MathDB

1

Find n for a tennis tourney with n women and 2n men

n

women and

2n

men play, and each player plays exactly one match with every other player. If there are no ties and the ratio of the number of matches won by women to the number of matches won by men is

7/5

, then

n

equals

<span class='latex-bold'>(A) </span>2\qquad<span class='latex-bold'>(B) </span>4\qquad<span class='latex-bold'>(C) </span>6\qquad<span class='latex-bold'>(D) </span>7\qquad <span class='latex-bold'>(E) </span>\text{none of these}

29

1

Find [A'B'C'D'] if [ABCD]=10

Sides

AB,~ BC, ~CD

and

DA

, respectively, of convex quadrilateral

ABCD

are extended past

B,~ C ,~ D

and

A

to points

B',~C',~ D'

and

A'

. Also,

AB = BB' = 6,~ BC = CC' = 7, ~CD = DD' = 8

and

DA = AA' = 9

; and the area of

ABCD

is 10. The area of

A 'B 'C'D'

is

<span class='latex-bold'>(A) </span>20\qquad<span class='latex-bold'>(B) </span>40\qquad<span class='latex-bold'>(C) </span>45\qquad<span class='latex-bold'>(D) </span>50\qquad <span class='latex-bold'>(E) </span>60

28

1

Find <A_44 A_45 A_43

[asy] import cse5; size(180); pathpen=black; pair A1=(0,0), A2=(1,0), A3=(0.5,sqrt(3)/2); D(MP("A_1",A1)--MP("A_2",A2)--MP("A_3",A3,N)--cycle); pair A4=(A1+A2)/2, A5 = (A3+A2)/2, A6 = (A4+A3)/2; D(MP("A_4",A4,S)--MP("A_6",A6,W)--A3); D(A6--MP("A_5",A5,NE)--A4); //Credit to chezbgone2 for the diagram[/asy]
If

\triangle A_1A_2A_3

is equilateral and

A_{n+3}

is the midpoint of line segment

A_nA_{n+1}

for all positive integers

n

, then the measure of

\measuredangle A_{44}A_{45}A_{43}

equals

<span class='latex-bold'>(A) </span>30^\circ\qquad<span class='latex-bold'>(B) </span>45^\circ\qquad<span class='latex-bold'>(C) </span>60^\circ\qquad<span class='latex-bold'>(D) </span>90^\circ\qquad <span class='latex-bold'>(E) </span>120^\circ

27

1

Numbers congruent to 1 mod k

There is more than one integer greater than

1

which, when divided by any integer

k

such that

2 \le k \le 11

, has a remainder of

1

. What is the difference between the two smallest such integers?

<span class='latex-bold'>(A) </span>2310\qquad<span class='latex-bold'>(B) </span>2311\qquad<span class='latex-bold'>(C) </span>27,720\qquad<span class='latex-bold'>(D) </span>27,721\qquad <span class='latex-bold'>(E) </span>\text{none of these}

26

1

Find the length of QR in circle P

[asy] import cse5; size(180); real a=4, b=3; pathpen=black; pair A=(a,0), B=(0,b), C=(0,0); D(MP("A",A)--MP("B",B,N)--MP("C",C,SW)--cycle); pair X=IP(B--A,(0,0)--(b,a)); D(CP((X+C)/2,C)); D(MP("R",IP(CP((X+C)/2,C),B--C),NW)--MP("Q",IP(CP((X+C)/2,C),A--C+(0.1,0)))); //Credit to chezbgone2 for the diagram[/asy] In

\triangle ABC

,

AB = 10~ AC = 8

and

BC = 6

. Circle

P

is the circle with smallest radius which passes through

C

and is tangent to

AB

. Let

Q

and

R

be the points of intersection, distinct from

C

, of circle

P

with sides

AC

and

BC

, respectively. The length of segment

QR

is

<span class='latex-bold'>(A) </span>4.75\qquad<span class='latex-bold'>(B) </span>4.8\qquad<span class='latex-bold'>(C) </span>5\qquad<span class='latex-bold'>(D) </span>4\sqrt{2}\qquad <span class='latex-bold'>(E) </span>3\sqrt{3}

25

1

Boundary of set S

Let

u

be a positive number. Consider the set

S

of all points whose rectangular coordinates

(x, y )

satisfy all of the following conditions:

\text{(i) }\frac{a}{2}\le x\le 2a\qquad\text{(ii) }\frac{a}{2}\le y\le 2a\qquad\text{(iii) }x+y\ge a \\ \\ \qquad\text{(iv) }x+a\ge y\qquad \text{(v) }y+a\ge x

