MathDB

1

Number of Real Solutions

\lfloor x \rfloor

be the greatest integer less than or equal to

x

. Then the number of real solutions to 4x^2 \minus{} 40 \lfloor x \rfloor \plus{} 51 \equal{} 0 is

<span class='latex-bold'>(A)</span>\ 0 \qquad <span class='latex-bold'>(B)</span>\ 1 \qquad <span class='latex-bold'>(C)</span>\ 2 \qquad <span class='latex-bold'>(D)</span>\ 3 \qquad <span class='latex-bold'>(E)</span>\ 4

29

1

Sum of the Digits of a Large Product

In their base

10

representation, the integer

a

consists of a sequence of

1985

eights and the integer

b

consists of a sequence of

1985

fives. What is the sum of the digits of the base 10 representation of

9ab

?

<span class='latex-bold'>(A)</span>\ 15880 \qquad <span class='latex-bold'>(B)</span>\ 17856 \qquad <span class='latex-bold'>(C)</span>\ 17865 \qquad <span class='latex-bold'>(D)</span>\ 17874 \qquad <span class='latex-bold'>(E)</span>\ 19851

28

1

Finding the Side Length of a Triangle

In

\triangle ABC

, we have \angle C \equal{} 3 \angle A, a \equal{} 27, and c \equal{} 48. What is

b

?
[asy]size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(14,0), C=(10,6); draw(A--B--C--cycle); label("

A

", A, SW); label("

B

", B, SE); label("

C

", C, N); label("

a

", B--C, dir(B--C)*dir(-90)); label("

b

", A--C, dir(C--A)*dir(-90)); label("

c

", A--B, dir(A--B)*dir(-90)); [/asy]

<span class='latex-bold'>(A)</span>\ 33 \qquad <span class='latex-bold'>(B)</span>\ 35 \qquad <span class='latex-bold'>(C)</span>\ 37 \qquad <span class='latex-bold'>(D)</span>\ 39 \qquad <span class='latex-bold'>(E)</span>\ \text{not uniquely determined}

27

1

Finding the Smallest Integer in a Sequence

Consider a sequence

x_1, x_2, x_3, ...

, defined by \begin{align*}x_1 &= \sqrt [3]{3}\\ x_2 &= \sqrt [3]{3} ^ {\sqrt [3]{3}},\end{align*}and in general x_n \equal{} (x_{n \minus{} 1}) ^ {\sqrt [3]{3}}\,\,\text{ for }\,\,n > 1. What is the smallest value of

n

for which

x_n

is an integer?

<span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ 3 \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 9 \qquad <span class='latex-bold'>(E)</span>\ 27

26

1

Finding the Smallest Reducible Fraction

Find the least positive integer

n

for which \frac{n\minus{}13}{5n\plus{}6} is non-zero reducible fraction.

<span class='latex-bold'>(A)</span>\ 45 \qquad <span class='latex-bold'>(B)</span>\ 68 \qquad <span class='latex-bold'>(C)</span>\ 155 \qquad <span class='latex-bold'>(D)</span>\ 226 \qquad <span class='latex-bold'>(E)</span>\ \text{none of these}

25

1

Dimensions of a Rectangular Solid

The volume of a certain rectangular solid is

8 \text{ cm}^3

, its total surface area is

32 \text{ cm}^3

, and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is

<span class='latex-bold'>(A)</span>\ 28 \qquad <span class='latex-bold'>(B)</span>\ 32 \qquad <span class='latex-bold'>(C)</span>\ 36 \qquad <span class='latex-bold'>(D)</span>\ 40 \qquad <span class='latex-bold'>(E)</span>\ 44

24

1

Logarithms

A non-zero digit is chosen in such a way that the probability of choosing digit

d

is \log_{10}(d\plus{}1) \minus{} \log_{10} d. The probability that the digit

2

is chosen is exactly

\frac12

the probability that the digit chosen is in the set

<span class='latex-bold'>(A)</span>\ \{2,3\} \qquad <span class='latex-bold'>(B)</span>\ \{3,4\} \qquad <span class='latex-bold'>(C)</span>\ \{4,5,6,7,8\} \qquad <span class='latex-bold'>(D)</span>\ \{5,6,7,8,9\} \qquad <span class='latex-bold'>(E)</span>\ \{4,5,6,7,8,9\}

