MathDB

1

Trig Series

w = a + bi

,

|w|

is defined to be the real number

\sqrt{a^2 + b^2}

. If

w = \cos{40^\circ} + i\sin{40^\circ}

, then

|w + 2w^2 + 3w^3 + \cdots + 9w^9|^{-1}

equals

<span class='latex-bold'>(A)</span>\ \frac{1}{9}\sin{40^\circ} \qquad <span class='latex-bold'>(B)</span>\ \frac{2}{9}\sin{20^\circ} \qquad <span class='latex-bold'>(C)</span>\ \frac{1}{9}\cos{40^\circ} \qquad <span class='latex-bold'>(D)</span>\ \frac{1}{18}\cos{20^\circ} \qquad <span class='latex-bold'>(E)</span>\text{ none of these}

29

1

Largest y/x

Find the largest value for

\frac{y}{x}

for pairs of real numbers

(x,y)

which satisfy

(x-3)^2 + (y-3)^2 = 6.

<span class='latex-bold'>(A) </span>3 + 2 \sqrt 2\qquad <span class='latex-bold'>(B) </span> 2 + \sqrt 3\qquad <span class='latex-bold'>(C ) </span>3 \sqrt 3\qquad <span class='latex-bold'>(D) </span>6\qquad <span class='latex-bold'>(E) </span>6 + 2 \sqrt 3

27

1

Triangles, triangles, triangles

In

\triangle ABC

,

D

is on

AC

and

F

is on

BC

. Also,

AB \perp AC, AF \perp BC

, and

BD = DC = FC = 1

. Find

AC

. A.

\sqrt{2}

B.

\sqrt{3}

C.

\sqrt[3] {2}

D.

\sqrt[3] {3}

E.

\sqrt[4] {3}

26

1

Obtuse triangle

In the obtuse triangle

ABC

,

AM = MB, MD \perp BC, EC \perp BC

. If the area of

\triangle ABC

is 24, then the area of

\triangle BED

is
[asy] size(200); defaultpen(linewidth(0.8)+fontsize(11pt)); pair A = (6.5,3.2), B = origin, C = (5.0), D = (3.3,0); pair Xc = (C.x,4), Xd = (D.x,4), E = intersectionpoint(A--B,C--Xc), M = intersectionpoint(D--Xd, A--B); draw(C--A--B--C--E--D--M); label("

A

",A,NE); label("

B

",B,W); label("

C

",C,SE); label("

D

",D,S); label("

E

",E,N); label("

M

",M,N); draw(rightanglemark(D,C,E,7)^^rightanglemark(B,D,M,7)); [/asy]

<span class='latex-bold'>(A) </span>9\qquad <span class='latex-bold'>(B) </span>12\qquad <span class='latex-bold'>(C) </span>15\qquad <span class='latex-bold'>(D) </span>18\qquad <span class='latex-bold'>(E) </span>\text{not uniquely determined}

22

1

Interesting set of points by t

Let

a

and

c

be fixed positive numbers. For each real number

t

let

(x_t, y_t)

be the vertex of the parabola

y = ax^2+bx+c

. If the set of vertices

(x_t, y_t)

for all real values of

t

is graphed in the plane, the graph is A. a straight line B. a parabola C. part, but not all, of a parabola D. one branch of a hyperbola E. None of these

19

1

6 randomly chosen numbers and odd sum

A box contains 11 balls, numbered 1,2,3,....,11. If 6 balls are drawn simultaneously at random, what is the probability that the sum of the numbers on the balls drawn is odd? A.

\frac{100}{231}

B.

\frac{115}{231}

C.

\frac{1}{2}

D.

\frac{118}{231}

E.

\frac{6}{11}

18

1

Chosen point

A point

(x,y)

is to be chosen in the coordinate plane so that it is equally distant from the x-axis, the y-axis, and the line

x+y = 2

. Then

x

is A.

\sqrt{2} - 1

B.

\frac{1}{2}

C.

2 - \sqrt{2}

D. 1 E. Not uniquely determined

17

1

Altitude CH and area of triangle ABC

A right triangle

ABC

with hypotenuse

AB

has side

AC = 15

. Altitude

CH

divides

AB

into segments

AH

And

HB

, with

HB = 16

. The area of

\triangle ABC

is: [asy] size(200); defaultpen(linewidth(0.8)+fontsize(11pt)); pair A = origin, H = (5,0), B = (13,0), C = (5,6.5); draw(C--A--B--C--H^^rightanglemark(C,H,B,16)); label("

A

",A,W); label("

B

",B,E); label("

C

",C,N); label("

H

",H,S); label("

15

",C/2,NW); label("

16

",(H+B)/2,S); [/asy]

<span class='latex-bold'>(A) </span>120\qquad <span class='latex-bold'>(B) </span>144\qquad <span class='latex-bold'>(C) </span>150\qquad <span class='latex-bold'>(D) </span>216\qquad <span class='latex-bold'>(E) </span>144\sqrt5

16

1

f(2+x) = f(2-x)

The function

f(x)

satisfies

f(2+x) = f(2-x)

for all real numbers

x

. If the equation

f(x) = 0

has exactly four distinct real roots, then the sum of these roots is A. 0 B. 2 C. 4 D. 6 E. 8

14

1

Log problem

The product of all real roots of the equation

x^{\log_{10} x} = 10

is A. 1 B. -1 C. 10 D.