The boundary of set

S

is a polygon with

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

24

1

x,y,z form a geometric progression

If the distinct non-zero numbers

x ( y - z),~ y(z - x),~ z(x - y )

form a geometric progression with common ratio

r

, then

r

satisfies the equation

<span class='latex-bold'>(A) </span>r^2+r+1=0\qquad<span class='latex-bold'>(B) </span>r^2-r+1=0\qquad<span class='latex-bold'>(C) </span>r^4+r^2-1=0

\qquad<span class='latex-bold'>(D) </span>(r+1)^4+r=0\qquad <span class='latex-bold'>(E) </span>(r-1)^4+r=0

23

1

Find [ABF]

[asy] size(100); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((0,1)--(1,0)); draw((0,0)--(.5,sqrt(3)/2)--(1,0)); label("

A

",(0,0),SW); label("

B

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

C

",(1,1),NE); label("

D

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

E

",(.5,sqrt(3)/2),E); label("

F

",intersectionpoint((0,0)--(.5,sqrt(3)/2),(0,1)--(1,0)),2W); //Credit to chezbgone2 for the diagram[/asy]
Vertex

E

of equilateral triangle

ABE

is in the interior of square

ABCD

, and

F

is the point of intersection of diagonal

BD

and line segment

AE

. If length

AB

is

\sqrt{1+\sqrt{3}}

then the area of

\triangle ABF

is

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

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

20

1

Algebraic expressions with a,b,c,x

If

a,b,c

are non-zero real numbers such that

\frac{a+b-c}{c}=\frac{a-b+c}{b}=\frac{-a+b+c}{a},

and

x=\frac{(a+b)(b+c)(c+a)}{abc},

and

x<0

, then

x

equals

<span class='latex-bold'>(A) </span>-1\qquad<span class='latex-bold'>(B) </span>-2\qquad<span class='latex-bold'>(C) </span>-4\qquad<span class='latex-bold'>(D) </span>-6\qquad <span class='latex-bold'>(E) </span>-8

19

1

Probability of choosing n, n>50, is 3p

A positive integer

n

not exceeding

100

is chosen in such a way that if

n\le 50

, then the probability of choosing

n

is

p

, and if

n > 50

, then the probability of choosing

n

is

3p

. The probability that a perfect square is chosen is

<span class='latex-bold'>(A) </span>.05\qquad<span class='latex-bold'>(B) </span>.065\qquad<span class='latex-bold'>(C) </span>.08\qquad<span class='latex-bold'>(D) </span>.09\qquad <span class='latex-bold'>(E) </span>.1

18

1

n^0.5-(n-1)^0.5<0.01

What is the smallest positive integer

n

such that

\sqrt{n}-\sqrt{n-1}<.01

?

<span class='latex-bold'>(A) </span>2499\qquad<span class='latex-bold'>(B) </span>2500\qquad<span class='latex-bold'>(C) </span>2501\qquad<span class='latex-bold'>(D) </span>10,000\qquad <span class='latex-bold'>(E) </span>\text{There is no such integer}

17

1

Function such that f(x^2+1)^sqrt(x)=k

If

k

is a positive number and

f

is a function such that, for every positive number

x

,

\left[f(x^2+1)\right]^{\sqrt{x}}=k;

then, for every positive number

y

,

\left[f(\frac{9+y^2}{y^2})\right]^{\sqrt{\frac{12}{y}}}

is equal to

<span class='latex-bold'>(A) </span>\sqrt{k}\qquad<span class='latex-bold'>(B) </span>2k\qquad<span class='latex-bold'>(C) </span>k\sqrt{k}\qquad<span class='latex-bold'>(D) </span>k^2\qquad <span class='latex-bold'>(E) </span>y\sqrt{k}

16

1

Shaking hands in a room containing N people

In a room containing

N

people,

N > 3

, at least one person has not shaken hands with everyone else in the room. What is the maximum number of people in the room that could have shaken hands with everyone else?

