer: C

Concepts:order of operations

Difficulty rating: 730

Solution:

Starting from $10$ : José computes $9, 18, 20$ ; Thuy computes $20, 19, 21$ ; Kareem computes $9, 11, 22$ .

Kareem doubles last, so the $2$ he adds is doubled too, giving the largest result.

Thus, the correct answer is C .

3.

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

$130$

$131$

$132$

$133$

$134$

Answer: A

Concepts:arithmetic sequence

Difficulty rating: 560

Solution:

The first row is $1, 2, \ldots, 8$ and the last row is $57, 58, \ldots, 64$ . The four corners are $1, 8, 57,$ and $64$ .

Their sum is $1 + 8 + 57 + 64 = 130$ .

Thus, the correct answer is A .

4.

What is the value of the following expression?

$\frac{2 + 4 + 6 + \cdots + 34}{3 + 6 + 9 + \cdots + 51}$

$\dfrac{1}{3}$

$\dfrac{2}{3}$

$\dfrac{3}{2}$

$\dfrac{17}{3}$

$\dfrac{34}{3}$

Answer: B

Concepts:arithmetic sequence ratio and proportion

Difficulty rating: 800

Solution:

The numerator is $2(1 + 2 + \cdots + 17)$ and the denominator is $3(1 + 2 + \cdots + 17)$ .

The common factor cancels, leaving $\dfrac{2}{3}$ .

Thus, the correct answer is B .

5.

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?

$P - Q$

$P \cdot Q$

$\dfrac{S}{Q} \cdot P$

$\dfrac{R}{P \cdot Q}$

$\dfrac{S + T}{R}$

Answer: A

Concepts:inequality casework

Difficulty rating: 820

Solution:

From the number line, $P$ and $Q$ are negative and $R, S, T$ are positive.

Then $P \cdot Q$ is positive; $\dfrac{S}{Q} \cdot P$ has two negative factors, so it is positive; $\dfrac{R}{P \cdot Q}$ is positive; and $\dfrac{S + T}{R}$ is positive. Since $P$ is to the left of $Q$ , $P - Q$ is negative.

Thus, the correct answer is A .

6.

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 them, then multiply their sum by the third number.

$15$

$30$

$36$

$50$

$56$

Answer: C

Concepts:optimization casework

Difficulty rating: 820

Solution:

Use the three smallest numbers $3, 5, 7$ . The choices are $3(5 + 7) = 36$ , $5(3 + 7) = 50$ , and $7(3 + 5) = 56$ .

Making the smallest number the multiplier gives the least result, $36$ .

Thus, the correct answer is C .

7.

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?

$4$

$5$

$6$

$7$

$8$

Answer: B

Concepts:geometric sequence exponent

Difficulty rating: 860

Solution:

Brent's counts are $4, 16, 64, 256, 1024, 4096$ , and Gretel's are $128, 256, 512, 1024, 2048, 4096$ .

They are equal after $5$ months, when both have $4096$ .

Thus, the correct answer is B .

8.

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

$0$

$3$

$9$

$11$

$17$

Answer: B

Concepts:optimization

Difficulty rating: 930

Solution:

The distance is smallest when the points are collinear with $C$ and $D$ toward $A$ : take $A = 0, B = 10, C = 6, D = 3$ .

Then $AD = 10 - 4 - 3 = 3$ .

Thus, the correct answer is B .

9.

If $5$ times a number is $2$ , then $100$ times the reciprocal of the number is

$2.5$

$40$

$50$

$250$

$500$

Answer: D

Concepts:linear equation fraction

Difficulty rating: 730

Solution:

The number is $\dfrac{2}{5}$ , whose reciprocal is $\dfrac{5}{2}$ .

Then $100 \cdot \dfrac{5}{2} = 250$ .

Thus, the correct answer is D .

10.

When Walter drove up to the gasoline pump, he noticed that his gasoline tank was $\dfrac18$ full. He purchased $7.5$ gallons of gasoline for $10. With this additional gasoline, his gasoline tank was then $\dfrac58$ full. The number of gallons of gasoline his tank holds when it is full is

$8.75$

$10$

$11.5$

$15$

$22.5$

Answer: D

Concepts:fraction ratio and proportion

Difficulty rating: 860

Solution:

The increase is $\dfrac58 - \dfrac18 = \dfrac12$ of a tank, which equals $7.5$ gallons.

So a full tank holds $2 \cdot 7.5 = 15$ gallons.

Thus, the correct answer is D .

11.

Let $x$ be the number

$0.\underbrace{0000\ldots00001}_{1996 \text{ zeros}},$

where there are $1996$ zeros after the decimal point. Which of the following expressions represents the largest number?

$3 + x$

$3 - x$

$3 \cdot x$

$3 / x$

$x / 3$

Answer: D

Concepts:estimation

Difficulty rating: 860

Solution:

Since $x$ is a very small positive number, $3 + x$ and $3 - x$ are near $3$ , while $3 \cdot x$ and $x / 3$ are near $0$ .

But $3 / x$ is $3$ followed by $1997$ zeros, far larger than any other choice.

Thus, the correct answer is D .

12.

What 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 $6.1?$

$4$

$5$

$6$

$7$

$8$

Answer: B

Concepts:mean work backwards

Difficulty rating: 820

Solution:

The sum of $1$ through $11$ is $66$ . For ten numbers to average $6.1$ , their sum must be $10 \cdot 6.1 = 61$ .

So the removed number is $66 - 61 = 5$ .

Thus, the correct answer is B .

13.

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

$1200$

$1500$

$2000$

$2400$

$2700$

Answer: E

Concepts:percentage geometric sequence

Difficulty rating: 930

Solution:

Each year multiplies the count by $1.5$ : $800 \to 1200 \to 1800 \to 2700$ .

So $800 \cdot 1.5^3 = 2700$ participants are expected in $1999$ .

Thus, the correct answer is E .

14.

Six different digits from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ are placed in a figure made of a vertical column of three squares and a horizontal row of four squares that overlap in one shared square, so that the sum of the three entries in the vertical column is $23$ and the sum of the four entries in the horizontal row is $12$ . The sum of the six digits used is

$27$

$29$

$31$

$33$

$35$

Answer: B

Concepts:digits logical deduction

Difficulty rating: 1090

Solution:

Three distinct digits from $1$ through $9$ summing to $23$ must be $6, 8, 9$ . The row's other three digits are at least $1 + 2 + 3 = 6$ , so the shared square (belonging to both the column and the row) is at most $12 - 6 = 6$ . Hence the shared digit is $6$ .

The six digits are then $6, 8, 9$ and $1, 2, 3$ , whose sum is $29$ . Equivalently, $23 + 12 - 6 = 29$ .

Thus, the correct answer is B .

15.

The remainder when the product $1492 \cdot 1776 \cdot 1812 \cdot 1996$ is divided by $5$ is

$0$

$1$

$2$

$3$

$4$

Answer: E

Concepts:units digit modular arithmetic

Difficulty rating: 930

Solution:

The units digit of the product equals the units digit of $2 \cdot 6 \cdot 2 \cdot 6 = 144$ , which is $4$ .

A number ending in $4$ leaves remainder $4$ when divided by $5$ .

Thus, the correct answer is E .

16.

What is the value of the following expression?

$1 - 2 - 3 + 4 + 5 - 6 - 7 + 8 + 9 - 10 - 11 + 12 + 13 - \cdots + 1992 + 1993 - 1994 - 1995 + 1996$

$-998$

$-1$

$0$

$1$

$998$

Answer: C

Concepts:summation pairing and grouping

Difficulty rating: 1060

Solution:

Grouping in blocks of four gives $1 - 2 - 3 + 4 = 0$ , $5 - 6 - 7 + 8 = 0$ , and so on.

There are $1996 / 4 = 499$ such blocks, each equal to $0$ , so the total is $0$ .

Thus, the correct answer is C .

17.

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 $OPQR?$

$(-6, 0)$

$(-4, 0)$

$(-2, 0)$

$(2, 0)$

$(4, 0)$

Answer: C

Concepts:coordinate geometry triangle area

Difficulty rating: 1090

Solution:

Since $OPQR$ is a square with $O = (0, 0)$ and $Q = (2, 2)$ , we have $P = (2, 0)$ and $R = (0, 2)$ , so the area is $2^2 = 4$ .

Triangle $PQT$ has vertical base $PQ$ of length $2$ , and $T = (t, 0)$ lies on the $x$ -axis. Its area is $\tfrac12 \cdot 2 \cdot (2 - t) = 2 - t$ . Setting $2 - t = 4$ gives $t = -2$ , so $T = (-2, 0)$ .

Thus, the correct answer is C .

18.

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

$1920

$1980

$2000

$2020

$2040

Answer: A

Concepts:percentage

Difficulty rating: 960

Solution:

After the raise, the salary is $2000 \cdot 1.2 = 2400$ .

After the cut, it is $2400 \cdot 0.8 = 1920$ .

Thus, the correct answer is A .

19.