23

1

Finding the False Equation

If x \equal{} \frac { \minus{} 1 \plus{} i\sqrt3}{2}\qquad\text{and}\qquad y \equal{} \frac { \minus{} 1 \minus{} i\sqrt3}{2}, where i^2 \equal{} \minus{} 1, then which of the following is not correct?
(A)\ x^5 \plus{} y^5 \equal{} \minus{} 1 \qquad (B)\ x^7 \plus{} y^7 \equal{} \minus{} 1 \qquad (C)\ x^9 \plus{} y^9 \equal{} \minus{} 1 (D)\ x^{11} \plus{} y^{11} \equal{} \minus{} 1 \qquad (E)\ x^{13} \plus{} y^{13} \equal{} \minus{} 1

22

1

Finding the Length of a Partial Chord

In a circle with center

O

,

AD

is a diameter,

ABC

is a chord, BO \equal{} 5, and

\angle ABO = \stackrel{\frown}{CD} = 60^{\circ}

. Then the length of

BC

is:
[asy]size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair O=origin, A=dir(35), C=dir(155), D=dir(215), B=intersectionpoint(dir(125)--O, A--C); draw(C--A--D^^B--O^^Circle(O,1)); pair point=O; label("

A

", A, dir(point--A)); label("

B

", B, dir(point--B)); label("

C

", C, dir(point--C)); label("

D

", D, dir(point--D)); label("

O

", O, dir(305));
label("

5

", B--O, dir(O--B)*dir(90)); label("

60^\circ

", dir(185), dir(185)); label("

60^\circ

", B+0.05*dir(-25), dir(-25));[/asy]
(A)\ 3 \qquad (B)\ 3 \plus{} \sqrt3 \qquad (C)\ 5 \minus{} \frac{\sqrt3}{2} \qquad (D)\ 5 \qquad (E)\ \text{none of the above}

21

1

Finding Integral Solutions

How many integers

x

satisfy the equation (x^2 \minus{} x \minus{} 1)^{x \plus{} 2} \equal{} 1

<span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ 3 \qquad <span class='latex-bold'>(C)</span>\ 4 \qquad <span class='latex-bold'>(D)</span>\ 5 \qquad <span class='latex-bold'>(E)</span>\ \text{none of these}

20

1

Cutting a Cube

A wooden cube with edge length

n

units (where

n

is an integer

>2

) is painted black all over. By slices parallel to its faces, the cube is cut into

n^3

smaller cubes each of unit length. If the number of smaller cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is

n

?

<span class='latex-bold'>(A)</span>\ 5 \qquad <span class='latex-bold'>(B)</span>\ 6 \qquad <span class='latex-bold'>(C)</span>\ 7 \qquad <span class='latex-bold'>(D)</span>\ 8 \qquad <span class='latex-bold'>(E)</span>\ \text{none of these}

19

1

Intersection Between Two Functions

Consider the graphs y \equal{} Ax^2 and and y^2 \plus{} 3 \equal{} x^2 \plus{} 4y, where

A

is a positive constant and

x

and

y

are real variables. In how many points do the two graphs intersect?

<span class='latex-bold'>(A)</span>\ \text{exactly } 4 \qquad <span class='latex-bold'>(B)</span>\ \text{exactly } 2

<span class='latex-bold'>(C)</span>\ \text{at least } 1, \text{ but the number varies for different positive values of } A

<span class='latex-bold'>(D)</span>\ 0 \text{ for at least one positive value of } A \qquad <span class='latex-bold'>(E)</span>\ \text{none of these}

18

1

Bags of Marbles

Six bags of marbles contain

18

,

19

,

21

,

23

,

25

, and

34

marbles, respectively. One bag contains chipped marbles only. The other

5

bags contain no chipped marbles. Jane takes three of the bags and George takes two of the others. Only the bag of chipped marbles remains. If Jane gets twice as many marbles as George, how many chipped marbles are there?

