1966 AMC 12/AHSME

Part of AMC 12/AHSME

Subcontests

(40)

1

Which Relation is Deduced Correctly?

[asy]draw(Circle((0,0), 1)); dot((0,0)); label("

O

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

A

", (-1,0), W); label("

B

", (1,0), E); label("

a

", (-0.5,0), S); draw((-1,-1.25)--(-1,1.25)); draw((1,-1.25)--(1,1.25)); draw((-1,0)--(1,0)); draw((-1,0)--(-1,0)+2.3*dir(30)); label("

C

", (-1,0)+2.3*dir(30), E); label("

D

", (-1,0)+1.8*dir(30), N); dot((-1,0)+.4*dir(30)); label("

E

", (-1,0)+.4*dir(30), N); [/asy] In this figure

AB

is a diameter of a circle, centered at

O

a

AD

is drawn and extended to meet the tangent to the circle at

B

C

E

AC

AE=DC

. Denoting the distances of

E

from the tangent through

A

and from the diameter

AB

x

y

, respectively, we can deduce the relation:

\text{(A)}\ y^2=\dfrac{x^3}{2a-x} \qquad \text{(B)}\ y^2=\frac{x^3}{2a+x}\qquad \text{(C)}\ y^4=\frac{x^2}{2-x}\qquad\\ \text{(D)}\ x^2=\dfrac{y^2}{2a-x}\qquad \text{(E)}\ x^2=\frac{y^2}{2a+x}

1

Sum of Two Bases

R_1

the expanded fraction

F_1

0.373737...

, and the expanded fraction

F_2

0.737373...

R_2

F_1

, when expanded, becomes

0.252525...

, while fraction

F_2

0.525252...

R_1

R_2

, each written in base ten is:

\text{(A)}\ 24 \qquad \text{(B)}\ 22\qquad \text{(C)}\ 21\qquad \text{(D)}\ 20\qquad \text{(E)}\ 19

1

Area of the Triangle

ABC

AM

CN

BC

AB

, respectively, intersect in point

O

P

is the midpoint of side

AC

MP

CN

Q

. If the area of triangle

OMQ

n

, then the area of triangle

ABC

\text{(A)}\ 16n\qquad \text{(B)}\ 18n\qquad \text{(C)}\ 21n\qquad \text{(D)}\ 24n\qquad \text{(E)}\ 27n

1

How Long does it Take?

Three men, Alpha, Beta, and Gamma, working together, do a job in

6

hours less time than Alpha alone, in

1

hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let

h

be the number of hours needed by Alpha and Beta, working together to do the job. Then

h

\text{(A)}\ \dfrac{5}{2}\qquad \text{(B)}\ \frac{3}{2}\qquad \text{(C)}\ \dfrac{4}{3}\qquad \text{(D)}\ \dfrac{5}{4}\qquad \text{(E)}\ \dfrac{3}{4}

1

Identity with Series

(1+x+x^2)^n=a_0+a_1x+a_2x^2+...+a_{2n}x^{2n}

be an identity in

x

s=a_0+a_2+a_4+...+a_{2n}

s

\text{(A)}\ 2^n\qquad \text{(B)}\ 2^n+1\qquad \text{(C)}\ \dfrac{3^n-1}{2}\qquad \text{(D)}\ \dfrac{3^n}{2}\qquad \text{(E)}\ \dfrac{3^n+1}{2}

1

Which is True?

O

be an interior point of triangle

ABC

s_1=OA+OB+OC

s_2=AB+AC+CA

\text{(A)}\ \text{for every triangle }s_2>2s_1,s_1\le s_2\qquad\\ \text{(B)}\ \text{for every triangle } s_2\ge2s_1,s_1<s_2\qquad\\ \text{(C)}\ \text{for every triangle } s_1>\tfrac{1}{2}s_2,s_1<s_2\qquad\\ \text{(D)}\ \text{for every triangle }s_2\ge2s_1,s_1\le s_2\qquad\\ \text{(E)}\ \text{neither (A) nor (B) nor (C) nor (D) applies to every triangle}

1

What's the Speed?

r

be the speed in miles per hour at which a wheel,

11

feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by

\tfrac{1}{4}

of a second, the speed

r

is increased by

5

miles per hour. The

r

\text{(A)}\ 9\qquad \text{(B)}\ 10\qquad \text{(C)}\ 10\tfrac{1}{2}\qquad \text{(D)}\ 11\qquad \text{(E)}\ 12

