Let f(x)=4x-x^2. Given x_0, consider the sequence defined by x_n=f(x_n-1) for all n≥1. For how many real numbers x_0 will the sequence x_0, x_1, x_2, … take on only a finite number of different values?

(D) more than 6 but finitely many

1988 AMC 12 — Problems & Online Timed Exam

1.

\sqrt8+\sqrt{18}=

\sqrt{20}

2(\sqrt2+\sqrt3)

7

5\sqrt2

2\sqrt{13}

Answer: D

Concepts:simplifying radicals like terms

Difficulty rating: 890

Small hint:

Extract the largest square factor from each radicand

Big hint:

After simplifying, the two radicals are like terms

Solution:

We have

\sqrt8=2\sqrt2

and

\sqrt{18}=3\sqrt2.

Their sum is

5\sqrt2.

Thus the correct answer is D.

2.

Triangles

ABC

and

XYZ

are similar, with

A

corresponding to

X

and

B

to

Y.

If

AB=3,

BC=4,

and

XY=5,

then

YZ

is

3\frac34

6

6\frac14

6\frac23

8

Answer: D

Difficulty rating: 920

Small hint:

Match

AB

with

XY

and

BC

with

YZ

Big hint:

Use the scale factor from the first triangle to the second

Solution:

The scale factor is

\frac{XY}{AB}=\frac{5}{3}.

Therefore

YZ=(\frac{5}{3})BC=(\frac{5}{3})(4)

and

YZ=\frac{20}{3}=6\frac23.

Thus the correct answer is D.

3.

Four rectangular paper strips of length

10

and width

1

are put flat on a table and overlap perpendicularly as shown. How much area of the table is covered?

36

40

44

98

100

Answer: A

Concepts:area inclusion-exclusion rectangles

Difficulty rating: 960

Small hint:

First add the areas of the four strips

Big hint:

Each perpendicular crossing is a unit square counted twice

Solution:

The strips have total area

4(10)=40.

There are four

1

-by-

1

overlaps, each counted twice, so the covered area is

40-4=36.

Thus the correct answer is A.

4.

The slope of the line

\frac{x}{3}+\frac{y}{2}=1

is

-\frac32

-\frac23

\frac13

\frac23

\frac32

Answer: B

Concepts:slope-intercept form linear equations

Difficulty rating: 1120

Small hint:

Solve the equation for

y

Big hint:

In

y=mx+b,

the coefficient

m

is the slope

Solution:

Multiplying by

2

after isolating

\frac{y}{2}

gives

y=2-\frac23x.

The slope is

-\frac{2}{3}.

Thus the correct answer is B.

5.

If

b

and

c

are constants and

(x+2)(x+b)=x^2+cx+6,

then

c

is

-5

-3

-1

3

5

Answer: E

Concepts:polynomial identities coefficient comparison

Difficulty rating: 1030

Small hint:

Expand the product on the left

Big hint:

Compare the constant term before comparing the coefficient of

x

Solution:

Expansion gives

x^2+(b+2)x+2b.

Thus

2b=6,

so

b=3,

and

c=b+2=5.

Thus the correct answer is E.

6.

A figure is an equiangular parallelogram if and only if it is a

rectangle

regular polygon

rhombus

square

trapezoid

Answer: A

Concepts:quadrilaterals parallelograms angle sum

Difficulty rating: 920

Small hint:

The four equal interior angles must add to

360^\circ

Big hint:

No condition here forces adjacent sides to have equal lengths

Solution:

Four equal angles summing to

360^\circ

are all right angles. A parallelogram with four right angles is exactly a rectangle.

Thus the correct answer is A.

7.

Estimate the time it takes to send

60

blocks of data over a communications channel if each block consists of

512

“chunks” and the channel can transmit

120

chunks per second.