<span class='latex-bold'>(A)</span>\ 18 \qquad <span class='latex-bold'>(B)</span>\ 19 \qquad <span class='latex-bold'>(C)</span>\ 21 \qquad <span class='latex-bold'>(D)</span>\ 23 \qquad <span class='latex-bold'>(E)</span>\ 25

17

1

Finding the Area of a Rectangle

Diagonal

DB

of rectangle

ABCD

is divided into

3

segments of length

1

by parallel lines

L

and

L'

that pass through

A

and

C

and are perpendicular to

DB

. The area of

ABCD

, rounded to the nearest tenth, is
[asy]size(200); defaultpen(linewidth(0.7)+fontsize(10)); real x=sqrt(6), y=sqrt(3), a=0.4; pair D=origin, A=(0,y), B=(x,y), C=(x,0), E=foot(C,B,D), F=foot(A,B,D); real r=degrees(B); pair M1=F+3*dir(r)*dir(90), M2=F+3*dir(r)*dir(-90), N1=E+3*dir(r)*dir(90), N2=E+3*dir(r)*dir(-90); markscalefactor=0.02; draw(B--C--D--A--B--D^^M1--M2^^N1--N2^^rightanglemark(A,F,B,6)^^rightanglemark(N1,E,B,6)); pair W=A+a*dir(135), X=B+a*dir(45), Y=C+a*dir(-45), Z=D+a*dir(-135);
label("A", A, NE); label("B", B, NE); label("C", C, dir(0)); label("D", D, dir(180)); label("

L

", (x/2,0), SW); label("

L^\prime

", C, SW);
label("1", D--F, NW); label("1", F--E, SE); label("1", E--B, SE); clip(W--X--Y--Z--cycle); [/asy]

<span class='latex-bold'>(A)</span>\ 4.1 \qquad <span class='latex-bold'>(B)</span>\ 4.2 \qquad <span class='latex-bold'>(C)</span>\ 4.3 \qquad <span class='latex-bold'>(D)</span>\ 4.4 \qquad <span class='latex-bold'>(E)</span>\ 4.5

16

1

Finding the Tangents of Angles

If A \equal{} 20^{\circ} and B \equal{} 25^{\circ}, then the value of (1 \plus{} \tan A)(1 \plus{} \tan B) is (A) \sqrt3 \qquad (B)\ 2 \qquad (C)\ 1 \plus{} \sqrt2 \qquad (D)\ 2(\tan A \plus{} \tan B)

<span class='latex-bold'>(E)</span>\ \text{ none of these}

14

1

Interior Angles of a Polygon

Exactly three of the interior angles of a convex polygon are obtuse. What is the maximum number of sides of such a polygon?

<span class='latex-bold'>(A)</span>\ 4 \qquad <span class='latex-bold'>(B)</span>\ 5 \qquad <span class='latex-bold'>(C)</span>\ 6 \qquad <span class='latex-bold'>(D)</span>\ 7 \qquad <span class='latex-bold'>(E)</span>\ 8

13

1

Area of a Quadrilateral

Pegs are put in a board

1

unit apart both horizontally and vertically. A reubber band is stretched over

4

pegs as shown in the figure, forming a quadrilateral. Its area in square units is
[asy] int i,j; for(i=0; i<5; i=i+1) { for(j=0; j<4; j=j+1) { dot((i,j)); }} draw((0,1)--(1,3)--(4,1)--(3,0)--cycle, linewidth(0.7)); [/asy]

<span class='latex-bold'>(A)</span>\ 4 \qquad <span class='latex-bold'>(B)</span>\ 4.5 \qquad <span class='latex-bold'>(C)</span>\ 5 \qquad <span class='latex-bold'>(D)</span>\ 5.5 \qquad <span class='latex-bold'>(E)</span>\ 6

10

1

Intersections Between a Circle and a Trig Equation

An arbitrary circle can intersect the graph y \equal{} \sin x in

<span class='latex-bold'>(A)</span> \text{ at most 2 points} \qquad <span class='latex-bold'>(B)</span> \text{ at most 4 points} \qquad

<span class='latex-bold'>(C)</span> \text{ at most 6 points} \qquad <span class='latex-bold'>(D)</span> \text{ at most 8 points} \qquad

<span class='latex-bold'>(E)</span> \text{ more than 16 points}

9

1

Arranging Numbers into Columns

The odd positive integers

1,3,5,7,\cdots,

are arranged into in five columns continuing with the pattern shown on the right. Counting from the left, the column in which

1985

appears in is the
[asy] int i,j; for(i=0; i<4; i=i+1) { label(string(16*i+1), (2*1,-2*i)); label(string(16*i+3), (2*2,-2*i)); label(string(16*i+5), (2*3,-2*i)); label(string(16*i+7), (2*4,-2*i)); } for(i=0; i<3; i=i+1) { for(j=0; j<4; j=j+1) { label(string(16*i+15-2*j), (2*j,-2*i-1)); }} dot((0,-7)^^(0,-9)^^(2*4,-8)^^(2*4,-10)); for(i=-10; i<-6; i=i+1) { for(j=1; j<4; j=j+1) { dot((2*j,i)); }} [/asy]

