MathDB

1

Circle, chords, and area of the circle

\overline{AB}

and

\overline{CD}

are diameters of the circle with center

O

,

\overline{AB} \perp \overline{CD}

, and chord

\overline{DF}

intersects

\overline{AB}

at

E

. If

DE = 6

and

EF = 2

, then the area of the circle is [asy] size(120); defaultpen(linewidth(0.7)); pair O=origin, A=(-5,0), B=(5,0), C=(0,5), D=(0,-5), F=5*dir(40), E=intersectionpoint(A--B, F--D); draw(Circle(O, 5)); draw(A--B^^C--D--F); dot(O^^A^^B^^C^^D^^E^^F); markscalefactor=0.05; draw(rightanglemark(B, O, D)); label("

A

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

B

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

C

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

D

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

F

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

O

", O, NW); label("

E

", E, SE);[/asy]

<span class='latex-bold'>(A)</span>\ 23\pi \qquad <span class='latex-bold'>(B)</span>\ \dfrac{47}{2}\pi \qquad <span class='latex-bold'>(C)</span>\ 24\pi \qquad <span class='latex-bold'>(D)</span>\ \dfrac{49}{2}\pi \qquad <span class='latex-bold'>(E)</span>\ 25\pi

22

1

Cutting triangle to form a pentagon!

A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths

13,19,20,25

and

31

, although this is not necessarily their order around the pentagon. The area of the pentagon is

<span class='latex-bold'>(A)</span>\ 459 \qquad <span class='latex-bold'>(B)</span>\ 600 \qquad <span class='latex-bold'>(C)</span>\ 680 \qquad <span class='latex-bold'>(D)</span>\ 720\qquad <span class='latex-bold'>(E)</span>\ 745

23

1

Triangle equality and obtuse triangle problem

The sides of a triangle have lengths

11

,

15

, and

k

, where

k

is an integer. For how many values of

k

is the triangle obtuse?

<span class='latex-bold'>(A)</span>\ 5 \qquad <span class='latex-bold'>(B)</span>\ 7 \qquad <span class='latex-bold'>(C)</span>\ 12 \qquad <span class='latex-bold'>(D)</span>\ 13 \qquad <span class='latex-bold'>(E)</span>\ 14

25

1

Range, mean, mode, and median

A list of five positive integers has mean

12

and range

18

. The mode and median are both

8

. How many different values are possible for the second largest element of the list?

<span class='latex-bold'>(A)</span>\ 4\qquad <span class='latex-bold'>(B)</span>\ 6 \qquad <span class='latex-bold'>(C)</span>\ 8\qquad <span class='latex-bold'>(D)</span>\ 10\qquad <span class='latex-bold'>(E)</span>\ 12

24

1

Logarithm equation

There exist positive integers

A,B

and

C

, with no common factor greater than

1

, such that

A \log_{200} 5 + B \log_{200} 2 = C.

What is

A+B+C

?

<span class='latex-bold'>(A)</span>\ 6 \qquad <span class='latex-bold'>(B)</span>\ 7\qquad <span class='latex-bold'>(C)</span>\ 8 \qquad <span class='latex-bold'>(D)</span>\ 9 \qquad <span class='latex-bold'>(E)</span>\ 10

28

1

Chord length of sqrt{a}

Two parallel chords in a circle have lengths

10

and

14

, and the distance between them is

6

. The chord parallel to these chords and midway between them is of length

\sqrt{a}

where

a

is
[asy] // note: diagram deliberately not to scale -- azjps void htick(pair A, pair B, real r){ D(A--B); D(A-(r,0)--A+(r,0)); D(B-(r,0)--B+(r,0)); } size(120); pathpen = linewidth(0.7); pointpen = black+linewidth(3); real min = -0.6, step = 0.5; pair[] A, B; D(unitcircle); for(int i = 0; i < 3; ++i) { A.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[0]); B.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[1]); D(D(A)--D(B)); } MP("10",(A[0]+B[0])/2,N); MP("\sqrt{a}",(A[1]+B[1])/2,N); MP("14",(A[2]+B[2])/2,N); htick((B[1].x+0.1,B[0].y),(B[1].x+0.1,B[2].y),0.06); MP("6",(B[1].x+0.1,B[0].y/2+B[2].y/2),E);[/asy]