0.04

seconds

0.4

seconds

4

seconds

4

minutes

4

hours

Answer: D

Concepts:unit rates estimation unit conversion

Difficulty rating: 1030

Small hint:

Compute the total number of chunks first

Big hint:

Convert the resulting number of seconds to minutes

Solution:

The transmission takes

\frac{60\cdot512}{120}=256

seconds, which is a little over

4

minutes.

Thus the correct answer is D.

8.

If

\frac ba=2

and

\frac cb=3,

what is the ratio of

a+b

to

b+c?

\frac13

\frac38

\frac35

\frac23

\frac34

Answer: B

Concepts:ratios substitution proportions

Difficulty rating: 1130

Small hint:

Write both

b

and

c

as multiples of

a

Big hint:

Substitute those expressions into

\frac{a+b}{b+c}

Solution:

We have

b=2a

and

c=3b=6a.

Therefore

\frac{a+b}{b+c}=\frac{3a}{8a}=\frac38.

Thus the correct answer is B.

9.

An

8'\times10'

table sits in the corner of a square room, as in Figure

1

below. The owners desire to move the table to the position shown in Figure

2.

The side of the room is

S

feet. What is the smallest integer value of

S

for which the table can be moved as desired without tilting it or taking it apart?

11

12

13

14

15

Answer: C

Concepts:rectangle diagonal rotation geometric bounds

Difficulty rating: 1540

Small hint:

During a quarter-turn, a diagonal becomes perpendicular to a pair of opposite walls

Big hint:

Compare the room side with the table’s diagonal

Solution:

The table diagonal is

\sqrt{8^2+10^2}=\sqrt{164},

which is between

12

and

13.

During the turn this diagonal must fit across the room, so

S\ge\sqrt{164}.

Conversely, a rectangle can rotate inside a circle whose diameter is its diagonal, so that bound is sufficient. The least integer

S

is

13.

Thus the correct answer is C.

10.

In an experiment, a scientific constant

C

is determined to be

2.43865

with an error of at most

\pm0.00312.

The experimenter wishes to announce a value for

C

in which every digit is significant. That is, whatever

C

is, the announced value must be the correct result when

C

is rounded to that number of digits. The most accurate value the experimenter can announce for

C

is

2

2.4

2.43

2.44

2.439

Answer: D

Concepts:error bounds rounding significant digits

Difficulty rating: 1410

Small hint:

Find the smallest and largest possible values of

C

Big hint:

Round both endpoints to increasing numbers of significant digits

Solution:

The possible interval is

[2.43553,2.44177].

Every number in it rounds to

2.44

at three significant digits, but the endpoints round differently at four significant digits. Hence

2.44

is the most accurate guaranteed announcement.

Thus the correct answer is D.

11.

On each horizontal line in the figure below, the five large dots indicate the populations of cities

A,

B,

C,

D,

and

E

in the year indicated. Which city had the greatest percentage increase in population from

1970

to

1980?

A

B

C

D

E

Answer: C

Concepts:percent change graph interpretation data comparison

Difficulty rating: 1290

Small hint:

Read each city’s two population values from the scale

Big hint:

Compare increase divided by the

1970

population, not just absolute increase

Solution:

The percentage increases for

A,B,C,D,E

are respectively

\frac{10}{40},\frac{20}{50},\frac{30}{70},\frac{30}{100},

and

\frac{40}{120}.

These are

25\%,40\%,42\frac67\%,30\%,

and

33\frac13\%,

so city

C

has the greatest percentage increase.

Thus the correct answer is C.

12.

Each integer

1

through

9

is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. Which digit is most likely to be the units digit of the sum of Jack’s integer and Jill’s integer?

0

1

8

9

each digit is equally likely

Answer: A

Concepts:probability modular arithmetic counting ordered pairs

Difficulty rating: 1290

Small hint:

There are

81

equally likely ordered pairs

Big hint:

Count pairs whose sum has each residue modulo

10

Solution:

For a units digit of

0,

the ordered pairs are

(1,9),(2,8),\ldots,(9,1),

giving

9

pairs. Each other units digit occurs

8

times among the

81

ordered pairs. Therefore

0

is most likely.

