AMC 12 2009a

Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took $h$ hours and $m$ minutes, with $0 < m < 60$, what is $h + m$?

\textbf{(A)}\ 46 \textbf{(B)}\ 47 \textbf{(C)}\ 50 \textbf{(D)}\ 53 \textbf{(E)}\ 54

Which of the following is equal to $1 + \frac {1}{1 + \frac {1}{1 + 1}}$?

\textbf{(A)}\ \frac {5}{4} \textbf{(B)}\ \frac {3}{2} \textbf{(C)}\ \frac {5}{3} \textbf{(D)}\ 2 \textbf{(E)}\ 3

What number is one third of the way from $\frac14$ to $\frac34$?

\textbf{(A)}\ \frac {1}{3} \textbf{(B)}\ \frac {5}{12} \textbf{(C)}\ \frac {1}{2} \textbf{(D)}\ \frac {7}{12} \textbf{(E)}\ \frac {2}{3}

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents?

\textbf{(A)}\ 15 \textbf{(B)}\ 25 \textbf{(C)}\ 35 \textbf{(D)}\ 45 \textbf{(E)}\ 55

One dimension of a cube is increased by $1$, another is decreased by $1$, and the third is left unchanged. The volume of the new rectangular solid is $5$ less than that of the cube. What was the volume of the cube?

\textbf{(A)}\ 8 \textbf{(B)}\ 27 \textbf{(C)}\ 64 \textbf{(D)}\ 125 \textbf{(E)}\ 216

Suppose that $P = 2^m$ and $Q = 3^n$. Which of the following is equal to $12^{mn}$ for every pair of integers $(m,n)$?

\textbf{(A)}\ P^2Q \textbf{(B)}\ P^nQ^m \textbf{(C)}\ P^nQ^{2m} \textbf{(D)}\ P^{2m}Q^n \textbf{(E)}\ P^{2n}Q^m

The first three terms of an arithmetic sequence are $2x - 3$, $5x - 11$, and $3x + 1$ respectively. The $n$th term of the sequence is $2009$. What is $n$?

\textbf{(A)}\ 255 \textbf{(B)}\ 502 \textbf{(C)}\ 1004 \textbf{(D)}\ 1506 \textbf{(E)}\ 8037

Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?

unitsize(6mm);
defaultpen(linewidth(.8pt));

path p=(1,1)--(-2,1)--(-2,2)--(1,2);
draw(p);
draw(rotate(90)*p);
draw(rotate(180)*p);
draw(rotate(270)*p);

\textbf{(A)}\ 3 \textbf{(B)}\ \sqrt {10} \textbf{(C)}\ 2 + \sqrt2 \textbf{(D)}\ 2\sqrt3 \textbf{(E)}\ 4

Suppose that $f(x+3)=3x^2 + 7x + 4$ and $f(x)=ax^2 + bx + c$. What is $a+b+c$?

\textbf{(A)}\ -1 \textbf{(B)}\ 0 \textbf{(C)}\ 1 \textbf{(D)}\ 2 \textbf{(E)}\ 3

In quadrilateral $ABCD$, $AB = 5$, $BC = 17$, $CD = 5$, $DA = 9$, and $BD$ is an integer. What is $BD$?

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dotfactor=4;

pair C=(0,0), B=(17,0);
pair D=intersectionpoints(Circle(C,5),Circle(B,13))[0];
pair A=intersectionpoints(Circle(D,9),Circle(B,5))[0];
pair[] dotted={A,B,C,D};

draw(D--A--B--C--D--B);
dot(dotted);
label("$D$",D,NW);
label("$C$",C,W);
label("$B$",B,E);
label("$A$",A,NE);

\textbf{(A)}\ 11 \textbf{(B)}\ 12 \textbf{(C)}\ 13 \textbf{(D)}\ 14 \textbf{(E)}\ 15

The figures $F_1$, $F_2$, $F_3$, and $F_4$ shown are the first in a sequence of figures. For $n\ge3$, $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outside square. For example, figure $F_3$ has $13$ diamonds. How many diamonds are there in figure $F_{20}$?

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path d=(1/2,0)--(0,sqrt(3)/2)--(-1/2,0)--(0,-sqrt(3)/2)--cycle;
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path f2=(0,0)--(-1,1)--(1,1)--(1,-1)--(-1,-1);
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path f3=f2--(-2,-2)--(-2,0)--(-2,2)--(0,2)--(2,2)--(2,0)--(2,-2)--(0,-2);
path[] g3=g2^^(-2,-2)--(0,-2)^^(2,-2)--(1,-1)^^(1,1)--(2,2)^^(-1,1)--(-2,2);
path[] f4=f3^^(-3,-3)--(-3,-1)--(-3,1)--(-3,3)--(-1,3)--(1,3)--(3,3)--
(3,1)--(3,-1)--(3,-3)--(1,-3)--(-1,-3);
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(-2,2)--(-3,3);

draw(f1,m);
draw(shift(5,0)*f2,m);
draw(shift(5,0)*g2);
draw(shift(12,0)*f3,m);
draw(shift(12,0)*g3);
draw(shift(21,0)*f4,m);
draw(shift(21,0)*g4);
label("$F_1$",(0,-4));
label("$F_2$",(5,-4));
label("$F_3$",(12,-4));
label("$F_4$",(21,-4));

\textbf{(A)}\ 401 \textbf{(B)}\ 485 \textbf{(C)}\ 585 \textbf{(D)}\ 626 \textbf{(E)}\ 761

How many positive integers less than $1000$ are $6$ times the sum of their digits?