<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>1\qquad<span class='latex-bold'>(C) </span>N-1\qquad<span class='latex-bold'>(D) </span>N\qquad <span class='latex-bold'>(E) </span>\text{none of these}

15

1

sin x+cos x=1/5 ==> find tan x

If

\sin x+\cos x=1/5

and

0\le x<\pi

, then

\tan x

is

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

<span class='latex-bold'>(E) </span>\text{not completely determined by the given information}

13

1

Find a+b+c+d

If

a,b,c,

and

d

are non-zero numbers such that

c

and

d

are the solutions of

x^2+ax+b=0

and

a

and

b

are the solutions of

x^2+cx+d=0

, then

a+b+c+d

equals

<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>-2\qquad<span class='latex-bold'>(C) </span>2\qquad<span class='latex-bold'>(D) </span>4\qquad <span class='latex-bold'>(E) </span>(-1+\sqrt{5})/2

12

1

Find <EAD

In

\triangle ADE

,

\measuredangle ADE=140^\circ

, points

B

and

C

lie on sides

AD

and

AE

, respectively, and points

A,~B,~C,~D,~E

are distinct.* If lengths

AB,~BC,~CD,

and

DE

are all equal, then the measure of

\measuredangle EAD

is

<span class='latex-bold'>(A) </span>5^\circ\qquad<span class='latex-bold'>(B) </span>6^\circ\qquad<span class='latex-bold'>(C) </span>7.5^\circ\qquad<span class='latex-bold'>(D) </span>8^\circ\qquad <span class='latex-bold'>(E) </span>10^\circ

* The specification that points

A,B,C,D,E

be distinct was not included in the original statement of the problem. If

B=D

, then

C=E

and

\measuredangle EAD=20^\circ

.

11

1

Line and tangent circle

If

r

is positive and the line whose equation is

x + y = r

is tangen to the circle whose equation is

x^2 + y ^2 = r

, then

r

equals

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

21

1

Divisibility of a factorial

p

and

q

are distinct prime numbers. Prove that the number

\frac {(pq-1)!} {p^{q-1}q^{p-1}(p-1)!(q-1)!}

is an integer.

10

1

Distance between A and B ≤ distance between A and other poin

If

\mathit{B}

is a point on circle

\mathit{C}

with center

\mathit{P}

, then the set of all points

\mathit{A}

in the plane of circle

\mathit{C}

such that the distance between

\mathit{A}

and

\mathit{B}

is less than or equal to the distance between

\mathit{A}

and any other point on circle

\mathit{C}

is

<span class='latex-bold'>(A) </span>\text{the line segment from }P \text{ to }B\qquad

<span class='latex-bold'>(B) </span>\text{the ray beginning at }P \text{ and passing through }B\qquad

<span class='latex-bold'>(C) </span>\text{a ray beginning at }B\qquad

<span class='latex-bold'>(D) </span>\text{a circle whose center is }P\qquad

<span class='latex-bold'>(E) </span>\text{a circle whose center is }B

9

1

Simplify absolute value expression

If

x<0

, then

\left|x-\sqrt{(x-1)^2}\right|

equals

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

8

1

Find (a_2-a_1)/(b_2-b_1)

If

x\neq y

and the sequences

x,a_1,a_2,y

and

x,b_1,b_2,b_3,y

each are in arithmetic progression, then

(a_2-a_1)/(b_2-b_1)

equals

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

7

1

12 inch distance between opposite sides of a regular hexagon

Opposite sides of a regular hexagon are

12

inches apart. The length of each side, in inches, is

<span class='latex-bold'>(A) </span>7.5\qquad<span class='latex-bold'>(B) </span>6\sqrt{2}\qquad<span class='latex-bold'>(C) </span>5\sqrt{2}\qquad<span class='latex-bold'>(D) </span>\frac{9}{2}\sqrt{3}\qquad <span class='latex-bold'>(E) </span>4\sqrt{3}

6

1

Number of Distinct Pairs

The number of distinct pairs

(x,y)

of real numbers satisfying both of the following equations: \begin{align*}x&=x^2+y^2, \\ y&=2xy\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

5

1

Find the amount the fourth boy paid

Four boys bought a boat for \textdollar 60. The first boy paid one half of the sum of the amounts paid by the other boys; the second boy paid one third of the sum of the amounts paid by the other boys; and the third boy paid one fourth of the sum of the amounts paid by the other boys. How much did the fourth boy pay?
(A) \textdollar 10\qquad(B) \textdollar 12\qquad(C) \textdollar 13\qquad(D) \textdollar 14\qquad (E) \textdollar 15

4

1

Substituting powers of 10

If

a = 1,~ b = 10, ~c = 100

, and

d = 1000

, then

(a+ b+ c-d) + (a + b- c+ d) +(a-b+ c+d)+ (-a+ b+c+d)

is equal to

<span class='latex-bold'>(A) </span>1111\qquad<span class='latex-bold'>(B) </span>2222\qquad<span class='latex-bold'>(C) </span>3333\qquad<span class='latex-bold'>(D) </span>1212\qquad <span class='latex-bold'>(E) </span>4242

