MathDB

1

We All Scream For Ice Cream

An ice-cream novelty item consists of a cup in the shape of a

4

-inch-tall frustum of a right circular cone, with a

2

-inch-diameter base at the bottom and a

4

-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height

4

inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches?

<span class='latex-bold'>(A)</span>\ 8\pi\qquad<span class='latex-bold'>(B)</span>\ \frac{28\pi}{3}\qquad<span class='latex-bold'>(C)</span>\ 12\pi\qquad<span class='latex-bold'>(D)</span>\ 14\pi\qquad<span class='latex-bold'>(E)</span>\ \frac{44\pi}{3}

5

1

Outliers in Data Set

The data set

[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]

has median

Q_2=40

, first quartile

Q_1=33

, and third quartile

Q_3=43

. An outlier in a data set is a value that is more than

1.5

times the interquartile range below the first quartile (

Q_1

) or more than

1.5

times the interquartile range above the third quartile (

Q_3

), where the interquartile range is defined as

Q_3-Q_1

. How many outliers does this data set have?

<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

6

1

The circle and the x-axis

The circle having

(0,0)

and

(8,6)

as the endpoints of a diameter intersects the

x

-axis at a second point. What is the

x

-coordinate of this point?

<span class='latex-bold'>(A)</span>\ 4\sqrt2 \qquad <span class='latex-bold'>(B)</span>\ 6\qquad <span class='latex-bold'>(C)</span>\ 5\sqrt2 \qquad <span class='latex-bold'>(D)</span>\ 8 \qquad <span class='latex-bold'>(E)</span>\ 6\sqrt2

9

1

Too Beautiful to be Wrong

A circle has center

(-10,-4)

and radius

13

. Another circle has center

(3,9)

and radius

\sqrt{65}

. The line passing through the two points of intersection of the two circles has equation

x+y=c

. What is

c

?

<span class='latex-bold'>(A)</span> \text{ 3} \qquad <span class='latex-bold'>(B)</span> \text{ } 3 \sqrt{3} \qquad <span class='latex-bold'>(C)</span> \text{ } 4\sqrt{2} \qquad <span class='latex-bold'>(D)</span> \text{ 6} \qquad <span class='latex-bold'>(E)</span> \text{ }\frac{13}{2}

8

1

Please read carefully

The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?

<span class='latex-bold'>(A)</span> \text{ } \frac{\sqrt{3}-1}{2} \qquad <span class='latex-bold'>(B)</span> \text{ } \frac{1}{2} \qquad <span class='latex-bold'>(C)</span> \text{ } \frac{\sqrt{5}-1}{2} \qquad <span class='latex-bold'>(D)</span> \text{ } \frac{\sqrt{2}}{2} \qquad <span class='latex-bold'>(E)</span> \text{ } \frac{\sqrt{6}-1}{2}

12

1

Complex Dodecagon

What is the sum of the roots of

z^{12} = 64

that have a positive real part?

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

23

1

This polynomial is so extra

The graph of

y=f(x)

, where

f(x)

is a polynomial of degree

3

, contains points

A(2,4)

,

B(3,9)

, and

C(4,16)

. Lines

AB

,

AC

, and

BC

intersect the graph again at points

D

,

E

, and

F

, respectively, and the sum of the

x

-coordinates of

D

,

E

, and

F

is

24

. What is

f(0)

?

<span class='latex-bold'>(A) </span> -2 \qquad <span class='latex-bold'>(B) </span> 0 \qquad <span class='latex-bold'>(C) </span> 2 \qquad <span class='latex-bold'>(D) </span> \frac{24}{5} \qquad <span class='latex-bold'>(E) </span> 8

1

Comic Book Collections

Kymbrea's comic book collection currently has

30

comic books in it, and she is adding to her collection at the rate of

2

comic books per month. LaShawn's comic book collection currently has

10

comic books in it, and he is adding to his collection at the rate of

6

comic books per month. After how many months will LaShawn's collection have twice as many comic books as Kymbrea's?

