AMC 12 2001

The sum of two numbers is $S$. Suppose $3$ is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers?

$\text{(A)}\ 2S + 3 \text{(B)}\ 3S + 2 \text{(C)}\ 3S + 6 \text{(D)} 2S + 6 \text{(E)}\ 2S + 12$

Let $P(n)$ and $S(n)$ denote the product and the sum, respectively, of the digits of the integer $n$. For example, $P(23) = 6$ and $S(23) = 5$. Suppose $N$ is a two-digit number such that $N = P(N)+S(N)$. What is the units digit of $N$?

$\text{(A)}\ 2 \text{(B)}\ 3 \text{(C)}\ 6 \text{(D)}\ 8 \text{(E)}\ 9$

The mean of three numbers is $10$ more than the least of the numbers and $15$ less than the greatest. The median of the three numbers is $5$. What is their sum?

$\text{(A)}\ 5 \text{(B)}\ 20 \text{(C)}\ 25 \text{(D)}\ 30 \text{(E)}\ 36$

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

$\text{(A)}\ \dfrac{10000!}{(5000!)^2} \text{(B)}\ \dfrac{10000!}{2^{5000}} \text{(C)}\ \dfrac{9999!}{2^{5000}} \text{(D)}\ \dfrac{10000!}{2^{5000} \cdot 5000!} \text{(E)}\ \dfrac{5000!}{2^{5000}}$

A telephone number has the form $\text{ABC-DEF-GHIJ}$, where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore, $D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, and $J$ are consecutive odd digits; and $A + B + C = 9$. Find $A$.

$\text{(A)}\ 4 \text{(B)}\ 5 \text{(C)}\ 6 \text{(D)}\ 7 \text{(E)}\ 8$

Which of the cones listed below can be formed from a $252^\circ$ sector of a circle of radius $10$ by aligning the two straight sides?

import graph;
unitsize(1.5cm);
defaultpen(fontsize(8pt));

draw(Arc((0,0),1,-72,180),linewidth(.8pt));
draw(dir(288)--(0,0)--(-1,0),linewidth(.8pt));
label("$10$",(-0.5,0),S);
draw(Arc((0,0),0.1,-72,180));
label("$252^{\circ}$",(0.05,0.05),NE);

$\text{(A) A cone with slant height of } 10 \text{ and radius } 6$

$\text{(B) A cone with height of } 10 \text{ and radius } 6$

$\text{(C) A cone with slant height of } 10 \text{ and radius } 7$

$\text{(D) A cone with height of } 10 \text{ and radius } 7$

$\text{(E) A cone with slant height of } 10 \text{ and radius } 8$

Let $f$ be a function satisfying $f(xy) = \frac{f(x)}y$ for all positive real numbers $x$ and $y$. If $f(500) =3$, what is the value of $f(600)$?

$\text{(A)}\ 1 \text{(B)}\ 2 \text{(C)}\ \frac52 \text{(D)}\ 3 \text{(E)}\ \frac{18}5$

The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to

$\text{(A) }50 \text{(B) }52 \text{(C) }54 \text{(D) }56 \text{(E) }58$

unitsize(3mm);
defaultpen(linewidth(0.8pt));

path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0);
path p2=(0,1)--(1,1)--(1,0);
path p3=(2,0)--(2,1)--(3,1);
path p4=(3,2)--(2,2)--(2,3);
path p5=(1,3)--(1,2)--(0,2);
path p6=(1,1)--(2,2);
path p7=(2,1)--(1,2);
path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7;
for(int i=0; i<3; ++i)
{
for(int j=0; j<3; ++j)
{
draw(shift(3*i,3*j)*p);
}
}

A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?

$\text{(A) }\frac {3}{10} \text{(B) }\frac {2}{5} \text{(C) }\frac {1}{2} \text{(D) }\frac {3}{5} \text{(E) }\frac {7}{10}$

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

$\text{(A) }768 \text{(B) }801 \text{(C) }934 \text{(D) }1067 \text{(E) }1167$

The parabola with equation $y=ax^2+bx+c$ and vertex $(h,k)$ is reflected about the line $y=k$. This results in the parabola with equation $y=dx^2+ex+f$. Which of the following equals $a+b+c+d+e+f$?