10^{-1}

E. None of these

8

1

Trapezoid

Figure

ABCD

is a trapezoid with

AB || DC, AB = 5, BC = 3 \sqrt 2, \measuredangle BCD = 45^\circ

, and

\measuredangle CDA = 60^\circ

. The length of

DC

is

<span class='latex-bold'>(A) </span>7 + \frac{2}{3} \sqrt{3}\qquad <span class='latex-bold'>(B) </span>8\qquad <span class='latex-bold'>(C) </span>9 \frac{1}{2}\qquad <span class='latex-bold'>(D) </span>8 + \sqrt 3\qquad <span class='latex-bold'>(E) </span>8 + 3 \sqrt 3

6

1

Boys, girls, and teachers

In a certain school, there are three times as many boys as girls and nine times as many girls as teachers. Using the letters

b,g,t

to represent the number of boys, girls, and teachers, respectively, then the total number of boys, girls, and teachers can be represented by the expression

<span class='latex-bold'>(A) </span>31b\qquad <span class='latex-bold'>(B) </span>\frac{37b}{27}\qquad <span class='latex-bold'>(C) </span>13g\qquad <span class='latex-bold'>(D) </span>\frac{37g}{27}\qquad <span class='latex-bold'>(E) </span>\frac{37t}{27}

4

1

Rectangle and circle

A rectangle intersects a circle as shown:

AB=4

,

BC=5

, and

DE=3

. Then

EF

equals:
[asy]size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair D=origin, E=(3,0), F=(10,0), G=(12,0), H=(12,1), A=(0,1), B=(4,1), C=(9,1), O=circumcenter(B,C,F); draw(D--G--H--A--cycle); draw(Circle(O, abs(O-C))); label("

A

", A, NW); label("

B

", B, NW); label("

C

", C, NE); label("

D

", D, SW); label("

E

", E, SE); label("

F

", F, SW);
label("4", (2,0.85), N); label("3", D--E, S); label("5", (6.5,0.85), N); [/asy]

\mathbf{(A)}\; 6\qquad \mathbf{(B)}\; 7\qquad \mathbf{(C)}\; \frac{20}3\qquad \mathbf{(D)}\; 8\qquad \mathbf{(E)}\; 9

28

1

square root sum

The number of distinct pairs of integers

(x,y)

such that 0 < x < y \text{and} \sqrt{1984} = \sqrt{x} + \sqrt{y} 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>7

24

1

smallest a+b

If

a

and

b

are positive real numbers and each of the equations x^2+ax+2b = 0 \text{and} x^2+2bx+a = 0 has real roots, then the smallest possible value of

a+b

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

11

1

Calculator problem

A calculator has a key which replaces the displayed entry with its square, and another key which replaces the displayed entry with its reciprocal. Let

y

be the final result if one starts with an entry

x \neq 0

and alternately squares and reciprocates

n

times each. Assuming the calculator is completely accurate (e.g., no roundoff or overflow), then

y

equals A.

x^{((-2)^n)}

B.

x^{2n}

C.

x^{-2n}

D.

x^{-(2^n)}

E.

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

25

1

internal diagonal length

The total area of all the faces of a rectangular solid is

22 \text{cm}^2

, and the total length of all its edges is

24 \text{cm}

. Then the length in

\text{cm}

of any one of its internal diagonal is A.

\sqrt{11}

B.

\sqrt{12}

C.

\sqrt{13}

D.

\sqrt{14}

E. Not uniquely determined

23

1

sin 10,sin 20, cos 10, cos 20

\frac{\sin 10^\circ + \sin 20^\circ}{\cos 10^\circ + \cos 20^\circ}

equals A.

\tan 10^\circ + \tan 20^\circ

B.

\tan 30^\circ

C.

\frac{1}{2} (\tan 10^\circ + \tan 20^\circ)

D.

\tan 15^\circ

E.

\frac{1}{4} \tan 60^\circ

21

1

triple (a,b,c)

The number of triples

(a,b,c)

of positive integers which satisfy the simultaneous equations \begin{align*} ab+bc &= 44,\\ ac+bc &= 23, \end{align*} 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

20

1

Absolute value equation

The number of distinct solutions of the equation

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

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

5

1

n^200 < 5^300

The largest integer

n

for which

n^{200} < 5^{300}

is

<span class='latex-bold'>(A) </span>8\qquad <span class='latex-bold'>(B) </span>9\qquad <span class='latex-bold'>(C) </span>10\qquad <span class='latex-bold'>(D) </span>11\qquad <span class='latex-bold'>(E) </span>12

3

1

nonprime in interval

Let

n

be the smallest nonprime integer greater than 1 with no prime factor less than 10. Then A.