Thus the correct answer is A.

13.

If

\sin x=3\cos x,

then what is

\sin x\cos x?

\frac16

\frac15

\frac29

\frac14

\frac3{10}

Answer: E

Concepts:trigonometric identities algebraic substitution

Difficulty rating: 1330

Small hint:

Square the given relation and use

\sin^2x+\cos^2x=1

Big hint:

The given relation also determines the sign of the product

Solution:

Squaring gives

\sin^2x=9\cos^2x.

Hence

10\cos^2x=1.

Multiplying

\sin x=3\cos x

by

\cos x

gives

\sin x\cos x=3\cos^2x=\frac{3}{10}.

Thus the correct answer is E.

14.

For any real number

a

and positive integer

k,

define

\binom ak= \frac{\begin{gathered} a(a-1)(a-2)\cdots\\ {}\cdot(a-(k-1)) \end{gathered}} {\begin{gathered} k(k-1)(k-2)\cdots\\ {}\cdot(2)(1) \end{gathered}}.

What is

\binom{-\frac12}{100}\div\binom{\frac12}{100}\ ?

-199

-197

-1

197

199

Answer: A

Concepts:generalized binomial coefficients products cancellation

Difficulty rating: 2090

Small hint:

Cancel the identical denominator

100!

Big hint:

Write the remaining factors as consecutive odd integers and look for telescoping cancellation

Solution:

After canceling

100!,

the numerator factors have magnitudes

1,3,\ldots,199,

while the denominator factors have magnitudes

1,1,3,\ldots,197.

All common odd factors cancel, leaving magnitude

199.

The numerator has

100

negative factors and the denominator has

99,

so the quotient is

-199.

Thus the correct answer is A.

15.

If

a

and

b

are integers such that

x^2-x-1

is a factor of

ax^3+bx^2+1,

then

b

is

-2

-1

0

1

2

Answer: A

Concepts:polynomial divisibility modular reduction coefficient comparison

Difficulty rating: 1760

Small hint:

Reduce powers using

x^2\equiv x+1

Big hint:

A polynomial of degree below

2

divisible by the quadratic must be zero

Solution:

Modulo

x^2-x-1,

we have

x^2\equiv x+1

and

x^3\equiv2x+1.

Thus the polynomial is congruent to

(2a+b)x+(a+b+1).

Both coefficients must vanish, so

2a+b=0

and

a+b=-1.

Hence

a=1

and

b=-2.

Thus the correct answer is A.

16.

ABC

and

A'B'C'

are equilateral triangles with parallel sides and the same center, as in the figure. The distance between side

BC

and side

B'C'

is

\frac16

the altitude of

\triangle ABC.

The ratio of the area of

\triangle A'B'C'

to the area of

\triangle ABC

is

\frac1{36}

\frac16

\frac14

\frac{\sqrt3}{4}

\frac{9+8\sqrt3}{36}

Answer: C

Concepts:equilateral triangles centroid area scale factor

Difficulty rating: 1660

Small hint:

A centroid lies one-third of the altitude above the base

Big hint:

Relate the two altitudes using the distance between their parallel bases

Solution:

Let the outer and inner altitudes be

h

and

h'.

Their common center is

\frac{h}{3}

above

BC

and

\frac{h'}{3}

above

B'C'.

Hence

\frac{h-h'}{3}=\frac{h}{6},

so

h'=\frac{h}{2}.

Areas scale as the square of corresponding lengths, giving

(\frac{h'}{h})^2=\frac{1}{4}.

Thus the correct answer is C.

17.

If

\lvert x\rvert+x+y=10

and

x+\lvert y\rvert-y=12,

find

x+y.

