1980 AMC 12/AHSME

Part of AMC 12/AHSME

Subcontests

(30)

1

Squarish or Not?

A six digit number (base 10) is squarish if it satisfies the following conditions: (i) none of its digits are zero; (ii) it is a perfect square; and (iii) the first of two digits, the middle two digits and the last two digits of the number are all perfect squares when considered as two digit numbers. How many squarish numbers are there?

\text{(A)} \ 0 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ 8 \qquad \text{(E)} \ 9

1

System of Equations

How many ordered triples

(x,y,z)

of integers satisfy the system of equations below?

\begin{array}{l} x^2-3xy+2yz-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \end{array}

\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{a finite number greater than 2} \qquad \text{(E)} \ \text{infinately many}

1

Polynomial Divisiblity

x^{2n}+1+(x+1)^{2n}

is not divisible by

x^2+x+1

n

\text{(A)} \ 17 \qquad \text{(B)} \ 20 \qquad \text{(C)} \ 21 \qquad \text{(D)} \ 64 \qquad \text{(E)} \ 65

1

Cubic Roots

\sqrt[3] {5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}

\text{(A)} \ \frac 32 \qquad \text{(B)} \ \frac{\sqrt[3]{65}}{4} \qquad \text{(C)} \ \frac{1+\sqrt[6]{13}}{2} \qquad \text{(D)} \ \sqrt[3]{2} \qquad \text{(E)} \ \text{none of these}

1

Tetrahedron

Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tetrahedron, each of whose edges have length

s

, is circumscribed around the balls. Then

s

\text{(A)} \ 4\sqrt 2 \qquad \text{(B)} \ 4\sqrt 3 \qquad \text{(C)} \ 2\sqrt 6 \qquad \text{(D)} \ 1+2\sqrt 6 \qquad \text{(E)} \ 2+2\sqrt 6

1

Odd Integer Sequence

In the non-decreasing sequence of odd integers

\{a_1,a_2,a_3,\ldots \}=\{1,3,3,3,5,5,5,5,5,\ldots \}

each odd positive integer

k

k

times. It is a fact that there are integers

b

c

d

such that for all positive integers

n

a_n=b\lfloor \sqrt{n+c} \rfloor +d,

\lfloor x \rfloor

denotes the largest integer not exceeding

x

b+c+d

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

1

Intriguing Geometry/Trig Problem

Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths

\sin x

\cos x

x

is a real number such that

0<x<\frac{\pi}2

. The length of the hypotenuse is

\text{(A)} \ \frac 43 \qquad \text{(B)} \ \frac 32 \qquad \text{(C)} \ \frac{3\sqrt{5}}{5} \qquad \text{(D)} \ \frac{2\sqrt{5}}{3} \qquad \text{(E)} \ \text{not uniquely determined}

1

Polynomial Remainders

For some real number

r

, the polynomial

8x^3-4x^2-42x+45

is divisible by

(x-r)^2

. Which of the following numbers is closest to

r

\text{(A)} \ 1.22 \qquad \text{(B)} \ 1.32 \qquad \text{(C)} \ 1.42 \qquad \text{(D)} \ 1.52 \qquad \text{(E)} \ 1.62

1

Maximum Value for Linear Equations

For each real number

x

f(x)

be the minimum of the numbers

4x+1

x+2

-2x+4

. Then the maximum value of

f(x)

\text{(A)} \ \frac 13 \qquad \text{(B)} \ \frac 12 \qquad \text{(C)} \ \frac 23 \qquad \text{(D)} \ \frac 52 \qquad \text{(E)} \ \frac 83

1

Area Ratios

ABC

\measuredangle CBA=72^\circ

E

is the midpoint of side

AC

D

is a point on side

BC

2BD=DC

AD

BE

F

. The ratio of the area of triangle

BDF

to the area of quadrilateral

FDCE

is
[asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, C=(15,3), D=(5,1), A=7*dir(72)*dir(B--C), E=midpoint(A--C), F=intersectionpoint(A--D, B--E); draw(E--B--A--C--B^^A--D); label("

A

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

B

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

C

", C, dir(0)); label("

D

", D, SE); label("

E

", E, N); label("

F

", F, dir(80));[/asy]

\text{(A)} \ \frac 15 \qquad \text{(B)} \ \frac 14 \qquad \text{(C)} \ \frac 13 \qquad \text{(D)} \ \frac 25 \qquad \text{(E)} \ \text{none of these}

1

Easy Probability

A box contains 2 pennies, 4 nickels, and 6 dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least 50 cents?

