MathDB

1

1990 AHSME #10: Cubes from a point

11\times 11\times 11

wooden cube is formed by gluing together

11^3

unit cubes. What is the greatest number of unit cubes that can be seen from a single point?

<span class='latex-bold'>(A) </span>328\qquad <span class='latex-bold'>(B) </span>329\qquad <span class='latex-bold'>(C) </span>330\qquad <span class='latex-bold'>(D) </span>331\qquad <span class='latex-bold'>(E) </span>332\qquad

9

1

1990 AHSME #9: Edges of a cube

Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest possible number of black edges is

<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>6\qquad

8

1

1990 AHSME #8: Number of Real solutions

The number of real solutions of the equation

|x-2|+|x-3|=1

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>\text{more than }3\qquad

7

1

1990 AHSME #7: Triangle with integral sides

A triangle with integral sides has perimeter

8

. The area of the triangle is

<span class='latex-bold'>(A) </span>2\sqrt{2}\qquad <span class='latex-bold'>(B) </span>\dfrac{16}{9}\sqrt{3}\qquad <span class='latex-bold'>(C) </span>2\sqrt{3}\qquad <span class='latex-bold'>(D) </span>4\qquad <span class='latex-bold'>(E) </span>4\sqrt{2}

6

1

1990 AHSME #6: Points in a Plane

Points

A

and

B

are

5

units apart. How many lines in a given plane containing

A

and

B

are

2

units from

A

and

3

units from

B

?

<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>\text{more than }3

5

1

1990 AHSME #5: Largest of Numbers

Which of these numbers is the largest?

<span class='latex-bold'>(A)</span> \sqrt{\sqrt[3]{5\cdot 6}}\qquad <span class='latex-bold'>(B)</span> \sqrt{6\sqrt[3]{5}}\qquad <span class='latex-bold'>(C)</span> \sqrt{5\sqrt[3]{6}}\qquad <span class='latex-bold'>(D)</span> \sqrt[3]{5\sqrt{6}}\qquad <span class='latex-bold'>(E)</span> \sqrt[3]{6\sqrt{5}}

4

1

1990 AHSME #4: Parallelogram

Let

ABCD

be a parallelogram with

\angle ABC=120^\circ

,

AB=16

and

BC=10

. Extend

\overline{CD}

through

D

to

E

so that

DE=4

. If

\overline{BE}

intersects

\overline{AD}

at

F

, then

FD

is closest to

<span class='latex-bold'>(A) </span>1\qquad <span class='latex-bold'>(B) </span>2\qquad <span class='latex-bold'>(C) </span>3\qquad <span class='latex-bold'>(D) </span>4\qquad <span class='latex-bold'>(E) </span>5

[asy] size(200); defaultpen(linewidth(0.8)); pair A=origin,B=(16,0),C=(26,10*sqrt(3)),D=(10,10*sqrt(3)),E=(0,10*sqrt(3)); draw(A--B--C--E--B--A--D); label("

A

",A,S); label("

B

",B,S); label("

C

",C,N); label("

D

",D,N); label("

E

",E,N); label("

F

",extension(A,D,B,E),W); label("

4

",(D+E)/2,N); label("

16

",(8,0),S); label("

10

",(B+C)/2,SE); [/asy]

3

1

1990 AHSME #3: Consecutive Angles in a Trapezoid

The consecutive angles of a trapezoid form an arithmetic sequence. If the smallest angle is

75^\circ

, then the largest angle is

<span class='latex-bold'>(A) </span>95^\circ\qquad <span class='latex-bold'>(B) </span>100^\circ\qquad <span class='latex-bold'>(C) </span>105^\circ\qquad <span class='latex-bold'>(D) </span>110^\circ\qquad <span class='latex-bold'>(E) </span>115^\circ

2

1

1990 AHSME #2: Fractional Exponents

\left(\dfrac{1}{4}\right)^{-\frac{1}{4}}=

<span class='latex-bold'>(A) </span>-16\qquad <span class='latex-bold'>(B) </span>-\sqrt{2}\qquad <span class='latex-bold'>(C) </span>-\dfrac{1}{16}\qquad <span class='latex-bold'>(D) </span>-\dfrac{1}{256}\qquad <span class='latex-bold'>(E) </span>\sqrt{2}