1

Number of Values that Satisfy the Equation

ab\ne0

|a|\ne|b|

the number of distinct values of

x

satisfying the equation

\dfrac{x-a}{b}+\dfrac{x-b}{a}=\dfrac{b}{x-a}+\dfrac{a}{x-b}

\text{(A)}\ \text{zero}\qquad \text{(B)}\ \text{one}\qquad \text{(C)}\ \text{two}\qquad \text{(D)}\ \text{three}\qquad \text{(E)}\ \text{four}

1

Ratios

M

be the midpoint of side

AB

of the triangle

ABC

P

AB

A

M

MD

be drawn parallel to

PC

and intersecting

BC

D

. If the ratio of the area of the triangle

BPD

to that of triangle

ABC

r

\text{(A)}\ \tfrac{1}{2}<r<1\text{ depending upon the position of }P \qquad\\ \text{(B)}\ r=\tfrac{1}{2}\text{ independent of the position of }P\qquad\\ \text{(C)}\ \tfrac{1}{2}\le r<1\text{ depending upon the position of }P \qquad\\ \text{(D)}\ \tfrac{1}{3}<r<\tfrac{2}{3}\text{ depending upon the position of }P \qquad\\ \text{(E)}\ r=\tfrac{1}{3} \text{ independent of the position of }P

1

Which One is True?

ABC

is inscribed in a circle with center

O'

. A circle with center

O

is inscribed in triangle

ABC

AO

is drawn, and extended to intersect the larger circle in

D

. Then, we must have:

\text{(A)}\ CD=BD=O'D \qquad \text{(B)}\ AO=CO=OD \qquad \text{(C)}\ CD=CO=BD \qquad\\ \text{(D)}\ CD=OD=BD \qquad \text{(E)}\ O'B=O'C=OD

[asy] size(200); defaultpen(linewidth(0.8)+fontsize(12pt)); pair A=origin,B=(15,0),C=(5,9),O=incenter(A,B,C),Op=circumcenter(A,B,C); path incirc = incircle(A,B,C),circumcirc = circumcircle(A,B,C),line=A--3*O; pair D[]=intersectionpoints(circumcirc,line); draw(A--B--C--A--D[0]^^incirc^^circumcirc); dot(O^^Op,linewidth(4)); label("

A

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

B

",B,dir(355)); label("

C

",C,dir(95)); label("

D

",D[0],dir(O--D[0])); label("

O

",O,NW); label("

O'

1

Value of a+c

If three of the roots of

x^4+ax^2+bx+c=0

1

2

3

, then the value of

a+c

\text{(A)}\ 35 \qquad \text{(B)}\ 24\qquad \text{(C)}\ -12\qquad \text{(D)}\ -61 \qquad \text{(E)}\ -63

1

Integers indivisible by both 5 and 7

The number of postive integers less than

1000

divisible by neither

5

7

\text{(A)}\ 688 \qquad \text{(B)}\ 686\qquad \text{(C)}\ 684 \qquad \text{(D)}\ 658\qquad \text{(E)}\ 630

1

Find OP

O,A,B,C,D

are taken in order on a straight line with distances

OA=a

OB=b

OC=c

OD=d

P

is a point on the line between

B

C

AP:PD=BP:PC

OP

\text{(A)}\ \dfrac{b^2-bc}{a-b+c-d} \qquad \text{(B)}\ \dfrac{ac-b}{a-b+c-d} \qquad \text{(C)}\ -\dfrac{bd+c}{a-b+c-d}\qquad\\ \text{(D)}\ \dfrac{bc+ad}{a+b+c+d}\qquad \text{(E)}\ \dfrac{ac-bd}{a+b+c+d} \qquad

1

Classic Problem

At his usual rate a man rows

15

miles downstream in five hours less time than it takes him to return. If he doubles his usual rate, the time downstream is only one hour less than the time upstream. In miles per hour, the rate of the stream's current is:

\text{(A)} \ 2 \qquad \text{(B)} \ \frac52 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ \frac72 \qquad \text{(E)} \ 4

1

Two Linear Equations

m

be a positive integer and let the lines

13x+11y=700

y=mx-1

intersect in a point whose coordinates are integers. Then

m

\text{(A)} \ 4 \qquad \text{(B)} \ 5 \qquad \text{(C)} \ 6 \qquad \text{(D)} \ 7 \qquad \text{(E)} \ \text{one of the integers}~ 4,5,6,7~\text{and one other positive integer}

1

Funny Function

F(n+1)=\frac{2F(n)+1}{2}

n=1,2,\ldots

F(1)=2

F(101)

\text{(A)} \ 49 \qquad \text{(B)} \ 50 \qquad \text{(C)} \ 51 \qquad \text{(D)} \ 52 \qquad \text{(E)} \ 53

1

Logarithm

\log_MN=\log_NM

M\ne N

MN>0

M\ne 1

N\ne 1

MN

\text{(A)} \ \frac12 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 10 \qquad \text{(E)} \ \text{a number greater than 2 and less than 10}

1

what x's make y real?

x

4y^2+4xy+x+6=0

, then the complete set of values of

x

y

\text{(A)} \ x\le -2~\text{or}~x\ge3 \qquad \text{(B)} \ x\le 2~\text{or}~x\ge3 \qquad \text{(C)} \ x\le -3 ~\text{or}~x\ge 2

\text{(D)} \ -3\le x \le 2\qquad \text{(E)} \ \-2\le x \le 3

1

Which of the following are true?