<span class='latex-bold'>(A)</span>\ 144 \qquad <span class='latex-bold'>(B)</span>\ 156 \qquad <span class='latex-bold'>(C)</span>\ 168 \qquad <span class='latex-bold'>(D)</span>\ 176 \qquad <span class='latex-bold'>(E)</span>\ 184

27

1

Triangular array of numbers and 100th row

Consider the triangular array of numbers with

0,1,2,3,...

along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows

1

through

6

are shown.
\begin{tabular}{ccccccccccc} & & & & & 0 & & & & & \\ & & & & 1 & & 1 & & & & \\ & & & 2 & & 2 & & 2 & & & \\ & & 3 & & 4 & & 4 & & 3 & & \\ & 4 & & 7 & & 8 & & 7 & & 4 & \\ 5 & & 11 & & 15 & & 15 & & 11 & & 5 \end{tabular}
Let

f(n)

denote the sum of the numbers in row

n

. What is the remainder when

f(100)

is divided by

100

?

<span class='latex-bold'>(A)</span>\ 12\qquad <span class='latex-bold'>(B)</span>\ 30 \qquad <span class='latex-bold'>(C)</span>\ 50 \qquad <span class='latex-bold'>(D)</span>\ 62 \qquad <span class='latex-bold'>(E)</span>\ 74

29

1

Set a,b,c

For how many three-element sets of positive integers

\{a,b,c\}

is it true that

a \times b \times c = 2310

?

<span class='latex-bold'>(A)</span>\ 32 \qquad <span class='latex-bold'>(B)</span>\ 36 \qquad <span class='latex-bold'>(C)</span>\ 40 \qquad <span class='latex-bold'>(D)</span>\ 43 \qquad <span class='latex-bold'>(E)</span>\ 45

21

1

Rectangles and nonadjacent given points

Two nonadjacent vertices of a rectangle are

(4,3)

and

(-4,-3)

, and the coordinates of the other two vertices are integers. The number of such rectangles is

<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

20

1

Parity probability of ab+c

If

a,b

and

c

are three (not necessarily different) numbers chosen randomly and with replacement from the set

\{1,2,3,4,5 \}

, the probability that

ab+c

is even is

<span class='latex-bold'>(A)</span>\ \dfrac{2}{5} \qquad <span class='latex-bold'>(B)</span>\ \dfrac{59}{125} \qquad <span class='latex-bold'>(C)</span>\ \dfrac{1}{2} \qquad <span class='latex-bold'>(D)</span>\ \dfrac{64}{125} \qquad <span class='latex-bold'>(E)</span>\ \dfrac{3}{5}

18

1

Two rays

Two rays with common endpoint

O

forms a

30^\circ

angle. Point

A

lies on one ray, point

B

on the other ray, and

AB = 1

. The maximum possible length of

OB

is

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

19

1

Ratio of two equilateral triangles' area

Equilateral triangle

DEF

is inscribed in equilateral triangle

ABC

such that

\overline{DE} \perp \overline{BC}

. The ratio of the area of

\triangle DEF

to the area of

\triangle ABC

is [asy] size(180); pathpen = linewidth(0.7); pointpen = black; pointfontpen = fontsize(10); pair B = (0,0), C = (1,0), A = dir(60), D = C*2/3, E = (2*A+C)/3, F = (2*B+A)/3; D(D("A",A,N)--D("B",B,SW)--D("C",C,SE)--cycle); D(D("D",D)--D("E",E,NE)--D("F",F,NW)--cycle); D(rightanglemark(C,D,E,1.5));[/asy]

<span class='latex-bold'>(A)</span>\ \dfrac{1}{6}\qquad <span class='latex-bold'>(B)</span>\ \dfrac{1}{4} \qquad <span class='latex-bold'>(C)</span>\ \dfrac{1}{3} \qquad <span class='latex-bold'>(D)</span>\ \dfrac{2}{5} \qquad <span class='latex-bold'>(E)</span>\ \dfrac{1}{2}