-2

2

\frac{18}{5}

\frac{22}{3}

22

Answer: C

Concepts:absolute value systems of equations case analysis

Difficulty rating: 1570

Small hint:

Use each equation to rule out one sign possibility

Big hint:

After determining the signs of

x

and

y,

solve a linear system

Solution:

If

x\le0,

the first equation gives

y=10,

contradicting the second; hence

x>0.

If

y\ge0,

the second gives

x=12,

contradicting the first; hence

y<0.

The equations become

2x+y=10

and

x-2y=12.

Thus

x=\frac{32}{5},\ y=-\frac{14}{5},

and

x+y=\frac{18}{5}.

Thus the correct answer is C.

18.

At the end of a professional bowling tournament, the top

5

bowlers have a play-off. First #

5

bowls #

4.

The loser receives

5

th prize and the winner bowls #

3

in another game. The loser of this game receives

4

th prize and the winner bowls #

2.

The loser of this game receives

3

rd prize and the winner bowls #

1.

The winner of this game gets

1

st prize and the loser gets

2

nd prize. In how many orders can bowlers #

1

through #

5

receive the prizes?

10

16

24

120

none of these

Answer: B

Concepts:counting binary choices tournament outcomes

Difficulty rating: 1500

Small hint:

Exactly four games are played

Big hint:

Each sequence of winners determines a different prize order

Solution:

Each of the four games has two possible winners. A sequence of four outcomes uniquely determines the five prize positions, so there are

2^4=16

orders.

Thus the correct answer is B.

19.

Simplify

\frac{\begin{gathered} bx(a^2x^2+2a^2y^2+b^2y^2)\\ {}+ay(a^2x^2+2b^2x^2+b^2y^2) \end{gathered}}{bx+ay}.

a^2x^2+b^2y^2

(ax+by)^2

(ax+by)(bx+ay)

2(a^2x^2+b^2y^2)

(bx+ay)^2

Answer: B

Concepts:factoring polynomial identities algebraic simplification

Difficulty rating: 1660

Small hint:

Separate the

2a^2y^2

and

2b^2x^2

terms

Big hint:

Try to factor the numerator first by

bx+ay

Solution:

Regrouping the numerator gives

\begin{aligned} &(bx+ay)(a^2x^2+b^2y^2)\\ &\quad+2abxy(ay+bx)\\ &=(bx+ay)(ax+by)^2. \end{aligned}

Dividing by

bx+ay

yields

(ax+by)^2.

Thus the correct answer is B.

20.

In one of the adjoining figures a square of side

2

is dissected into four pieces so that

E

and

F

are the midpoints of opposite sides and

AG

is perpendicular to

BF.

These four pieces can then be reassembled into a rectangle as shown in the second figure. The ratio of height to base,

\frac{XY}{YZ},

in this rectangle is

4

1+2\sqrt3

2\sqrt5

\frac{8+4\sqrt3}{3}

5

Answer: E

Concepts:dissection area preservation Pythagorean theorem

Difficulty rating: 2010

Small hint:

Find

BF

and

DE

from the side length and midpoint conditions

Big hint:

Use the unchanged area to determine the rectangle’s base

Solution:

Both

BF

and

DE

have horizontal and vertical changes

1

and

2,

so each has length

\sqrt5.

Thus

XY=BF+DE=2\sqrt5.

The rectangle has area

4,

so

YZ=\frac{4}{2\sqrt5}=\frac{2}{\sqrt5}.

Therefore

\frac{XY}{YZ}=5.

Thus the correct answer is E.

21.

The complex number

z

satisfies

z+\lvert z\rvert=2+8i.

What is

\lvert z\rvert^2?

Note: if

z=a+bi,

then

\lvert z\rvert=\sqrt{a^2+b^2}.