3

1

Simplify for x=1/y

For all non-zero numbers

x

and

y

such that

x = 1/y

,

\left(x-\frac{1}{x}\right)\left(y+\frac{1}{y}\right)

equals

<span class='latex-bold'>(A) </span>2x^2\qquad<span class='latex-bold'>(B) </span>2y^2\qquad<span class='latex-bold'>(C) </span>x^2+y^2\qquad<span class='latex-bold'>(D) </span>x^2-y^2\qquad <span class='latex-bold'>(E) </span>y^2-x^2

2

1

Reciprocal of circumference

If four times the reciprocal of the circumference of a circle equals the diameter of the circle, then the area of the circle is

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

1

Find 2/x

If

1-\frac{4}{x}+\frac{4}{x^2}=0

, then

\frac{2}{x}

equals

<span class='latex-bold'>(A) </span>-1\qquad<span class='latex-bold'>(B) </span>1\qquad<span class='latex-bold'>(C) </span>2\qquad<span class='latex-bold'>(D) </span>-1\text{ or }2\qquad <span class='latex-bold'>(E) </span>-1\text{ or }-2

22

1

4 True/False Statements on a Card

The following four statements, and only these are found on a card: [asy] pair A,B,C,D,E,F,G; A=(0,1); B=(0,5); C=(11,5); D=(11,1); E=(0,4); F=(0,3); G=(0,2); draw(A--B--C--D--cycle); label("On this card exactly one statement is false.", B, SE); label("On this card exactly two statements are false.", E, SE); label("On this card exactly three statements are false.", F, SE); label("On this card exactly four statements are false.", G, SE); [/asy] (Assume each statement is either true or false.) Among them the number of false statements is exactly

<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

14

1

1978 AHSME Problem 14

If an integer

n > 8

is a solution of the equation

x^2 - ax+b=0

and the representation of

a

in the base-

n

number system is

18

, then the base-

n

representation of

b

is

<span class='latex-bold'>(A)</span>\ 18 \qquad <span class='latex-bold'>(B)</span>\ 20 \qquad <span class='latex-bold'>(C)</span>\ 80 \qquad <span class='latex-bold'>(D)</span>\ 81 \qquad <span class='latex-bold'>(E)</span>\ 280

1978 AMC 12/AHSME

Subcontests

Find n for a tennis tourney with n women and 2n men

Find [A'B'C'D'] if [ABCD]=10

Find <A_44 A_45 A_43

Numbers congruent to 1 mod k

Find the length of QR in circle P

Boundary of set S

x,y,z form a geometric progression

Find [ABF]

Algebraic expressions with a,b,c,x

Probability of choosing n, n>50, is 3p

n^0.5-(n-1)^0.5<0.01

Function such that f(x^2+1)^sqrt(x)=k

Shaking hands in a room containing N people

sin x+cos x=1/5 ==> find tan x

Find a+b+c+d

Find <EAD

Line and tangent circle

Divisibility of a factorial

Distance between A and B ≤ distance between A and other poin

Simplify absolute value expression

Find (a_2-a_1)/(b_2-b_1)

12 inch distance between opposite sides of a regular hexagon

Number of Distinct Pairs

Find the amount the fourth boy paid

Substituting powers of 10

Simplify for x=1/y

Reciprocal of circumference

Find 2/x

4 True/False Statements on a Card

1978 AHSME Problem 14

1978 AMC 12/AHSME

Subcontests

Find n for a tennis tourney with n women and 2n men

Find [A'B'C'D'] if [ABCD]=10

Find &lt;A_44 A_45 A_43

Numbers congruent to 1 mod k

Find the length of QR in circle P

Boundary of set S

x,y,z form a geometric progression

Find [ABF]

Algebraic expressions with a,b,c,x

Probability of choosing n, n&gt;50, is 3p

n^0.5-(n-1)^0.5&lt;0.01

Function such that f(x^2+1)^sqrt(x)=k

Shaking hands in a room containing N people

sin x+cos x=1/5 ==&gt; find tan x

Find a+b+c+d

Find &lt;EAD

Line and tangent circle

Divisibility of a factorial

Distance between A and B ≤ distance between A and other poin

Simplify absolute value expression

Find (a_2-a_1)/(b_2-b_1)

12 inch distance between opposite sides of a regular hexagon

Number of Distinct Pairs

Find the amount the fourth boy paid

Substituting powers of 10

Simplify for x=1/y

Reciprocal of circumference

Find 2/x

4 True/False Statements on a Card

1978 AHSME Problem 14

Find <A_44 A_45 A_43

Probability of choosing n, n>50, is 3p

n^0.5-(n-1)^0.5<0.01

sin x+cos x=1/5 ==> find tan x

Find <EAD