1

1990 AHSME #1: Equation in X

If

\dfrac{x/4}{2}=\dfrac{4}{x/2}

then

x=

<span class='latex-bold'>(A) </span>\pm 1/2\qquad <span class='latex-bold'>(B) </span>\pm 1\qquad <span class='latex-bold'>(C) </span>\pm 2\qquad <span class='latex-bold'>(D) </span>\pm 4\qquad <span class='latex-bold'>(E) </span>\pm 8

24

1

1990 AMC 12 #24

All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined? \begin{tabular}{c c c c} {} & Adams & Baker & Adams and Baker \\ Boys: & 71 & 81 & 79 \\ Girls: & 76 & 90 & ? \\ Boys and Girls: & 74 & 84 & \\ \end{tabular} (A)\ 81 \qquad(B)\ 82 \qquad(C)\ 83 \qquad(D)\ 84 (E)\ 85

30

1

1990 AMC 12 #30

If

R_n=\frac{1}{2}(a^n+b^n)

where

a=3+2\sqrt{2}

,

b=3-2\sqrt{2}

, and

n=0,1,2, ...,

then

R_{12345}

is an integer. Its units digit is

<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 5 \qquad<span class='latex-bold'>(D)</span>\ 7 \qquad<span class='latex-bold'>(E)</span>\ 9

29

1

1990 AMC 12 #29

A subset of the integers

1, 2, ..., 100

has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?

<span class='latex-bold'>(A)</span>\ 50 \qquad<span class='latex-bold'>(B)</span>\ 66 \qquad<span class='latex-bold'>(C)</span>\ 67 \qquad<span class='latex-bold'>(D)</span>\ 76 \qquad<span class='latex-bold'>(E)</span>\ 78

28

1

1990 AMC 12 #28

A quadrilateral that has consecutive sides of lengths

70, 90, 130

and

110

is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length

130

divides that side into segments of lengths

x

and

y

. Find

|x-y|

.

<span class='latex-bold'>(A)</span>\ 12 \qquad<span class='latex-bold'>(B)</span>\ 13 \qquad<span class='latex-bold'>(C)</span>\ 14 \qquad<span class='latex-bold'>(D)</span>\ 15 \qquad<span class='latex-bold'>(E)</span>\ 16

27

1

1990 AMC 12 #27

Which of these triples could not be the lengths of the three altitudes of a triangle?

<span class='latex-bold'>(A)</span>\ 1,\sqrt{3},2 \qquad<span class='latex-bold'>(B)</span>\ 3,4,5 \qquad<span class='latex-bold'>(C)</span>\ 5,12,13 \qquad<span class='latex-bold'>(D)</span>\ 7,8,\sqrt{113} \qquad<span class='latex-bold'>(E)</span>\ 8,15,17

26

1

1990 AMC 12 #26

Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to him in the circle. Then each person computes and announces the average of the numbers of his two neighbors. The figure shows the average announced by each person (not the original number the person picked). The number picked by the person who announced the average

6

was [asy] label("(1)", (0,.9)); label("(2)", (.4,.65)); label("(3)", (.8,.25)); label("(4)", (.8,-.2)); label("(5)", (.4,-.65)); label("(6)", (0,-.9)); label("(7)", (-.4,-.65)); label("(8)", (-.8,-.2)); label("(9)", (-.8,.25)); label("(10)", (-.4,.65)); [/asy]

<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 5 \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 10 \qquad<span class='latex-bold'>(E)</span>\ \text{not uniquely determined from the given information}

25

1

1990 AMC 12 #25

Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of the others is tangent to the center sphere and to three faces of the cube. What is the radius of each sphere?