\textbf{(A)}\ 0 \textbf{(B)}\ 1 \textbf{(C)}\ 2 \textbf{(D)}\ 4 \textbf{(E)}\ 12

A ship sails $10$ miles in a straight line from $A$ to $B$, turns through an angle between $45^{\circ}$ and $60^{\circ}$, and then sails another $20$ miles to $C$. Let $AC$ be measured in miles. Which of the following intervals contains $AC^2$?

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pair B=(0,0), A=(-10,0), C=20*dir(50);

draw(A--B--C);
draw(A--C,linetype("4 4"));

dot(A);
dot(B);
dot(C);
label("$10$",midpoint(A--B),S);
label("$20$",midpoint(B--C),SE);
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);

\textbf{(A)}\ [400,500] \textbf{(B)}\ [500,600] \textbf{(C)}\ [600,700] \textbf{(D)}\ [700,800] \textbf{(E)}\ [800,900]

A triangle has vertices $(0,0)$, $(1,1)$, and $(6m,0)$, and the line $y = mx$ divides the triangle into two triangles of equal area. What is the sum of all possible values of $m$?

\textbf{(A)} - !\frac {1}{3} \textbf{(B)} - !\frac {1}{6} \textbf{(C)}\ \frac {1}{6} \textbf{(D)}\ \frac {1}{3} \textbf{(E)}\ \frac {1}{2}

For what value of $n$ is $i + 2i^2 + 3i^3 + \cdots + ni^n = 48 + 49i$?

Note: here $i = \sqrt { - 1}$.

\textbf{(A)}\ 24 \textbf{(B)}\ 48 \textbf{(C)}\ 49 \textbf{(D)}\ 97 \textbf{(E)}\ 98

A circle with center $C$ is tangent to the positive $x$ and $y$-axes and externally tangent to the circle centered at $(3,0)$ with radius $1$. What is the sum of all possible radii of the circle with center $C$?

\textbf{(A)}\ 3 \textbf{(B)}\ 4 \textbf{(C)}\ 6 \textbf{(D)}\ 8 \textbf{(E)}\ 9

Let $a + ar_1 + ar_1^2 + ar_1^3 + \cdots$ and $a + ar_2 + ar_2^2 + ar_2^3 + \cdots$ be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is $r_1$, and the sum of the second series is $r_2$. What is $r_1 + r_2$?

\textbf{(A)}\ 0 \textbf{(B)}\ \frac {1}{2} \textbf{(C)}\ 1 \textbf{(D)}\ \frac {1 + \sqrt {5}}{2} \textbf{(E)}\ 2

For $k > 0$, let $I_k = 10\ldots 064$, where there are $k$ zeros between the $1$ and the $6$. Let $N(k)$ be the number of factors of $2$ in the prime factorization of $I_k$. What is the maximum value of $N(k)$?

\textbf{(A)}\ 6 \textbf{(B)}\ 7 \textbf{(C)}\ 8 \textbf{(D)}\ 9 \textbf{(E)}\ 10

Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon (7 sides). The areas of the two regions were $A$ and $B$, respectively. Each polygon had a side length of $2$. Which of the following is true?

\textbf{(A)}\ A = \frac {25}{49}B \textbf{(B)}\ A = \frac {5}{7}B \textbf{(C)}\ A = B \textbf{(D)}\ A$$= \frac {7}{5}B \textbf{(E)}\ A = \frac {49}{25}B

Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$?

\textbf{(A)}\ \frac {9}{2} \textbf{(B)}\ \frac {50}{11} \textbf{(C)}\ \frac {21}{4} \textbf{(D)}\ \frac {17}{3} \textbf{(E)}\ 6

Let $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are complex numbers. Suppose that

What is the number of nonreal zeros of $x^{12} + ax^8 + bx^4 + c$?

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

A regular octahedron has side length $1$. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $\frac {a\sqrt {b}}{c}$, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. What is $a + b + c$?

\textbf{(A)}\ 10 \textbf{(B)}\ 11 \textbf{(C)}\ 12 \textbf{(D)}\ 13 \textbf{(E)}\ 14

Functions $f$ and $g$ are quadratic, $g(x) = - f(100 - x)$, and the graph of $g$ contains the vertex of the graph of $f$. The four $x$-intercepts on the two graphs have $x$-coordinates $x_1$, $x_2$, $x_3$, and $x_4$, in increasing order, and $x_3 - x_2 = 150$. Then $x_4 - x_1 = m + n\sqrt p$, where $m$, $n$, and $p$ are positive integers, and $p$ is not divisible by the square of any prime. What is $m + n + p$?

\textbf{(A)}\ 602 \textbf{(B)}\ 652 \textbf{(C)}\ 702 \textbf{(D)}\ 752 \textbf{(E)}\ 802

The ''tower function of twos'' is defined recursively as follows: $T(1) = 2$ and $T(n + 1) = 2^{T(n)}$ for $n\ge1$. Let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A$. What is the largest integer $k$ such that

$$\underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}$$

is defined?

\textbf{(A)}\ 2009 \textbf{(B)}\ 2010 \textbf{(C)}\ 2011 \textbf{(D)}\ 2012 \textbf{(E)}\ 2013

The first two terms of a sequence are $a_1 = 1$ and $a_2 = \frac {1}{\sqrt3}$. For $n\ge1$,

What is $|a_{2009}|$?

\textbf{(A)}\ 0 \textbf{(B)}\ 2 - \sqrt3 \textbf{(C)}\ \frac {1}{\sqrt3} \textbf{(D)}\ 1 \textbf{(E)}\ 2 + \sqrt3