Consider the statements:

\text{(I)}~~\sqrt{a^2+b^2}=0

\text{(II)}~~\sqrt{a^2+b^2}=ab

\text{(III)}~~\sqrt{a^2+b^2}=a+b

\text{(IV)}~~\sqrt{a^2+b^2}=a-b

,
where we allow

a

b

to be real or complex numbers. Those statements for which there exist solutions other than

a=0

b=0

\text{(A)} \ \text{(I)},\text{(II)},\text{(III)},\text{(IV)} \qquad \text{(B)} \ \text{(II)},\text{(III)},\text{(IV)} \qquad \text{(C)} \ \text{(I)},\text{(III)},\text{(IV)} \qquad \text{(D)} \ \text{(III)},\text{(IV)} \qquad \text{(E)} \ \text{(I)}

1

N-Pointed Star

An "n-pointed star" is formed as follows: the sides of a convex polygon are numbered consecutively

1,2,\cdots,k,\cdots,n

n\geq 5

n

k

k

k+2

are non-parallel, sides

n+1

n+2

being respectively identical with sides

1

2

n

pairs of sides numbered

k

k+2

until they meet. (A figure is shown for the case

n=5

) http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=704&sid=8da93909c5939e037aa99c429b2d157a Let

S

be the degree-sum of the interior angles at the

n

points of the star; then

S

\text{(A)} \ 180 \qquad \text{(B)} \ 360 \qquad \text{(C)} \ 180(n+2) \qquad \text{(D)} \ 180(n-2) \qquad \text{(E)} \ 180(n-4)

1

Negation of a Proposition

The negation of the proposition "For all pairs of real numbers

a

b

a=0

ab=0

" is: There are real numbers

a,b

\text{(A)} \ a\ne0,ab\ne 0 ~~\text{(B)} \ a\ne 0, ab=0 ~~ \text{(C)} \ a=0,ab\ne 0

\text{(D)} \ ab\ne0,a\ne0 ~~\text{(E)} \ ab=0,a\ne0

1

Arithmetic Series

s_1

be the sum of the first

n

terms of the arithmetic sequence

8,12,\cdots

s_2

be the sum of the first

n

terms of the arithmetic sequence

17,19\cdots

n\ne 0

s_1=s_2

\text{(A)} \ \text{no value of n} \qquad \text{(B)} \ \text{one value of n} \qquad \text{(C)} \ \text{two values of n}

\text{(D)} \ \text{four values of n} \qquad \text{(E)} \ \text{more than four values of n}

1

Arithmetic Sequence

In a given arithmetic sequence the first term is

2

, the last term is

29

, and the sum of all the terms is

155

. The common difference is:

\text{(A)} \ 3 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ \frac{27}{19} \qquad \text{(D)} \ \frac{13}9 \qquad \text{(E)} \ \frac{23}{38}

1

Examining two Graphs

The number of distinct points common to the curves

x^2+4y^2=1

4x^2+y^2=5

\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ 4

1

More Exponents

\frac{4^x}{2^{x+y}}=8

\frac{9^{x+y}}{3^{5y}}=243

x

y

are real numbers, then

xy

\text{(A)} \ \frac{12}{5} \qquad \text{(B)} \ 4 \qquad \text{(C)} \ 6 \qquad \text{(D)} \ 12 \qquad \text{(E)} \ -4

1

Basic Inequalities

x-y>x

x+y<y

\text{(A)} \ y<x \qquad \text{(B)} \ x<y \qquad \text{(C)} \ x<y<0 \qquad \text{(D)} \ x<0,y<0

\text{(E)} \ x<0,y>0

1

Areas

The length of rectangle

ABCD

5

inches and its width is

3

inches. Diagonal

AC

is dibided into three equal segments by points

E

F

. The area of triangle

BEF

, expressed in square inches, is:

\text{(A)} \ \frac 32 \qquad \text{(B)} \ \frac 53 \qquad \text{(C)} \ \frac 52 \qquad \text{(D)} \ \frac13\sqrt{34} \qquad \text{(E)} \ \frac13\sqrt{68}