$\text{(A) }2b \text{(B) }2c \text{(C) }2a+2b \text{(D) }2h \text{(E) }2k$

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 $\{A_1,A_2,\dots,A_9\}$?

$\text{(A) }30 \text{(B) }36 \text{(C) }63 \text{(D) }66 \text{(E) }72$

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.)

$\text{(A) }\frac {1}{2} \sqrt {3} \text{(B) }1 \text{(C) }\sqrt {2} \text{(D) }\frac {3}{2} \text{(E) }2$

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?

$\text{(A) }8! \text{(B) }2^8 \cdot 8! \text{(C) }(8!)^2 \text{(D) }\frac {16!}{2^8} \text{(E) }16!$

A point $P$ is selected at random from the interior of the pentagon with vertices $A = (0,2)$, $B = (4,0)$, $C = (2 \pi + 1, 0)$, $D = (2 \pi + 1,4)$, and $E=(0,4)$. What is the probability that $\angle APB$ is obtuse?

$\text{(A) }\frac {1}{5} \text{(B) }\frac {1}{4} \text{(C) }\frac {5}{16} \text{(D) }\frac {3}{8} \text{(E) }\frac {1}{2}$

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

unitsize(0.75cm);
pair A=(0,1), B=(4,4);
dot(A); dot(B);
draw( circle(A,1) );
draw( circle(B,4) );
draw( (-1.5,0)--(8.5,0) );
draw( A -- (A+(-1,0)) );
label("$1$", A -- (A+(-1,0)), N );
draw( B -- (B+(4,0)) );
label("$4$", B -- (B+(4,0)), N );
label("$A$",A,E);
label("$B$",B,W);

filldraw( circle( (12/9,4/9), 4/9 ), lightgray, black );
dot( (12/9,4/9) );

$\text{(A) }\frac {1}{3} \text{(B) }\frac {2}{5} \text{(C) }\frac {5}{12} \text{(D) }\frac {4}{9} \text{(E) }\frac {1}{2}$

The polynomial $P(x)=x^3+ax^2+bx+c$ 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 $y=P(x)$ is 2, what is $b$?

$\text{(A) }-11 \text{(B) }-10 \text{(C) }-9 \text{(D) }1 \text{(E) }5$

Points $A = (3,9)$, $B = (1,1)$, $C = (5,3)$, and $D=(a,b)$ 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 $D$?

$\text{(A) }7 \text{(B) }9 \text{(C) }10 \text{(D) }12 \text{(E) }16$

Four positive integers $a$, $b$, $c$, and $d$ have a product of $8!$ and satisfy:

$$\begin{align*} ab + a + b & = 524 \\ bc + b + c & = 146 \\ cd + c + d & = 104 \end{align*}$$

What is $a-d$?

$\text{(A) }4 \text{(B) }6 \text{(C) }8 \text{(D) }10 \text{(E) }12$

In rectangle $ABCD$, points $F$ and $G$ lie on $AB$ so that $AF=FG=GB$ 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 $EHJ$.

$\text{(A) }\frac {5}{2} \text{(B) }\frac {35}{12} \text{(C) }3 \text{(D) }\frac {7}{2} \text{(E) }\frac {35}{8}$

A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros, both of which are integers. Which of the following can also be a zero of the polynomial?

$\text{(A) }\frac {1 + i \sqrt {11}}{2} \text{(B) }\frac {1 + i}{2} \text{(C) }\frac {1}{2} + i \text{(D) }1 + \frac {i}{2} \text{(E) }\frac {1 + i \sqrt {13}}{2}$

In $\triangle ABC$, $\angle ABC=45^\circ$. Point $D$ is on $\overline{BC}$ so that $2\cdot BD=CD$ and $\angle DAB=15^\circ$. Find $\angle ACB$.

$\text{(A) }54^\circ \text{(B) }60^\circ \text{(C) }72^\circ \text{(D) }75^\circ \text{(E) }90^\circ$

Consider sequences of positive real numbers of the form $x, 2000, y, \dots$ 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?

$\text{(A) }1 \text{(B) }2 \text{(C) }3 \text{(D) }4 \text{(E) more than }4$