68

100

169

208

289

Answer: E

Concepts:complex numbers modulus equations with radicals

Difficulty rating: 1850

Small hint:

Write

z=a+bi

and compare imaginary parts

Big hint:

Use the real part to relate

a

to

\sqrt{a^2+b^2}

Solution:

Writing

z=a+bi

gives

b=8

and

a+\sqrt{a^2+64}=2.

Thus

\sqrt{a^2+64}=2-a.

Squaring yields

a=-15.

Therefore

\lvert z\rvert^2=a^2+b^2=225+64=289.

Thus the correct answer is E.

22.

For how many integers

x

does a triangle with side lengths

10,

24,

and

x

have all its angles acute?

4

5

6

7

more than

7

Answer: A

Concepts:acute triangles Pythagorean inequalities integer counting

Difficulty rating: 1760

Small hint:

Apply the strict Pythagorean inequality to the largest side in each possible ordering

Big hint:

The angle opposite

24

and the angle opposite

x

give the restrictive bounds

Solution:

Acuteness requires

x^2<10^2+24^2=676

and

24^2<10^2+x^2,

so

x<26

and

x^2>476.

For integer

x,

this gives

22\le x\le25.

All four values also satisfy the triangle inequality, so there are

4

values.

Thus the correct answer is A.

23.

The six edges of tetrahedron

ABCD

measure

7,

13,

18,

27,

36,

and

41

units. If the length of edge

AB

is

41,

then the length of edge

CD

is

7

13

18

27

36

Answer: B

Concepts:tetrahedron edges triangle inequality case analysis

Difficulty rating: 2160

Small hint:

The edge of length

7

belongs to two triangular faces

Big hint:

In either face containing that edge, the other two edge lengths must differ by less than

7

Solution:

For a face containing the edge

7,

its other two sides must differ by less than

7.

Among the remaining lengths, the only disjoint pairs with this property are

(13,18)

and

(36,41),

so edge

27

is opposite edge

7.

With

AB=41,

the two possible placements of these pairs can be checked by the triangle inequalities; the placement making the edge opposite

AB

equal to

18

fails, while the valid placement has

CD=13.

Thus the correct answer is B.

24.

An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is

16,

and one of the base angles is

\arcsin(0.8).

Find the area of the trapezoid.

72

75

80

90

not uniquely determined

Answer: C

Concepts:tangential quadrilateral isosceles trapezoid trigonometry area

Difficulty rating: 2160

Small hint:

For a circumscribed quadrilateral, the sums of opposite side lengths are equal

Big hint:

Use

\sin\alpha=\frac{4}{5}

and

\cos\alpha=\frac{3}{5}

to relate the leg, height, and difference of the bases

Solution:

Let the shorter base be

b

and each leg be

\ell.

Tangency gives

16+b=2\ell.

If the base angle is

\alpha,

then

\sin\alpha=\frac{4}{5},\ \cos\alpha=\frac{3}{5},

and

16-b=2\ell\cos\alpha=(\frac{6}{5})\ell.

Solving gives

b=4,\ \ell=10,

and height

\ell\sin\alpha=8.

The area is

\frac12(16+4)(8)=80.

Thus the correct answer is C.

25.

X,

Y,

and

Z

are pairwise disjoint sets of people. The average ages of people in the sets

X,

Y,

Z,

X\cup Y,

X\cup Z,

and

Y\cup Z

are given in the table below.

Set	$X$	$Y$	$Z$	$X\cup Y$	$X\cup Z$	$Y\cup Z$
Average age of people in the set	$37$	$23$	$41$	$29$	$39.5$	$33$

Find the average age of the people in the set

X\cup Y\cup Z.

33

33.5

33.66

33.833

34

Answer: E

Concepts:weighted averages systems of equations set cardinality

Difficulty rating: 2110

Small hint:

Let

x,y,z

be the sizes of the three sets

Big hint:

Each union average gives a linear relation among

x,y,z

Solution:

Let the set sizes be

x,y,z.

The three union averages give

\begin{aligned} 37x+23y&=29(x+y),\\ 37x+41z&=39.5(x+z), \end{aligned}

and

23y+41z=33(y+z).