100 < n \leq 110

B.

110 < n \leq 120

C.

120 < n \leq 130

D.

130 < n \leq 140

E.

140 < n \leq 150

1

Square and difference of square

\frac{1000^2}{252^2 - 248^2}

equals

<span class='latex-bold'>(A) </span>62,500\qquad <span class='latex-bold'>(B) </span>1000\qquad<span class='latex-bold'>(C) </span>500\qquad<span class='latex-bold'>(D) </span>250\qquad<span class='latex-bold'>(E) </span> \frac{1}{2}

7

1

"classic" walking problem

When Dave walks to school, he averages 90 steps per minute, each of his steps 75cm long. It takes him 16 minutes to get to school. His brother, Jack, going to the same school by the same route, averages 100 steps per minute, but his steps are only 60 cm long. How long does it take Jack to get to school?

<span class='latex-bold'>(A) </span>14 \frac{2}{9}\qquad <span class='latex-bold'>(B) </span>15\qquad <span class='latex-bold'>(C) </span>18\qquad <span class='latex-bold'>(D) </span>20\qquad <span class='latex-bold'>(E) </span>22 \frac{2}{9}

15

1

equal x values of sine and cosine

If

\sin 2x \sin 3x = \cos 2x \cos 3x

, then one value for

x

is A.

18^\circ

B.

30^\circ

C.

36^\circ

D.

45^\circ

E.

60^\circ

13

1

Fraction simplification

\frac{2 \sqrt 6}{\sqrt 2 + \sqrt 3 + \sqrt 5}

equals A.

\sqrt 2 + \sqrt 3 - \sqrt 5

B.

4 - \sqrt 2 - \sqrt 3

C.

\sqrt 2 + \sqrt 3 + \sqrt 6 - 5

D.

\frac{1}{2} (\sqrt 2 + \sqrt 5 - \sqrt 3)

E.

\frac{1}{3} (\sqrt 3 + \sqrt 5 - \sqrt 2)

12

1

{a_n} sequence

If the sequence

\{a_n\}

is defined by \begin{align*}a_1 &= 2,\\ a_{n+1} &= a_n + 2n\qquad (n\geq 1),\end{align*} then

a_{100}

equals

<span class='latex-bold'>(A) </span>9900\qquad <span class='latex-bold'>(B) </span>9902\qquad <span class='latex-bold'>(C) </span>9904\qquad <span class='latex-bold'>(D) </span>10100\qquad <span class='latex-bold'>(E) </span>10102

10

1

Fourth complex number

Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are

1+2i,-2+i

and

-1-2i

. The fourth number is A.

2+i

B.

2-i

C.

1-2i

D.

-1+2i

E.

-2-i

9

1

Number of digits

The number of digits in

4^{16} 5^{25}

(when written in the usual base 10 form) is A. 31 B. 30 C. 29 D. 28 E. 27

2

1

Simplification

If

x,y

and

y - \frac{1}{x}

are not 0, then

\frac{x - \frac{1}{y}}{y - \frac{1}{x}}

equals

<span class='latex-bold'>(A) </span>1\qquad<span class='latex-bold'>(B) </span> \frac{x}{y}\qquad<span class='latex-bold'>(C) </span>\frac{y}{x}\qquad<span class='latex-bold'>(D) </span>\frac{x}{y} - \frac{y}{x}\qquad<span class='latex-bold'>(E) </span> xy - \frac{1}{xy}

1984 AMC 12/AHSME

Subcontests

Trig Series

Largest y/x

Triangles, triangles, triangles

Obtuse triangle

Interesting set of points by t

6 randomly chosen numbers and odd sum

Chosen point

Altitude CH and area of triangle ABC

f(2+x) = f(2-x)

Log problem

Trapezoid

Boys, girls, and teachers

Rectangle and circle

square root sum

smallest a+b

Calculator problem

internal diagonal length

sin 10,sin 20, cos 10, cos 20

triple (a,b,c)

Absolute value equation

n^200 < 5^300

nonprime in interval

Square and difference of square

"classic" walking problem

equal x values of sine and cosine

Fraction simplification

{a_n} sequence

Fourth complex number

Number of digits

Simplification

1984 AMC 12/AHSME

Subcontests

Trig Series

Largest y/x

Triangles, triangles, triangles

Obtuse triangle

Interesting set of points by t

6 randomly chosen numbers and odd sum

Chosen point

Altitude CH and area of triangle ABC

f(2+x) = f(2-x)

Log problem

Trapezoid

Boys, girls, and teachers

Rectangle and circle

square root sum

smallest a+b

Calculator problem

internal diagonal length

sin 10,sin 20, cos 10, cos 20

triple (a,b,c)

Absolute value equation

n^200 &lt; 5^300

nonprime in interval

Square and difference of square

&quot;classic&quot; walking problem

equal x values of sine and cosine

Fraction simplification

{a_n} sequence

Fourth complex number

Number of digits

Simplification

n^200 < 5^300

"classic" walking problem