<span class='latex-bold'>(A)</span> \text{ first} \qquad <span class='latex-bold'>(B)</span> \text{ second} \qquad <span class='latex-bold'>(C)</span> \text{ third} \qquad <span class='latex-bold'>(D)</span> \text{ fourth} \qquad <span class='latex-bold'>(E)</span> \text{ fifth}

8

1

Comparing Solutions of Equations

Let

a

,

a'

,

b

, and

b'

be real numbers with

a

and

a'

nonzero. The solution to ax \plus{} b \equal{} 0 is less than the solution to a'x \plus{} b' \equal{} 0 if and only if

<span class='latex-bold'>(A)</span>\ a'b < ab' \qquad <span class='latex-bold'>(B)</span>\ ab' < a'b \qquad <span class='latex-bold'>(C)</span>\ ab < a'b' \qquad <span class='latex-bold'>(D)</span>\ \frac {b}{a} < \frac {b'}{a'}

<span class='latex-bold'>(E)</span>\ \frac {b'}{a'} < \frac {b}{a}

7

1

Algebraic Notation

In some computer languages (such as APL), when there are no parentheses in an algebraic expression, the operations are grouped from left to right. Thus, a \times b \minus{} c in such languages means the same as a(b\minus{}c) in ordinary algebraic notation. If a \div b \minus{} c \plus{} d is evaluated in such a language, the result in ordinary algebraic notation would be (A)\ \frac{a}{b} \minus{} c \plus{} d \qquad (B)\ \frac{a}{b} \minus{} c \minus{} d \qquad (C)\ \frac{d \plus{} c \minus{} b}{a} \qquad (D)\ \frac{a}{b \minus{} c \plus{} d} \qquad (E)\ \frac{a}{b\minus{}c\minus{}d}

6

1

Choosing a Class Representative

One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is

\frac23

of the probability that a girl is chosen. The ratio of the number of boys to the total number of boys and girls is

<span class='latex-bold'>(A)</span>\ \frac13 \qquad <span class='latex-bold'>(B)</span>\ \frac25 \qquad <span class='latex-bold'>(C)</span>\ \frac12 \qquad <span class='latex-bold'>(D)</span>\ \frac35 \qquad <span class='latex-bold'>(E)</span>\ \frac23

5

1

Removing Numbers from an Addition Problem

Which terms must be removed from the sum \frac12 \plus{} \frac14 \plus{} \frac16 \plus{} \frac18 \plus{} \frac1{10} \plus{} \frac1{12} if the sum of the remaining terms is equal to

1

?

<span class='latex-bold'>(A)</span>\ \frac14\text{ and }\frac18 \qquad <span class='latex-bold'>(B)</span>\ \frac14\text{ and }\frac1{12} \qquad <span class='latex-bold'>(C)</span>\ \frac18\text{ and }\frac1{12} \qquad <span class='latex-bold'>(D)</span>\ \frac16\text{ and }\frac1{10} \qquad <span class='latex-bold'>(E)</span>\ \frac18\text{ and }\frac1{10}

4

1

Bag of Coins

A large bag of coins contains pennies, dimes, and quarters. There are twice as many dimes as pennies and three times as many quarters as dimes. An amount of money which could be in the bag is

<span class='latex-bold'>(A)</span>\ \$306 \qquad <span class='latex-bold'>(B)</span>\ \$333 \qquad <span class='latex-bold'>(C)</span>\ \$342 \qquad <span class='latex-bold'>(D)</span>\ \$348 \qquad <span class='latex-bold'>(E)</span>\ \$360

3

1

Finding the Length between Two Circular Arcs

In right

\triangle ABC

with legs

5

and

12

, arcs of circles are drawn, one with center

A

and radius

12

, the other with center

B

and radius

5

. They intersect the hypotenuse at

M

and

N

. Then,

MN

has length:
[asy]size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(12,7), C=(12,0), M=12*dir(A--B), N=B+B.y*dir(B--A); real r=degrees(B); draw(A--B--C--cycle^^Arc(A,12,0,r)^^Arc(B,B.y,180+r,270)); pair point=incenter(A,B,C); label("

A

", A, dir(point--A)); label("