<span class='latex-bold'>(A)</span>\ 1-\frac{\sqrt{3}}{2} \qquad<span class='latex-bold'>(B)</span>\ \frac{2\sqrt{3}-3}{2} \qquad<span class='latex-bold'>(C)</span>\ \frac{\sqrt{2}}{6} \qquad<span class='latex-bold'>(D)</span>\ \frac{1}{4} \qquad<span class='latex-bold'>(E)</span>\ \frac{\sqrt{3}(2-\sqrt{2})}{4}

23

1

1990 AMC 12 #23

If

x,y>0

,

\log_yx+\log_xy=\frac{10}{3}

and

xy=144

, then

\frac{x+y}{2}=

<span class='latex-bold'>(A)</span>\ 12\sqrt{2} \qquad<span class='latex-bold'>(B)</span>\ 13\sqrt{3} \qquad<span class='latex-bold'>(C)</span>\ 24 \qquad<span class='latex-bold'>(D)</span>\ 30 \qquad<span class='latex-bold'>(E)</span>\ 36

22

1

Complex Roots

If the six solutions of

x^6=-64

are written in the form

a+bi

, where

a

and

b

are real, then the product of those solutions with

a>0

is

\text{(A)} \ -2 \qquad \text{(B)} \ 0 \qquad \text{(C)} \ 2i \qquad \text{(D)} \ 4 \qquad \text{(E)} \ 16

21

1

Pyramid and Trig

Consider a pyramid

P-ABCD

whose base

ABCD

is a square and whose vertex

P

is equidistant from

A

,

B

,

C

, and

D

. If

AB=1

and

\angle APD=2\theta

then the volume of the pyramid is

\text{(A)} \ \frac{\sin \theta}{6} \qquad \text{(B)} \ \frac{\cot \theta}{6} \qquad \text{(C)} \ \frac1{6\sin \theta} \qquad \text{(D)} \ \frac{1-\sin 2\theta}{6} \qquad \text{(E)} \ \frac{\sqrt{\cos 2\theta}}{6\sin \theta}

20

1

Quadrilateral ABCD

ABCD

is a quadrilateral with right angles at

A

and

C

. Points

E

and

F

are on

AC

, and

DE

and

BF

are perpendicular to

AC

. If

AE=3

,

DE=5

, and

CE=7

, then

BF=

[asy] draw((0,0)--(10,0)--(3,-5)--(0,0)--(6.5,3)--(10,0)); draw((6.5,0)--(6.5,3)); draw((3,0)--(3,-5)); dot((0,0)); dot((10,0)); dot((3,0)); dot((3,-5)); dot((6.5,0)); dot((6.5,3)); label("A", (0,0), W); label("B", (6.5,3), N); label("C", (10,0), E); label("D", (3,-5), S); label("E", (3,0), N); label("F", (6.5,0), S);[/asy]

\text{(A)} \ 3.6 \qquad \text{(B)} \ 4 \qquad \text{(C)} \ 4.2 \qquad \text{(D)} \ 4.5 \qquad \text{(E)} \ 5

19

1

Analyzing improper fraction

For how many integers

N

between

1

and

1990

is the improper fraction

\frac{N^2+7}{N+4}

not in lowest terms?

\text{(A)} \ 0 \qquad \text{(B)} \ 86 \qquad \text{(C)} \ 90 \qquad \text{(D)} \ 104 \qquad \text{(E)} \ 105

18

1

Random Integers

First

a

is chosen at random from the set

\{1,2,3,\ldots,99,100 \}

, and then

b

is chosen at random from the same set. The probability that the integer

3^a+7^b

has units digit

8

is

\text{(A)} \ \frac1{16} \qquad \text{(B)} \ \frac18 \qquad \text{(C)} \ \frac{3}{16}\qquad \text{(D)} \ \frac15 \qquad \text{(E)} \ \frac14

17

1

Special 3-digit numbers

How many of the numbers,

100,101,\ldots,999

, have three different digits in increasing order or in decreasing order?

\text{(A)} \ 120 \qquad \text{(B)} \ 168 \qquad \text{(C)} \ 204 \qquad \text{(D)} \ 216 \qquad \text{(E)} \ 240

16

1

Handshakes

At one of George Washington's parties, each man shook hands with everyone except his spouse, and no handshakes took place between women. If

13

married couples attended, how many handshakes were there among these

26

people?