\text{(A)} \ \frac{37}{924} \qquad \text{(B)} \ \frac{91}{924} \qquad \text{(C)} \ \frac{127}{924} \qquad \text{(D)} \ \frac{132}{924} \qquad \text{(E)} \ \text{none of these}

1

Chords in a Circle

C_1

C_2

C_3

be three parallel chords of a circle on the same side of the center. The distance between

C_1

C_2

is the same as the distance between

C_2

C_3

. The lengths of the chords are 20, 16, and 8. The radius of the circle is

\text{(A)} \ 12 \qquad \text{(B)} \ 4\sqrt{7} \qquad \text{(C)} \ \frac{5\sqrt{65}}{3} \qquad \text{(D)} \ \frac{5\sqrt{22}}{2} \qquad \text{(E)} \ \text{not uniquely determined}

1

Trig Log

b>1

\sin x>0

\cos x>0

\log_b \sin x = a

\log_b \cos x

\text{(A)} \ 2\log_b(1-b^{a/2}) ~~\text{(B)} \ \sqrt{1-a^2} ~~\text{(C)} \ b^{a^2} ~~\text{(D)} \ \frac 12 \log_b(1-b^{2a}) ~~\text{(E)} \ \text{none of these}

1

Working with i

i^2=-1

, for how many integers

n

(n+i)^4

\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ 4

1

Surface Area Ratios

Four of the eight vertices of a cube are the vertices of a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron.

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

1

Item Price

A store prices an item in dollars and cents so that when 4% sales tax is added, no rounding is necessary because the result is exactly

n

n

is a positive integer. The smallest value of

n

\text{(A)} \ 1 \qquad \text{(B)} \ 13 \qquad \text{(C)} \ 25 \qquad \text{(D)} \ 26 \qquad \text{(E)} \ 100

1

Function of x

If the function

f

f(x)=\frac{cx}{2x+3} , ~~~x\neq -\frac 32 ,

x=f(f(x))

for all real numbers

x

-\frac 32

c

\text{(A)} \ -3 \qquad \text{(B)} \ - \frac{3}{2} \qquad \text{(C)} \ \frac{3}{2} \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{not uniquely determined}

1

A "Cartesian Bug"

A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to

(1,0)

. Then it makes a

90^\circ

counterclockwise and travels

\frac 12

\left(1, \frac 12 \right)

. If it continues in this fashion, each time making a

90^\circ

degree turn counterclockwise and traveling half as far as the previous move, to which of the following points will it come closest?

\text{(A)} \ \left(\frac 23, \frac 23 \right) \qquad \text{(B)} \ \left( \frac 45, \frac 25 \right) \qquad \text{(C)} \ \left( \frac 23, \frac 45 \right) \qquad \text{(D)} \ \left(\frac 23, \frac 13 \right) \qquad \text{(E)} \ \left(\frac 25, \frac 45 \right)

1

Lines are Fun!

The equations of

L_1

L_2

y=mx

y=nx

, respectively. Suppose

L_1

makes twice as large of an angle with the horizontal (measured counterclockwise from the positive x-axis ) as does

L_2

L_1

has 4 times the slope of

L_2

L_1

is not horizontal, then

mn

\text{(A)} \ \frac{\sqrt{2}}{2} \qquad \text{(B)} \ -\frac{\sqrt{2}}{2} \qquad \text{(C)} \ 2 \qquad \text{(D)} \ -2 \qquad \text{(E)} \ \text{not uniquely determined}

1

Meshed Gears

The number of teeth in three meshed gears

A

B

C

x

y

z

, respectively. (The teeth on all gears are the same size and regularly spaced.) The angular speeds, in revolutions per minutes of

A

B

C

are in the proportion

\text{(A)} \ x: y: z ~~\text{(B)} \ z: y: x ~~ \text{(C)} \ y: z: x~~ \text{(D)} \ yz: xz: xy ~~ \text{(E)} \ xz: yx: zy

1

Man Gone Walking

x

miles due west, turns

150^\circ

to his left and walks 3 miles in the new direction. If he finishes a a point

\sqrt{3}

from his starting point, then

x

\text{(A)} \ \sqrt 3 \qquad \text{(B)} \ 2\sqrt{5} \qquad \text{(C)} \ \frac 32 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{not uniquely determined}

1

Arithmetic Progressions

If the sum of the first 10 terms and the sum of the first 100 terms of a given arithmetic progression are 100 and 10, respectively, then the sum of first 110 terms is:

\text{(A)} \ 90 \qquad \text{(B)} \ -90 \qquad \text{(C)} \ 110 \qquad \text{(D)} \ -110 \qquad \text{(E)} \ -100

1

Number of Solutions

(a,b)

of non-zero real numbers satisfy the equation

\frac{1}{a} + \frac{1}{b} = \frac{1}{a+b}?