B

", B, dir(point--B)); label("

C

", C, dir(point--C)); label("

M

", M, dir(point--M)); label("

N

", N, dir(point--N));
label("

12

", (6,0), S); label("

5

", (12,3.5), E);[/asy]

<span class='latex-bold'>(A)</span>\ 2 \qquad <span class='latex-bold'>(B)</span>\ \frac {13}{5} \qquad <span class='latex-bold'>(C)</span>\ 3 \qquad <span class='latex-bold'>(D)</span>\ 4 \qquad <span class='latex-bold'>(E)</span>\ \frac {24}{5}

2

1

Shape of a Monster

In an arcade game, the "monster" is the shaded sector of a circle of radius

1

cm, as shown in the figure. The missing piece (the mouth) has central angle

60^{\circ}

. What is the perimeter of the monster in cm?
[asy]size(100); defaultpen(linewidth(0.7)); filldraw(Arc(origin,1,30,330)--dir(330)--origin--dir(30)--cycle, yellow, black); label("1", (sqrt(3)/4, 1/4), NW); label("

60^\circ

", (1,0)); [/asy]
(A)\ \pi \plus{} 2 \qquad (B)\ 2\pi \qquad (C)\ \frac53 \pi \qquad (D)\ \frac56 \pi \plus{} 2 \qquad (E)\ \frac53 \pi \plus{} 2

1

Algebraic Equation

If 2x \plus{} 1 \equal{} 8, then 4x \plus{} 1 \equal{}

<span class='latex-bold'>(A)</span>\ 15 \qquad <span class='latex-bold'>(B)</span>\ 16 \qquad <span class='latex-bold'>(C)</span>\ 17 \qquad <span class='latex-bold'>(D)</span>\ 18 \qquad <span class='latex-bold'>(E)</span>\ 19

15

1

a and b

If

a

and

b

are positive numbers such that a^b \equal{} b^a and b \equal{} 9a, then the value of

a

is:

<span class='latex-bold'>(A)</span>\ 9\qquad <span class='latex-bold'>(B)</span>\ \frac {1}{9}\qquad <span class='latex-bold'>(C)</span>\ \sqrt [9] {9}\qquad <span class='latex-bold'>(D)</span>\ \sqrt [3] {9}\qquad <span class='latex-bold'>(E)</span>\ \sqrt [4] {3}

12

1

Three Primes

Let's write p,q, and r as three distinct prime numbers, where 1 is not a prime. Which of the following is the smallest positive perfect cube leaving n \equal{} pq^2r^4 as a divisor?

<span class='latex-bold'>(A)</span>\ p^8q^8r^8\qquad <span class='latex-bold'>(B)</span>\ (pq^2r^2)^3\qquad <span class='latex-bold'>(C)</span>\ (p^2q^2r^2)^3\qquad <span class='latex-bold'>(D)</span>\ (pqr^2)^3\qquad <span class='latex-bold'>(E)</span>\ 4p^3q^3r^3

11

1

Arrangements

How many distinguishable rearrangements of the letters in CONTEST have both the vowels first? (For instance, OETCNST is a one such arrangements but OTETSNC is not.)

<span class='latex-bold'>(A)</span>\ 60\qquad <span class='latex-bold'>(B)</span>\ 120\qquad <span class='latex-bold'>(C)</span>\ 240\qquad <span class='latex-bold'>(D)</span>\ 720\qquad <span class='latex-bold'>(E)</span>\ 2520

1985 AMC 12/AHSME

Subcontests

Number of Real Solutions

Sum of the Digits of a Large Product

Finding the Side Length of a Triangle

Finding the Smallest Integer in a Sequence

Finding the Smallest Reducible Fraction

Dimensions of a Rectangular Solid

Logarithms

Finding the False Equation

Finding the Length of a Partial Chord

Finding Integral Solutions

Cutting a Cube

Intersection Between Two Functions

Bags of Marbles

Finding the Area of a Rectangle

Finding the Tangents of Angles

Interior Angles of a Polygon

Area of a Quadrilateral

Intersections Between a Circle and a Trig Equation

Arranging Numbers into Columns

Comparing Solutions of Equations

Algebraic Notation

Choosing a Class Representative

Removing Numbers from an Addition Problem

Bag of Coins

Finding the Length between Two Circular Arcs

Shape of a Monster

Algebraic Equation

a and b

Three Primes

Arrangements