Thus

y=\frac{4x}{3}

and

z=\frac{5x}{3}.

The total average is

\frac{\begin{gathered} 37x+23(\frac{4x}{3})\\ {}+41(\frac{5x}{3}) \end{gathered}} {x+\frac{4x}{3}+\frac{5x}{3}}=34.

Thus the correct answer is E.

26.

Suppose that

p

and

q

are positive numbers for which

\begin{aligned} \log_9(p)&=\log_{12}(q)\\ &=\log_{16}(p+q). \end{aligned}

What is the value of

\frac{q}{p}?

\frac43

\frac12(1+\sqrt3)

\frac85

\frac12(1+\sqrt5)

\frac{16}{9}

Answer: D

Concepts:logarithms exponential equations quadratic equations

Difficulty rating: 2360

Small hint:

Call the common logarithm value

t

Big hint:

If

r=\frac{q}{p},

compare

(\frac{16}{9})^t

with

(\frac{12}{9})^t

Solution:

Let the common value be

t.

Then

p=9^t,\ q=12^t,

and

p+q=16^t.

Put

r=\frac{q}{p}=(\frac{4}{3})^t.

Dividing the last equation by

p

gives

\begin{aligned} 1+r&=(\frac{16}{9})^t\\ &=((\frac{4}{3})^t)^2=r^2. \end{aligned}

Since

r>0,

r=\frac{1+\sqrt5}{2}.

Thus the correct answer is D.

27.

In the figure,

AB\perp BC,

BC\perp CD,

and

BC

is tangent to the circle with center

O

and diameter

AD.

In which one of the following cases is the area of

ABCD

an integer?

AB=3,

CD=1

AB=5,

CD=2

AB=7,

CD=3

AB=9,

CD=4

AB=11,

CD=5

Answer: D

Concepts:circle tangency power of a point trapezoid area

Difficulty rating: 2280

Small hint:

Let the tangent point be

M

and use the diameter to form a rectangle inside the trapezoid

Big hint:

Power of point

B

relates half of

BC

to

AB

and

CD

Solution:

The tangent point is the midpoint of

BC,

and the tangent-secant relation gives

(\frac{BC}{2})^2=AB\cdot CD.

Thus

BC=2\sqrt{AB\cdot CD},

and the trapezoid area is

\begin{aligned} \text{Area} &=\frac{AB+CD}{2}BC\\ &=(AB+CD)\sqrt{AB\cdot CD}. \end{aligned}

Only

AB=9,\ CD=4

makes the product under the radical a square; the area is

13\cdot6=78.

Thus the correct answer is D.

28.

An unfair coin has probability

p

of coming up heads on a single toss. Let

w

be the probability that, in

5

independent tosses of this coin, heads come up exactly

3

times. If

w=\frac{144}{625},

then

p

must be

\frac{2}{5}

p

must be

\frac{3}{5}

p

must be greater than

\frac{3}{5}

p

is not uniquely determined

there is no value of

p

for which

w=\frac{144}{625}

Answer: D

Concepts:binomial probability continuity nonunique solutions

Difficulty rating: 2360

Small hint:

Write

w

as a function of

p

using the binomial coefficient

Big hint:

Check one simple value, then compare the function at

p=\frac{3}{5}

and

p=1

Solution:

Here

w(p)=10p^3(1-p)^2.

At

p=\frac{2}{5},

w=10\left(\frac25\right)^3\left(\frac35\right)^2 =\frac{144}{625}.

Also

w(\frac{3}{5})=\frac{216}{625}>\frac{144}{625},

while

w(1)=0.

By continuity there is another solution between

\frac{3}{5}

and

1.

Hence

p

is not unique.

Thus the correct answer is D.

29.

You plot weight

(y)

against height

(x)

for three of your friends and obtain the points

(x_1,y_1),

(x_2,y_2),

(x_3,y_3).