The percent of students who prefer golf, bowling, or tennis at East Junior High School and West Middle School is as follows. At East ( $2000$ students): golf $30\%$ , bowling $48\%$ , tennis $22\%$ . At West ( $2500$ students): golf $24\%$ , bowling $36\%$ , tennis $40\%$ . In the two schools combined, the percent of students who prefer tennis is

$30\%$

$31\%$

$32\%$

$33\%$

$34\%$

Answer: C

Concepts:percentage mean

Difficulty rating: 1090

Solution:

East has $22\%$ of $2000 = 440$ tennis fans, and West has $40\%$ of $2500 = 1000$ .

Together $1440$ of the $4500$ students prefer tennis, which is $\dfrac{1440}{4500} = 32\%$ .

Thus, the correct answer is C .

20.

Suppose there is a special key on a calculator that replaces the number $x$ currently displayed with the number given by the formula $1/(1 - x)$ . For example, if the calculator is displaying $2$ and the special key is pressed, then the calculator will display $-1$ since $1/(1 - 2) = -1$ . Now suppose that the calculator is displaying $5$ . After the special key is pressed $100$ times in a row, the calculator will display

$-0.25$

$0$

$0.8$

$1.25$

$5$

Answer: A

Concepts:function modular arithmetic

Difficulty rating: 1280

Solution:

Starting from $5$ : $1/(1 - 5) = -0.25$ , then $1/(1 + 0.25) = 0.8$ , then $1/(1 - 0.8) = 5$ . The values repeat with period $3$ .

Since $100 = 3 \cdot 33 + 1$ , the $100$ th press gives the same result as the first press, $-0.25$ .

Thus, the correct answer is A .

21.

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?

$6$

$8$

$10$

$12$

$24$

Answer: D

Concepts:parity combinations

Difficulty rating: 1200

Solution:

The set has $4$ odd numbers ( $89, 95, 99, 173$ ) and $2$ even numbers ( $132, 166$ ). A sum of three is even only with two odds and one even, since three evens is impossible with just two available.

The count is $\binom{4}{2} \cdot \binom{2}{1} = 6 \cdot 2 = 12$ .

Thus, the correct answer is D .

22.

The horizontal and vertical distances between adjacent points equal $1$ unit. The area of triangle $ABC$ is

$\dfrac{1}{4}$

$\dfrac{1}{2}$

$\dfrac{3}{4}$

$1$

$\dfrac{5}{4}$

Answer: B

Concepts:Pick’s Theorem triangle area

Difficulty rating: 1140

Solution:

Taking $A = (0, 0)$ , $B = (3, 2)$ , and $C = (4, 3)$ , the enclosing $4 \times 3$ rectangle has area $12$ ; subtracting the surrounding regions of areas $6, 3, 2,$ and $\tfrac12$ leaves $\tfrac12$ .

Equivalently, by Pick's theorem with no interior lattice points and $3$ boundary points, the area is $0 + \tfrac32 - 1 = \tfrac12$ .

Thus, the correct answer is B .

23.

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

$945

$950

$955

$990

$995

Answer: E

Concepts:linear equation

Difficulty rating: 1090

Solution:

Let $n$ be the number of employees. The fund is $50n - 5$ (five dollars short of $50$ each) and also $45n + 95$ .

Setting $50n - 5 = 45n + 95$ gives $5n = 100$ , so $n = 20$ . The fund is $45 \cdot 20 + 95 = 995$ .

Thus, the correct answer is E .

24.

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

$90^\circ$

$100^\circ$

$115^\circ$

$122.5^\circ$

$125^\circ$

Answer: C

Concepts:angle chasing angle bisector

Difficulty rating: 1150

Solution:

In triangle $ABC$ , $\angle BAC + \angle BCA = 180^\circ - 50^\circ = 130^\circ$ .

The bisectors give $\angle DAC + \angle DCA = \tfrac{130^\circ}{2} = 65^\circ$ . In triangle $ADC$ , $\angle ADC = 180^\circ - 65^\circ = 115^\circ$ .

Thus, the correct answer is C .

25.

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?

$\dfrac{1}{4}$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

$\dfrac{2}{3}$

$\dfrac{3}{4}$

Answer: A

Concepts:geometric probability area ratio

Difficulty rating: 1260

Solution:

Take the radius to be $1$ . A point at distance $r$ from the center is closer to the center than to the boundary when $r \lt 1 - r$ , i.e. $r \lt \tfrac12$ .

The favorable region is a circle of radius $\tfrac12$ , with area $\pi(\tfrac12)^2 = \tfrac{\pi}{4}$ , out of the total area $\pi$ . The probability is $\dfrac{\pi/4}{\pi} = \dfrac14$ .

Thus, the correct answer is A .