<span class='latex-bold'>(A)</span>\ 1\qquad<span class='latex-bold'>(B)</span>\ 4\qquad<span class='latex-bold'>(C)</span>\ 5\qquad<span class='latex-bold'>(D)</span>\ 20\qquad<span class='latex-bold'>(E)</span>\ 25

16

1

Circle Inscribed in Semicircle

In the figure below, semicircles with centers at

A

and

B

and with radii

2

and

1

, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter

\overline{JK}

. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at

P

is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at

P

?
[asy] size(8cm); draw(arc((0,0),3,0,180)); draw(arc((2,0),1,0,180)); draw(arc((-1,0),2,0,180)); draw((-3,0)--(3,0)); pair P = (-1,0)+(2+6/7)*dir(36.86989); draw(circle(P,6/7)); dot((-1,0)); dot((2,0)); dot((-3,0)); dot((3,0)); dot(P); label("

J

",(-3,0),W); label("

A

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

B

",(2,0),NE); label("

K

",(3,0),E); label("

P

",P,NW); [/asy]

<span class='latex-bold'>(A)</span>\ \frac{3}{4} \qquad <span class='latex-bold'>(B)</span>\ \frac{6}{7} \qquad<span class='latex-bold'>(C)</span>\ \frac{1}{2}\sqrt{3} \qquad<span class='latex-bold'>(D)</span>\ \frac{5}{8}\sqrt{2} \qquad<span class='latex-bold'>(E)</span>\ \frac{11}{12}

20

2

Solving for the Log

How many ordered pairs

(a, b)

such that

a

is a real positive number and

b

is an integer between

2

and

200

, inclusive, satisfy the equation

(\log_b a)^{2017} = \log_b (a^{2017})

?

<span class='latex-bold'>(A) \ </span>198\qquad <span class='latex-bold'>(B) \ </span> 199 \qquad <span class='latex-bold'>(C) \ </span> 398 \qquad <span class='latex-bold'>(D) \ </span>399\qquad <span class='latex-bold'>(E) \ </span> 597

A Proof Without Words

Real numbers

x

and

y

are chosen independently and uniformly at random from the interval

(0,1)

. What is the probability that

\lfloor \log_2{x} \rfloor=\lfloor \log_2{y} \rfloor

, where

\lfloor r \rfloor

denotes the greatest integer less than or equal to the real number

r

?

<span class='latex-bold'>(A)</span>\ \frac{1}{8}\qquad<span class='latex-bold'>(B)</span>\ \frac{1}{6}\qquad<span class='latex-bold'>(C)</span>\ \frac{1}{4}\qquad<span class='latex-bold'>(D)</span>\ \frac{1}{3}\qquad<span class='latex-bold'>(E)</span>\ \frac{1}{2}

7

2

Recursive Function

Define a function on the positive integers recursively by

f(1) = 2

,

f(n) = f(n-1) + 1

if

n

is even, and

f(n) = f(n-2) + 2

if

n

is odd and greater than

1

. What is

f(2017)

?

<span class='latex-bold'>(A) </span> 2017 \qquad <span class='latex-bold'>(B) </span> 2018 \qquad <span class='latex-bold'>(C) </span> 4034 \qquad <span class='latex-bold'>(D) </span> 4035 \qquad <span class='latex-bold'>(E) </span> 4036

Period of Trigonometric Function

The functions

\sin(x)

and

\cos(x)

are periodic with least period

2\pi

. What is the least period of the function

\cos(\sin(x))

?

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

It's not periodic.

21

1

Set Construction

A set

S

is constructed as follows. To begin,

S=\{0,10\}

. Repeatedly, as long as possible, if

x

is an integer root of some polynomial

a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0

for some

n\geq 1

, all of whose coefficients

a_i

are elements of

S

, then

x

is put into

S

. When no more elements can be added to

S

, how many elements does

S

have?

<span class='latex-bold'>(A) </span> 4 \qquad <span class='latex-bold'>(B) </span> 5 \qquad <span class='latex-bold'>(C) </span> 7 \qquad <span class='latex-bold'>(D) </span> 9 \qquad <span class='latex-bold'>(E) </span> 11

11

1

Forgetful Claire

Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of

2017

. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?

<span class='latex-bold'>(A)</span>\ 37\qquad<span class='latex-bold'>(B)</span>\ 63\qquad<span class='latex-bold'>(C)</span>\ 117\qquad<span class='latex-bold'>(D)</span>\ 143\qquad<span class='latex-bold'>(E)</span>\ 163

22

2

An English Problem

A square is drawn in the Cartesian coordinate plane with vertices at

(2,2)

,

(-2,2)

,

(-2,-2)

, and

(2,-2)

. A particle starts at

(0,0)

. Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is

\frac{1}{8}

that the particle will move from

(x,y)

to each of

(x,y+1)

,

(x+1,y+1)

,

(x+1,y)

,

(x+1,y-1)

,

(x,y-1)

,

(x-1,y-1)

,

(x-1,y)

,

(x-1,y+1)

. The particle will eventually hit the square for the first time, either at one of the

4

corners of the square or one of the

12

lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is

\frac{m}{n}

, where

m

and

n

are relatively prime positive integers. What is

m+n

?