1

Number of Points

The number of points with positive rational coordinates selected from the set of points in the xy-plane such that

x+y\leq 5

\text{(A)} \ 9 \qquad \text{(B)} \ 10 \qquad \text{(C)} \ 14 \qquad \text{(D)} \ 15 \qquad \text{(E)} \ \text{infinite}

1

Exponents

The number of real values of

x

that satisfy the equation

(2^{6x+3})(4^{3x+6})=8^{4x+5}

\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{greater than 3}

1

Angle Bisector

The sides of triangle

BAC

are in the ratio

2: 3: 4

BD

is the angle-bisector drawn to the shortest side

AC

, dividing it into segments

AD

CD

. If the length of

AC

10

, then the length of the longer segment of

AC

\text{(A)} \ 3\frac12 \qquad \text{(B)} \ 5 \qquad \text{(C)} \ 5\frac57 \qquad \text{(D)} \ 6 \qquad \text{(E)} \ 7\frac12

1

Know your Cubes?

If the sum of two numbers is 1 and their product is 1, then the sum of their cubes is:

\text{(A)} \ 2 \qquad \text{(B)} \ -2-\frac{3i\sqrt{3}}{4} \qquad \text{(C)} \ 0 \qquad \text{(D)} \ -\frac{3i\sqrt{3}}{4} \qquad \text{(E)} \ -2

1

Log Powers

x=(\log_82)^{(\log_28)}

\log_3x

\text{(A)} \ -3 \qquad \text{(B)} \ -\frac13 \qquad \text{(C)} \ \frac13 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ 9

1

Circle Intersection

The length of the common chord of two intersecting circles is

16

feet. If the radii are

10

17

feet, a possible value for the distance between the centers of teh circles, expressed in feet, is:

\text{(A)} \ 27 \qquad \text{(B)} \ 21 \qquad \text{(C)} \ \sqrt{389} \qquad \text{(D)} \ 15 \qquad \text{(E)} \ \text{undetermined}

1

Partial Fraction Decomposition

\frac{35x-29}{x^2-3x+2}=\frac{N_1}{x-1}+\frac{N_2}{x-2}

be an identity in

x

. The numerical value of

N_1N_2

\text{(A)} \ -246 \qquad \text{(B)} \ -210 \qquad \text{(C)} \ -29 \qquad \text{(D)} \ 210 \qquad \text{(E)} \ 246

1

Length Chasing

AB

is the diameter of a circle centered at

O

C

is a point on the circle such that angle

BOC

60^\circ

. If the diameter of the circle is

5

inches, the length of chord

AC

, expressed in inches, is:

\text{(A)} \ 3 \qquad \text{(B)} \ \frac{5\sqrt{2}}{2} \qquad \text{(C)} \frac{5\sqrt3}{2} \ \qquad \text{(D)} \ 3\sqrt3 \qquad \text{(E)} \ \text{none of these}

1

Number of Solutions

The number of values of

x

satisfying the equation

\frac{2x^2-10x}{x^2-5x}=x-3

\text{(A)} \ \text{zero} \qquad \text{(B)} \ \text{one} \qquad \text{(C)} \ \text{two} \qquad \text{(D)} \ \text{three} \qquad \text{(E)} \ \text{an integer greater than 3}

1

Circumscribed Figures

Circle I is circumscribed about a given square and circle II is inscribed in the given square. If

r

is the ratio of the area of circle

I

to that of circle

II

r

\text{(A)} \ \sqrt 2 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ \sqrt 3 \qquad \text{(D)} \ 2\sqrt 2 \qquad \text{(E)} \ 2\sqrt 3

1

AM-GM

If the arithmetic mean of two numbers is

6

and thier geometric mean is

10

, then an equation with the given two numbers as roots is:

\text{(A)} \ x^2+12x+100=0 ~~ \text{(B)} \ x^2+6x+100=0 ~~ \text{(C)} \ x^2-12x-10=0

\text{(D)} \ x^2-12x+100=0 \qquad \text{(E)} \ x^2-6x+100=0

1

Area Ratios

When the base of a triangle is increased

10\%

and the altitude to this base is decreased

10\%

, the change in area is

\text{(A)} \ 1\%~ \text{increase} \qquad \text{(B)} \ \frac12 \%~ \text{increase} \qquad \text{(C)} \ 0\% \qquad \text{(D)} \ \frac12 \% ~\text{decrease} \qquad \text{(E)} \ 1\% ~\text{decrease}

1

Ratios

Given that the ratio of

3x-4

y+15

is constant, and

y=3

x=2

y=12

x

\text{(A)} \ \frac 18 \qquad \text{(B)} \ \frac 73 \qquad \text{(C)} \ \frac78 \qquad \text{(D)} \ \frac72 \qquad \text{(E)} \ 8