17

1

Pentagon and circle

Given regular pentagon

ABCDE

, a circle can be drawn that is tangent to

\overline{DC}

at

D

and to

\overline{AB}

at

A

. The number of degrees in minor arc

AD

is

<span class='latex-bold'>(A)</span>\ 72 \qquad <span class='latex-bold'>(B)</span>\ 108 \qquad <span class='latex-bold'>(C)</span>\ 120 \qquad <span class='latex-bold'>(D)</span>\ 135 \qquad <span class='latex-bold'>(E)</span>\ 144

[asy] size(100); defaultpen(linewidth(0.7)); draw(rotate(18)*polygon(5)); real x=0.6180339887; draw(Circle((-x,0), 1)); int i; for(i=0; i<5; i=i+1) { dot(origin+1*dir(36+72*i)); } label("

B

", origin+1*dir(36+72*0), dir(origin--origin+1*dir(36+72*0))); label("

A

", origin+1*dir(36+72*1), dir(origin--origin+1*dir(36+72))); label("

E

", origin+1*dir(36+72*2), dir(origin--origin+1*dir(36+144))); label("

D

", origin+1*dir(36+72*3), dir(origin--origin+1*dir(36+72*3))); label("

C

", origin+1*dir(36+72*4), dir(origin--origin+1*dir(36+72*4)));[/asy]

15

1

Five points and bug jumping!

Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the bug begins on point 5, after 1995 jumps it will be on point
[asy] size(80); defaultpen(linewidth(0.7)+fontsize(10)); draw(unitcircle); for(int i = 0; i < 5; ++i) { pair P = dir(90+i*72); dot(P); label("

"+string(i+1)+"

",P,1.4*P); }[/asy]

<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

13

1

Addition error

The addition below is incorrect. The display can be made correct by changing one digit

d

, wherever it occurs, to another digit

e

. Find the sum of

d

and

e

. \begin{tabular}{ccccccc}& 7 & 4 & 2 & 5 & 8 & 6 \\ + & 8 & 2 & 9 & 4 & 3 & 0\\ \hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}

<span class='latex-bold'>(A)</span>\ 4 \qquad <span class='latex-bold'>(B)</span>\ 6 \qquad <span class='latex-bold'>(C)</span>\ 8 \qquad <span class='latex-bold'>(D)</span>\ 10 \qquad <span class='latex-bold'>(E)</span>\ \text{More than } 10

14

1

Two function values

If

f(x) = ax^4-bx^2+x+5

and

f(-3) = 2

, then

f(3) =

<span class='latex-bold'>(A)</span>\ -5 \qquad <span class='latex-bold'>(B)</span>\ -2 \qquad <span class='latex-bold'>(C)</span>\ 1 \qquad <span class='latex-bold'>(D)</span>\ 3 \qquad <span class='latex-bold'>(E)</span>\ 8

11

1

Three conditions and number

How many base 10 four-digit numbers,

N = \underline{a} \underline{b} \underline{c} \underline{d}

, satisfy all three of the following conditions?
(i)

4,000 \leq N < 6,000;

(ii)

N

is a multiple of 5; (iii)

3 \leq b < c \leq 6

.

\mathbf{(A)}\; 10\qquad \mathbf{(B)}\; 18\qquad \mathbf{(C)}\; 24\qquad \mathbf{(D)}\; 36\qquad \mathbf{(E)}\; 48

10

1

Three lines and area formed by it.. "classic"

The area of the triangle bounded by the lines

y = x, y = -x

and

y = 6

is

\mathbf{(A)}\; 12\qquad \mathbf{(B)}\; 12\sqrt2\qquad \mathbf{(C)}\; 24\qquad \mathbf{(D)}\; 24\sqrt2\qquad \mathbf{(E)}\; 36

9

1

Number of triangles in drawing

Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is
[asy] size(100); defaultpen(linewidth(0.7)); draw(unitsquare^^(0,0)--(1,1)^^(0,1)--(1,0)^^(.5,0)--(.5,1)^^(0,.5)--(1,.5));[/asy]

<span class='latex-bold'>(A)</span>\ 10 \qquad <span class='latex-bold'>(B)</span>\ 12 \qquad <span class='latex-bold'>(C)</span>\ 14 \qquad <span class='latex-bold'>(D)</span>\ 16 \qquad <span class='latex-bold'>(E)</span>\ 18

8

1

Triangle and BD

In

\triangle ABC

,

\angle C = 90^\circ, AC = 6

and

BC = 8

. Points

D

and

E

are on

\overline{AB}

and

\overline{BC}

, respectively, and

\angle BED = 90^\circ

. If

DE = 4

, then

BD =

[asy] size(100); pathpen = linewidth(0.7); pointpen = black+linewidth(3); pair A = (0,0), C = (6,0), B = (6,8), D = (2*A+B)/3, E = (2*C+B)/3; D(D("A",A,SW)--D("B",B,NW)--D("C",C,SE)--cycle); D(D("D",D,NW)--D("E",E,plain.E)); D(rightanglemark(D,E,B,16)); D(rightanglemark(A,C,B,16));[/asy]

\mathbf{(A)}\;5\qquad \mathbf{(B)}\;\frac{16}{3}\qquad \mathbf{(C)}\; \frac{20}{3}\qquad \mathbf{(D)}\; \frac{15}{2}\qquad \mathbf{(E)}\; 8

16

1

Estimation

Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:
i. The actual attendance in Atlanta is within

10 \%

of Anita's estimate. ii. Bob's estimate is within

10 \%

of the actual attendance in Boston.
To the nearest 1,000, the largest possible difference between the numbers attending the two games is

<span class='latex-bold'>(A)</span>\ 10,000 \qquad <span class='latex-bold'>(B)</span>\ 11,000 \qquad <span class='latex-bold'>(C)</span>\ 20,000 \qquad <span class='latex-bold'>(D)</span>\ 21,000 \qquad <span class='latex-bold'>(E)</span>\ 22,000

30

1

Very hard plane intersecting

A large cube is formed by stacking

27

unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is [asy] size(120); defaultpen(linewidth(0.7)); pair slant = (2,1); for(int i = 0; i < 4; ++i) draw((0,i)--(3,i)^^(i,0)--(i,3)^^(3,i)--(3,i)+slant^^(i,3)--(i,3)+slant); for(int i = 1; i < 4; ++i) draw((0,3)+slant*i/3--(3,3)+slant*i/3^^(3,0)+slant*i/3--(3,3)+slant*i/3);[/asy]

<span class='latex-bold'>(A)</span>\ 16\qquad <span class='latex-bold'>(B)</span>\ 17 \qquad <span class='latex-bold'>(C)</span>\ 18 \qquad <span class='latex-bold'>(D)</span>\ 19 \qquad <span class='latex-bold'>(E)</span>\ 20

6

1

Cube and unknown opposite face

The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked

x

?
[asy] defaultpen(linewidth(0.7)); path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3); draw(p^^(2,3)--(4,3)^^shift(2,0)*p^^(2,0)--origin); draw(shift(1,0)*p, dashed); label("

x

", (0.3,0.5), E); label("

A

", (1.3,0.5), E); label("

B

", (1.3,1.5), E); label("

C

", (2.3,1.5), E); label("

D

", (2.3,2.5), E); label("

E

", (3.3,2.5), E);[/asy]

\mathbf{(A)}\; A\qquad \mathbf{(B)}\; B\qquad \mathbf{(C)}\; C\qquad \mathbf{(D)}\; D\qquad \mathbf{(E)}\; E

5

1

Unit conversion and ants

A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field is

<span class='latex-bold'>(A)</span>\ \text{500 thousand} \qquad <span class='latex-bold'>(B)</span>\ \text{5 million} \qquad <span class='latex-bold'>(C)</span>\ \text{50 million} \qquad <span class='latex-bold'>(D)</span>\ \text{500 million} \qquad <span class='latex-bold'>(E)</span>\ \text{5 billion}

4

1

M,q,p,n

If

M

is

30 \%

of

Q

,

Q

is

20 \%

of

P

, and

N

is

50 \%

of

P

, then

\frac{M}{N} =

<span class='latex-bold'>(A)</span>\ \frac{3}{250} \qquad <span class='latex-bold'>(B)</span>\ \frac{3}{25} \qquad <span class='latex-bold'>(C)</span>\ 1 \qquad <span class='latex-bold'>(D)</span>\ \displaystyle \frac{6}{5} \qquad <span class='latex-bold'>(E)</span>\ \displaystyle \frac{4}{3}

3

1

Buy in television commericial than store! :-d

The total in-store price for an appliance is

\$99.99

. A television commercial advertises the same product for three easy payments of

\$29.98

and a one-time shipping and handling charge of

\$9.98

. How much is saved by buying the appliance from the television advertiser?

<span class='latex-bold'>(A)</span>\ \text{6 cents} \qquad <span class='latex-bold'>(B)</span>\ \text{7 cents} \qquad <span class='latex-bold'>(C)</span>\ \text{8 cents} \qquad <span class='latex-bold'>(D)</span>\ \text{9 cents} \qquad <span class='latex-bold'>(E)</span>\ \text{10 cents}

2

1

Simple square root

If

\sqrt{2 + \sqrt{x}} = 3

, then

x =

<span class='latex-bold'>(A)</span>\ 1 \qquad <span class='latex-bold'>(B)</span>\ \sqrt{7} \qquad <span class='latex-bold'>(C)</span>\ 7 \qquad <span class='latex-bold'>(D)</span>\ 49 \qquad <span class='latex-bold'>(E)</span>\ 121

1

Kim's average score

Kim earned scores of 87,83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will

<span class='latex-bold'>(A)</span>\ \text{remain the same} \qquad <span class='latex-bold'>(B)</span>\ \text{increase by 1} \qquad <span class='latex-bold'>(C)</span>\ \text{increase by 2} \qquad <span class='latex-bold'>(D)</span>\ \text{increase by 3} \qquad <span class='latex-bold'>(E)</span>\ \text{increase by 4}

12

1

Problem of the Day

Hi guys, I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this: 1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though. 2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary. 3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions: A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh? B. Do NOT go back to the previous problem(s). This causes too much confusion. C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for. 4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving! Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D

7

1

Easy challenge

The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a neglibile height above the equator, then, among the following choices, the best estimate of the number of hours of flight is:

<span class='latex-bold'>(A)</span>\ 25 \qquad <span class='latex-bold'>(B)</span>\ 8 \qquad <span class='latex-bold'>(C)</span>\ 75 \qquad <span class='latex-bold'>(D)</span>\ 50 \qquad <span class='latex-bold'>(E)</span>\ 100

1995 AMC 12/AHSME

Subcontests

Circle, chords, and area of the circle

Cutting triangle to form a pentagon!

Triangle equality and obtuse triangle problem

Range, mean, mode, and median

Logarithm equation

Chord length of sqrt{a}

Triangular array of numbers and 100th row

Set a,b,c

Rectangles and nonadjacent given points

Parity probability of ab+c

Two rays

Ratio of two equilateral triangles' area

Pentagon and circle

Five points and bug jumping!

Addition error

Two function values

Three conditions and number

Three lines and area formed by it.. "classic"

Number of triangles in drawing

Triangle and BD

Estimation

Very hard plane intersecting

Cube and unknown opposite face

Unit conversion and ants

M,q,p,n

Buy in television commericial than store! :-d

Simple square root

Kim's average score

Problem of the Day

Easy challenge

1995 AMC 12/AHSME

Subcontests

Circle, chords, and area of the circle

Cutting triangle to form a pentagon!

Triangle equality and obtuse triangle problem

Range, mean, mode, and median

Logarithm equation

Chord length of sqrt{a}

Triangular array of numbers and 100th row

Set a,b,c

Rectangles and nonadjacent given points

Parity probability of ab+c

Two rays

Ratio of two equilateral triangles' area

Pentagon and circle

Five points and bug jumping!

Addition error

Two function values

Three conditions and number

Three lines and area formed by it.. &quot;classic&quot;

Number of triangles in drawing

Triangle and BD

Estimation

Very hard plane intersecting

Cube and unknown opposite face

Unit conversion and ants

M,q,p,n

Buy in television commericial than store! :-d

Simple square root

Kim's average score

Problem of the Day

Easy challenge

Three lines and area formed by it.. "classic"