\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{one pair for each} ~b \neq 0

\text{(E)} \ \text{two pairs for each} ~b \neq 0

1

Quadrilateral Area

AB,BC,CD

DA

of convex polygon

ABCD

have lengths 3,4,12, and 13, respectively, and

\measuredangle CBA

is a right angle. The area of the quadrilateral is
[asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); real r=degrees((12,5)), s=degrees((3,4)); pair D=origin, A=(13,0), C=D+12*dir(r), B=A+3*dir(180-(90-r+s)); draw(A--B--C--D--cycle); markscalefactor=0.05; draw(rightanglemark(A,B,C)); pair point=incenter(A,C,D); 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("

3

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

4

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

12

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

13

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

\text{(A)} \ 32 \qquad \text{(B)} \ 36 \qquad \text{(C)} \ 39 \qquad \text{(D)} \ 42 \qquad \text{(E)} \ 48

1

Inequality

A positive number

x

satisfies the inequality

\sqrt{x} < 2x

\text{(A)} \ x > \frac{1}{4} \qquad \text{(B)} \ x > 2 \qquad \text{(C)} x > 4 \qquad \text{(D)} \ x < \frac{1}{4}\qquad \text{(E)} x < 4

1

Length Chasing

AB

CD

are perpendicular diameters of circle

Q

P

\overline{AQ}

\measuredangle QPC = 60^\circ

, then the length of

PQ

divided by the length of

AQ

is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=(-1,0), B=(1,0), C=(0,1), D=(0,-1), Q=origin, P=(-0.5,0); draw(P--C--D^^A--B^^Circle(Q,1)); label("

A

", A, W); label("

B

", B, E); label("

C

", C, N); label("

D

", D, S); label("

P

", P, S); label("

Q

", Q, SE); label("

60^\circ

", P+0.0.5*dir(30), dir(30));[/asy]

<span class='latex-bold'>(A)</span> \ \frac{\sqrt{3}}{2} \qquad <span class='latex-bold'>(B)</span> \ \frac{\sqrt{3}}{3} \qquad <span class='latex-bold'>(C)</span> \ \frac{\sqrt{2}}{2} \qquad <span class='latex-bold'>(D)</span> \ \frac12 \qquad <span class='latex-bold'>(E)</span> \ \frac23

1

Angle Measure

In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares. The measure of

\angle GDA

\text{(A)} \ 90^\circ \qquad \text{(B)} \ 105^\circ \qquad \text{(C)} \ 120^\circ \qquad \text{(D)} \ 135^\circ \qquad \text{(E)} \ 150^\circ

[asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair D=origin, C=D+dir(240), E=D+dir(300), F=E+dir(30), G=D+dir(30), A=D+dir(150), B=C+dir(150); draw(E--D--G--F--E--C--D--A--B--C); pair point=(0,0.5); label("

A

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

B

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

C

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

D

", D, dir(-15)); label("

E

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

F

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

G

", G, dir(point--G));[/asy]

1

Ratios

If the ratio of

2x-y

x+y

\frac{2}{3}

, what is the ratio of

x

y

\text{(A)} \ \frac{1}{5} \qquad \text{(B)} \ \frac{4}{5} \qquad \text{(C)} \ 1 \qquad \text{(D)} \ \frac{6}{5} \qquad \text{(E)} \ \frac{5}{4}

1

Polynomial Degree

(x^2+1)^4 (x^3+1)^3

as a polynomial in

x

\text{(A)} \ 5 \qquad \text{(B)} \ 7 \qquad \text{(C)} \ 12 \qquad \text{(D)} \ 17 \qquad \text{(E)} \ 72

1

Find the Number

The largest whole number such that seven times the number is less than 100 is

\text{(A)} \ 12 \qquad \text{(B)} \ 13 \qquad \text{(C)} \ 14 \qquad \text{(D)} \ 15 \qquad \text{(E)} \ 16