<span class='latex-bold'>(A)</span>\ 4\qquad<span class='latex-bold'>(B)</span>\ 5\qquad<span class='latex-bold'>(C)</span>\ 7\qquad<span class='latex-bold'>(D)</span>\ 15\qquad<span class='latex-bold'>(E)</span>\ 39

Coin game

Abby, Bernardo, Carl, and Debra play a game in which each of them starts with four coins. The game consists of four rounds. In each round, four balls are placed in an urn - one green, one red, and two white. The players each draw a ball at random without replacement. Whoever gets the green ball gives one coin to whoever gets the red ball. What is the probability that, at the end of the fourth round, each of the players has four coins?

<span class='latex-bold'>(A)</span> \dfrac{7}{576} \qquad <span class='latex-bold'>(B)</span> \dfrac{5}{192} \qquad <span class='latex-bold'>(C)</span> \dfrac{1}{36} \qquad <span class='latex-bold'>(D)</span> \dfrac{5}{144} \qquad <span class='latex-bold'>(E)</span>\dfrac{7}{48}

15

1

T R I G g e r e d

Let

f(x)=\sin x+2\cos x+3\tan x

, using radian measure for the variable

x

. In what interval does the smallest positive value of

x

for which

f(x)=0

lie?

<span class='latex-bold'>(A) </span> (0,1) \qquad <span class='latex-bold'>(B) </span> (1,2) \qquad <span class='latex-bold'>(C) </span> (2,3) \qquad <span class='latex-bold'>(D) </span> (3,4) \qquad <span class='latex-bold'>(E) </span> (4,5)

13

1

Sharon's Long Drive

Driving at a constant speed, Sharon usually takes

180

minutes to drive from her house to her mother's house. One day Sharon begins the drive at her usual speed, but after driving

\frac{1}{3}

of the way, she hits a bad snowstorm and reduces her speed by

20

miles per hour. This time the trip takes her a total of

276

minutes. How many miles is it from Sharon's house to her mother's house?

<span class='latex-bold'>(A)</span>\ 132\qquad<span class='latex-bold'>(B)</span>\ 135\qquad<span class='latex-bold'>(C)</span>\ 138\qquad<span class='latex-bold'>(D)</span>\ 141\qquad<span class='latex-bold'>(E)</span>\ 144

25

2

Crystal Math

The vertices

V

of a centrally symmetric hexagon in the complex plane are given by

V=\left\{ \sqrt{2}i,-\sqrt{2}i, \frac{1}{\sqrt{8}}(1+i),\frac{1}{\sqrt{8}}(-1+i),\frac{1}{\sqrt{8}}(1-i),\frac{1}{\sqrt{8}}(-1-i) \right\}.

For each

j

,

1\leq j\leq 12

, an element

z_j

is chosen from

V

at random, independently of the other choices. Let

P={\prod}_{j=1}^{12}z_j

be the product of the

12

numbers selected. What is the probability that

P=-1

?

<span class='latex-bold'>(A) </span> \dfrac{5\cdot11}{3^{10}} \qquad <span class='latex-bold'>(B) </span> \dfrac{5^2\cdot11}{2\cdot3^{10}} \qquad <span class='latex-bold'>(C) </span> \dfrac{5\cdot11}{3^{9}} \qquad <span class='latex-bold'>(D) </span> \dfrac{5\cdot7\cdot11}{2\cdot3^{10}} \qquad <span class='latex-bold'>(E) </span> \dfrac{2^2\cdot5\cdot11}{3^{10}}

AJ basketball???

A set of

n

people participate in an online video basketball tournament. Each person may be a member of any number of

5

-player teams, but no two teams may have exactly the same

5

members. The site statistics show a curious fact: The average, over all subsets of size

9

of the set of

n

participants, of the number of complete teams whose members are among those

9

people is equal to the reciprocal of the average, over all subsets of size

8

of the set of

n

participants, of the number of complete teams whose members are among those

8

people. How many values

n

,

9\leq n\leq 2017

, can be the number of participants?