If

\begin{aligned} x_1&\lt x_2\lt x_3,\\ x_3-x_2&=x_2-x_1, \end{aligned}

which of the following is necessarily the slope of the line which best fits the data? “Best fits” means that the sum of the squares of the vertical distances from the data points to the line is smaller than for any other line.

\frac{y_3-y_1}{x_3-x_1}

\frac{(y_2-y_1)-(y_3-y_2)}{x_3-x_1}

\frac{2y_3-y_1-y_2}{2x_3-x_1-x_2}

\frac{y_2-y_1}{x_2-x_1}+\frac{y_3-y_2}{x_3-x_2}

none of these

Answer: A

Concepts:least squares slope symmetric coordinates

Difficulty rating: 2480

Small hint:

Translate and scale the

x

-coordinates to

-1,0,1

Big hint:

For a least-squares line through symmetric

x

-values, compute the slope from the covariance numerator

Solution:

Translate and scale so the

x

-coordinates are

-d,0,d.

Their mean is

0,

so the least-squares slope is

\begin{aligned} \frac{(-d)y_1+0y_2+dy_3}{d^2+0+d^2} &=\frac{y_3-y_1}{2d}\\ &=\frac{y_3-y_1}{x_3-x_1}. \end{aligned}

Thus the correct answer is A.

30.

Let

f(x)=4x-x^2.

Given

x_0,

consider the sequence defined by

x_n=f(x_{n-1})

for all

n\ge1.

For how many real numbers

x_0

will the sequence

x_0,

x_1,

x_2,

\ldots

take on only a finite number of different values?

0

1

or

2

3,

4,

5

or

6

more than

6

but finitely many

infinitely many

Answer: E

Concepts:function iteration preimages induction finite orbits

Difficulty rating: 2930

Small hint:

Start with

0,

then find numbers mapping successively to

0

Big hint:

For every

a\le4,

solve

4x-x^2=a

and choose a new real preimage

Solution:

The starting values

0,4,2

give finite orbits

0

,

4\to0

, and

2\to4\to0.

More generally, if

a_n

begins a finite chain ending at

0,

solve

4a_{n+1}-a_{n+1}^2=a_n.

Its real solutions are

a_{n+1}=2\pm\sqrt{4-a_n}.

Choosing a preimage not already in the chain extends it by one new value. Repeating produces infinitely many distinct starting values with finite orbits.

Thus the correct answer is E.

1988 AMC 12 Problems

Answer: D

Difficulty rating: 890

Small hint:

Big hint:

Solution:

Answer: D

Difficulty rating: 920

Small hint:

Big hint:

Solution:

Answer: A

Difficulty rating: 960

Small hint:

Big hint:

Solution:

Answer: B

Difficulty rating: 1120

Small hint:

Big hint:

Solution:

Answer: E

Difficulty rating: 1030

Small hint:

Big hint:

Solution:

Answer: A

Difficulty rating: 920

Small hint:

Big hint:

Solution:

Answer: D

Difficulty rating: 1030

Small hint:

Big hint:

Solution:

Answer: B

Difficulty rating: 1130

Small hint:

Big hint:

Solution:

Answer: C

Difficulty rating: 1540

Small hint:

Big hint:

Solution:

Answer: D

Difficulty rating: 1410

Small hint:

Big hint:

Solution:

Answer: C

Difficulty rating: 1290

Small hint:

Big hint:

Solution:

Answer: A

Difficulty rating: 1290

Small hint:

Big hint:

Solution:

Answer: E

Difficulty rating: 1330

Small hint:

Big hint:

Solution:

Answer: A

Difficulty rating: 2090

Small hint:

Big hint:

Solution:

Answer: A

Difficulty rating: 1760

Small hint:

Big hint:

Solution:

Answer: C

Difficulty rating: 1660

Small hint:

Big hint: