MathDB

1

Sequence including 2001

Consider sequences of positive real numbers of the form

x,2000,y,...,

in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of

x

does the term 2001 appear somewhere in the sequence?

<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> \ \text{more than 4}

22

1

Geometry (Out of unique title for this one)

In rectangle

ABCD

, points

F

and

G

lie on

\overline{AB}

so that AF \equal{} FG \equal{} GB and

E

is the midpoint of

\overline{DC}

. Also,

\overline{AC}

intersects

\overline{EF}

at

H

and

\overline{EG}

at

J

. The area of the rectangle

ABCD

is

70

. Find the area of triangle

EHJ

. [asy] size(180); pair A, B, C, D, E, F, G, H, J; A = origin; real length = 6; real width = 3.5; B = length*dir(0); C = (length, width); D = width*dir(90); F = length/3*dir(0); G = 2*length/3*dir(0); E = (length/2, width); H = extension(A, C, E, F); J = extension(A, C, E, G);
draw(A--B--C--D--cycle); draw(G--E--F); draw(A--C);
label("

A

", A, dir(180)); label("

D

", D, dir(180)); label("

B

", B, dir(0)); label("

C

", C, dir(0)); label("

F

", F, dir(270)); label("

E

", E, dir(90)); label("

G

", G, dir(270)); label("

H

", H, dir(140)); label("

J

", J, dir(340)); [/asy]

\displaystyle <span class='latex-bold'>(A)</span> \ \frac {5}{2} \qquad <span class='latex-bold'>(B)</span> \ \frac {35}{12} \qquad <span class='latex-bold'>(C)</span> \ 3 \qquad <span class='latex-bold'>(D)</span> \ \frac {7}{2} \qquad <span class='latex-bold'>(E)</span> \ \frac {35}{8}

20

1

Square made by midpoints

Points A \equal{} (3,9), B \equal{} (1,1), C \equal{} (5,3), and D \equal{} (a,b) lie in the first quadrant and are the vertices of quadrilateral

ABCD

. The quadrilateral formed by joining the midpoints of

\overline{AB}, \overline{BC}, \overline{CD},

and

\overline{DA}

is a square. What is the sum of the coordinates of point

D

?

<span class='latex-bold'>(A)</span> \ 7 \qquad <span class='latex-bold'>(B)</span> \ 9 \qquad <span class='latex-bold'>(C)</span> \ 10 \qquad <span class='latex-bold'>(D)</span> \ 12 \qquad <span class='latex-bold'>(E)</span> \ 16

24

1

Triangle and simple line

In

\triangle ABC

, \angle ABC \equal{} 45^\circ. Point

D

is on

\overline{BC}

so that 2 \cdot BD \equal{} CD and \angle DAB \equal{} 15^\circ. Find

\angle ACB

. [asy] pair A, B, C, D; A = origin; real Bcoord = 3*sqrt(2) + sqrt(6); B = Bcoord/2*dir(180); C = sqrt(6)*dir(120); draw(A--B--C--cycle); D = (C-B)/2.4 + B; draw(A--D);
label("

A

", A, dir(0)); label("

B

", B, dir(180)); label("

C

", C, dir(110)); label("

D

", D, dir(130)); [/asy]

<span class='latex-bold'>(A)</span> \ 54^\circ \qquad <span class='latex-bold'>(B)</span> \ 60^\circ \qquad <span class='latex-bold'>(C)</span> \ 72^\circ \qquad <span class='latex-bold'>(D)</span> \ 75^\circ \qquad <span class='latex-bold'>(E)</span> \ 90^\circ

18

1

Third circle radius

A circle centered at

A

with a radius of 1 and a circle centered at

B

with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is [asy] size(220); real r1 = 1; real r2 = 3; real r = (r1*r2)/((sqrt(r1)+sqrt(r2))**2); pair A=(0,r1), B=(2*sqrt(r1*r2),r2); dot(A); dot(B); draw( circle(A,r1) ); draw( circle(B,r2) ); draw( (-1.5,0)--(7.5,0) ); draw( A -- (A+dir(210)*r1) ); label("

1

", A -- (A+dir(210)*r1), N ); draw( B -- (B+r2*dir(330)) ); label("

4

", B -- (B+r2*dir(330)), N ); label("

A

",A,dir(330)); label("

B

",B, dir(140));
draw( circle( (2*sqrt(r1*r),r), r )); [/asy]

\displaystyle <span class='latex-bold'>(A)</span> \ \frac {1}{3} \qquad <span class='latex-bold'>(B)</span> \ \frac {2}{5} \qquad <span class='latex-bold'>(C)</span> \ \frac {5}{12} \qquad <span class='latex-bold'>(D)</span> \ \frac {4}{9} \qquad <span class='latex-bold'>(E)</span> \ \frac {1}{2}

17

1

Very nice geometry and probability combination

A point

P

is selected at random from the interior of the pentagon with vertices A \equal{} (0,2), B \equal{} (4,0), C \equal{} (2 \pi \plus{} 1, 0), D \equal{} (2 \pi \plus{} 1,4), and E \equal{} (0,4). What is the probability that

\angle APB

is obtuse? [asy] size(150); pair A, B, C, D, E; A = (0,1.5); B = (3,0); C = (2 *pi + 1, 0); D = (2 * pi + 1,4); E = (0,4); draw(A--B--C--D--E--cycle); label("

A

", A, dir(180)); label("

B

", B, dir(270)); label("

C

", C, dir(0)); label("

D

", D, dir(0)); label("

E

", E, dir(180)); [/asy]

\displaystyle <span class='latex-bold'>(A)</span> \ \frac {1}{5} \qquad <span class='latex-bold'>(B)</span> \ \frac {1}{4} \qquad <span class='latex-bold'>(C)</span> \ \frac {5}{16} \qquad <span class='latex-bold'>(D)</span> \ \frac {3}{8} \qquad <span class='latex-bold'>(E)</span> \ \frac {1}{2}

19

1

Test your polynomial knowledge!

The polynomial P(x) \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the

y

-intercept of the graph of y \equal{} P(x) is 2, what is

b

? (A) \ \minus{} 11 \qquad (B) \ \minus{} 10 \qquad (C) \ \minus{} 9 \qquad (D) \ 1 \qquad (E) \ 5

23

1

Possible complex root

A polynomial of degree four with leading coefficient 1 and integer coefficients has two zeros, both of which are integers. Which of the following can also be a zero of the polynomial? (A) \ \frac {1 \plus{} i \sqrt {11}}{2} \qquad (B) \ \frac {1 \plus{} i}{2} \qquad (C) \ \frac {1}{2} \plus{} i \qquad (D) \ 1 \plus{} \frac {i}{2} \qquad (E) \ \frac {1 \plus{} i \sqrt {13}}{2}

21

1

Integers with Product 8!

Four positive integers

a,b,c,

and

d

have a product of 8! and satisfy\begin{align*}ab \plus{} a \plus{} b &\equal{} 524\\ bc \plus{} b \plus{} c &\equal{} 146\\ cd \plus{} c \plus{} d &\equal{} 104.\end{align*} What is a \minus{} d?

<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

16

1

Nice spider shoe and sock problem!

A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?

<span class='latex-bold'>(A)</span> \ 8! \qquad <span class='latex-bold'>(B)</span> \ 2^8 \cdot 8! \qquad <span class='latex-bold'>(C)</span> \ (8!)^2 \qquad <span class='latex-bold'>(D)</span> \ \frac {16!}{2^8} \qquad <span class='latex-bold'>(E)</span> \ 16!

15

1

Insect on tetrahedron

An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.)

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

14

1

Equilateral triangle picking..

Given the nine-sided regular polygon

A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8 A_9

, how many distinct equilateral triangles in the plane of the polygon have at least two vertices in the set

\{A_1,A_2,...A_9\}

?

<span class='latex-bold'>(A)</span> \ 30 \qquad <span class='latex-bold'>(B)</span> \ 36 \qquad <span class='latex-bold'>(C)</span> \ 63 \qquad <span class='latex-bold'>(D)</span> \ 66 \qquad <span class='latex-bold'>(E)</span> \ 72

13

1

Parabola and its reflection

The parabola with equation y \equal{} ax^2 \plus{} bx \plus{} c and vertex

(h,k)

is reflected about the line y \equal{} k. This results in the parabola with equation y \equal{} dx^2 \plus{} ex \plus{} f. Which of the following equals a \plus{} b \plus{} c \plus{} d \plus{} e \plus{} f? (A) \ 2b \qquad (B) \ 2c \qquad (C) \ 2a \plus{} 2b \qquad (D) \ 2h \qquad (E) \ 2k

12

1

Divisible by 3,4 but not 5

How many positive integers not exceeding 2001 are multiple of 3 or 4 but not 5?

<span class='latex-bold'>(A)</span> \ 768 \qquad <span class='latex-bold'>(B)</span> \ 801 \qquad <span class='latex-bold'>(C)</span> \ 934 \qquad <span class='latex-bold'>(D)</span> \ 1067 \qquad <span class='latex-bold'>(E)</span> \ 1167

9

1

Function problem

Let

f

be a function satisfying f(xy) \equal{} f(x)/y for all positive real numbers

x

and

y

. If f(500) \equal{} 3, what is the value of

f(600)

?

<span class='latex-bold'>(A)</span> \ 1 \qquad <span class='latex-bold'>(B)</span> \ 2 \qquad <span class='latex-bold'>(C)</span> \ \displaystyle \frac {5}{2} \qquad <span class='latex-bold'>(D)</span> \ 3 \qquad <span class='latex-bold'>(E)</span> \ \displaystyle \frac {18}{5}

5

1

Product of odd integers

What is the product of all odd positive integers less than 10000?

<span class='latex-bold'>(A)</span> \ \frac {10000!}{(5000!)^2} \qquad <span class='latex-bold'>(B)</span> \ \frac {10000!}{2^{5000}} \ \qquad <span class='latex-bold'>(C)</span> \ \frac {9999!}{2^{5000}} \qquad <span class='latex-bold'>(D)</span> \ \frac {10000!}{2^{5000} \cdot 5000!} \qquad <span class='latex-bold'>(E)</span> \ \frac {5000!}{2^{5000}}

3

1

Kristin and her state income tax

The state income tax where Kristin lives is levied at the rate of

p \%

of the first

\$28000

of annual income plus (p \plus{} 2) \% of any amount above

\$28000

. Kristin noticed that the state income tax she paid amounted to (p \plus{} 0.25) \% of her annual income. What was her annual income?

<span class='latex-bold'>(A)</span> \ \$28000 \qquad <span class='latex-bold'>(B)</span> \ \$32000 \qquad <span class='latex-bold'>(C)</span> \ \$35000 \qquad <span class='latex-bold'>(D)</span> \ \$42000 \qquad <span class='latex-bold'>(E)</span> \ \$56000

2

1

P(n) and S(n)

Let

P(n)

and

S(n)

denote the product and the sum, respectively, of the digits of the integer

n

. For example, P(23) \equal{} 6 and S(23) \equal{} 5. Suppose

N

is a two-digit number such that N \equal{} P(N) \plus{} S(N). What is the units digit of

N

?

<span class='latex-bold'>(A)</span> \ 2 \qquad <span class='latex-bold'>(B)</span> \ 3 \qquad <span class='latex-bold'>(C)</span> \ 6 \qquad <span class='latex-bold'>(D)</span> \ 8 \qquad <span class='latex-bold'>(E)</span> \ 9

2001 AMC 12/AHSME

Subcontests

Sequence including 2001

Geometry (Out of unique title for this one)

Square made by midpoints

Triangle and simple line

Third circle radius

Very nice geometry and probability combination

Test your polynomial knowledge!

Possible complex root

Integers with Product 8!

Nice spider shoe and sock problem!

Insect on tetrahedron

Equilateral triangle picking..

Parabola and its reflection

Divisible by 3,4 but not 5

Function problem

Product of odd integers

Kristin and her state income tax

P(n) and S(n)