<span class='latex-bold'>(A) </span> 477 \qquad <span class='latex-bold'>(B) </span> 482 \qquad <span class='latex-bold'>(C) </span> 487 \qquad <span class='latex-bold'>(D) </span> 557 \qquad <span class='latex-bold'>(E) </span> 562

24

2

Product of Lengths

Quadrilateral

ABCD

is inscribed in circle

O

and has sides

AB = 3

,

BC = 2

,

CD = 6

, and

DA = 8

. Let

X

and

Y

be points on

\overline{BD}

such that \frac{DX}{BD} = \frac{1}{4} \text{and} \frac{BY}{BD} = \frac{11}{36}. Let

E

be the intersection of intersection of line

AX

and the line through

Y

parallel to

\overline{AD}

. Let

F

be the intersection of line

CX

and the line through

E

parallel to

\overline{AC}

. Let

G

be the point on circle

O

other than

C

that lies on line

CX

. What is

XF \cdot XG

?

<span class='latex-bold'>(A) </span>17\qquad<span class='latex-bold'>(B) </span>\frac{59 - 5\sqrt{2}}{3}\qquad<span class='latex-bold'>(C) </span>\frac{91 - 12\sqrt{3}}{4}\qquad<span class='latex-bold'>(D) </span>\frac{67 - 10\sqrt{2}}{3}\qquad<span class='latex-bold'>(E) </span>18

funny trapezoid

Quadrilateral

ABCD

has right angles at

B

and

C

,

\triangle ABC \sim \triangle BCD

, and

AB > BC

. There is a point

E

in the interior of

ABCD

such that

\triangle ABC \sim \triangle CEB

and the area of

\triangle AED

is

17

times the area of

\triangle CEB

. What is

\tfrac{AB}{BC}

?

<span class='latex-bold'>(A) \ </span> 1+\sqrt{2} \qquad <span class='latex-bold'>(B) \ </span> 2+\sqrt{2}\qquad <span class='latex-bold'>(C) \ </span> \sqrt{17}\qquad <span class='latex-bold'>(D) \ </span> 2+\sqrt{5} \qquad <span class='latex-bold'>(E) \ </span> 1+2\sqrt{3}

17

2

I think I spoiled this one

There are 24 different complex numbers

z

such that

z^{24} = 1

. For how many of these is

z^6

a real number?

<span class='latex-bold'>(A) </span>1\qquad<span class='latex-bold'>(B) </span>3\qquad<span class='latex-bold'>(C) </span>6\qquad<span class='latex-bold'>(D) </span>12\qquad<span class='latex-bold'>(E) </span>24

Two Games

A coin is biased in such a way that on each toss the probability of heads is

\frac{2}{3}

and the probability of tails is

\frac{1}{3}

. The outcomes of the tosses are independent. A player has the choice of playing Game A or Game B. In Game A she tosses the coin three times and wins if all three outcomes are the same. In Game B she tosses the coin four times and wins if both the outcomes of the first and second tosses are the same and the outcomes of the third and fourth tosses are the same. How do the chances of winning Game A compare to the chances of winning Game B?

<span class='latex-bold'>(A)</span> \text{ The probability of winning Game A is }\frac{4}{81}\text{ less than the probability of winning Game B.}

<span class='latex-bold'>(B)</span> \text{ The probability of winning Game A is }\frac{2}{81}\text{ less than the probability of winning Game B.}

<span class='latex-bold'>(C)</span> \text{ The probabilities are the same.}

<span class='latex-bold'>(D)</span> \text{ The probability of winning Game A is }\frac{2}{81}\text{ greater than the probability of winning Game B.}

<span class='latex-bold'>(E)</span> \text{ The probability of winning Game A is }\frac{4}{81}\text{ greater than the probability of winning Game B.}

2017 AMC 12/AHSME

Subcontests

We All Scream For Ice Cream

Outliers in Data Set

The circle and the x-axis

Too Beautiful to be Wrong

Please read carefully

Complex Dodecagon

This polynomial is so extra

Comic Book Collections

Circle Inscribed in Semicircle

Solving for the Log

A Proof Without Words

Recursive Function

Period of Trigonometric Function

Set Construction

Forgetful Claire

An English Problem

Coin game

T R I G g e r e d

Sharon's Long Drive

Crystal Math

AJ basketball???

Product of Lengths

funny trapezoid

I think I spoiled this one

Two Games