\text{(A)} \ 78 \qquad \text{(B)} \ 185 \qquad \text{(C)} \ 234 \qquad \text{(D)} \ 312 \qquad \text{(E)} \ 325

15

1

Original Equation Problem

Four whole numbers, when added three at a time, give the sums

180

,

197

,

208

, and

222

. What is the largest of the four numbers?

\text{(A)} \ 77 \qquad \text{(B)} \ 83 \qquad \text{(C)} \ 89 \qquad \text{(D)} \ 95 \qquad \text{(E)} \ \text{cannot be determined}

14

1

Find the Angle Measure

An acute isosceles triangle,

ABC

is inscribed in a circle. Through

B

and

C

, tangents to the circle are drawn, meeting at point

D

. If

\angle ABC=\angle ACB=2\angle D

and

x

is the radian measure of

\angle A

, then

x=

[asy] defaultpen(linewidth(0.7)+fontsize(10)); real x=180/7; pair D=origin, B=dir(3x), C=dir(4x), A=intersectionpoint(C--C+dir(2x), B--B+dir(5x)), O=circumcenter(A,B,C); markscalefactor=0.015; draw(B--C--D--B--A--C^^Circle(O, abs(O-C))^^anglemark(C,A,B)); dot(A^^B^^C^^D); 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("

x

", A+0.1*dir(270), S);[/asy]

\text{(A)} \ \frac37\pi \qquad \text{(B)} \ \frac49\pi \qquad \text{(C)} \ \frac5{11}\pi \qquad \text{(D)} \ \frac6{13}\pi \qquad \text{(E)} \ \frac7{15}\pi

13

1

Computer Operations

If the following instructions are carried out by a computer, which of

X

will be printed because of instruction

5

?

1.

Start

X

at

3

and

S

at

0

2.

Increase the value of

X

by

2

.

3.

Increase the value of

S

by the value of

X

.

4.

If

S

is at least

10000

, then go to instsruction

5

; otherwise, go to instruction

2

and proceed from there.

5.

Print the value of

X

.

6.

Stop.

\text{(A)} \ 19 \qquad \text{(B)} \ 21 \qquad \text{(C)} \ 23 \qquad \text{(D)} \ 199 \qquad \text{(E)} \ 201

12

1

Function

Let

f

be the function defined by

f(x)=ax^2-\sqrt2

for some positive

a

. If

f(f(\sqrt2 ))=-\sqrt 2

, then

a=

\text{(A)} \ \frac{2-\sqrt2}{2} \qquad \text{(B)} \ \frac12 \qquad \text{(C)} \ 2-\sqrt2 \qquad \text{(D)} \ \frac{\sqrt{2}}{2} \qquad \text{(E)} \ \frac{2+\sqrt2}{2}

11

1

Number of such integers

How man y positive integers less than

50

have an odd number of positive integer divisors?

\text{(A)}\ 3 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 11

1990 AMC 12/AHSME

Subcontests

1990 AHSME #10: Cubes from a point

1990 AHSME #9: Edges of a cube

1990 AHSME #8: Number of Real solutions

1990 AHSME #7: Triangle with integral sides

1990 AHSME #6: Points in a Plane

1990 AHSME #5: Largest of Numbers

1990 AHSME #4: Parallelogram

1990 AHSME #3: Consecutive Angles in a Trapezoid

1990 AHSME #2: Fractional Exponents

1990 AHSME #1: Equation in X

1990 AMC 12 #24

1990 AMC 12 #30

1990 AMC 12 #29

1990 AMC 12 #28

1990 AMC 12 #27

1990 AMC 12 #26

1990 AMC 12 #25

1990 AMC 12 #23

Complex Roots

Pyramid and Trig

Quadrilateral ABCD

Analyzing improper fraction

Random Integers

Special 3-digit numbers

Handshakes

Original Equation Problem

Find the Angle Measure

Computer Operations

Function

Number of such integers