2009 AMC 12A

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 $0 \le m \lt 60,$ what is $h + m?$

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

What number is one third of the way from $\dfrac{1}{4}$ to $\dfrac{3}{4}?$

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?

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?

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

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 $n?$

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?

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

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

The figures $F_1,$ $F_2,$ $F_3,$ and $F_4$ shown are the first in a sequence of figures. For $n \ge 3,$ $F_n$ 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 $F_{20}?$

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

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 $AC^2?$

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 $m?$

For what value of $n$ is $$
\begin{aligned} &i + 2i^2 + 3i^3 + \cdots + ni^n \\ &= 48 + 49i? \end{aligned}
$$ Note: here $i = \sqrt{-1}.$

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 $C?$

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 $r_1 + r_2?$

For $k \gt 0,$ let $I_k = 10\ldots064,$ 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 $N(k)?$

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?

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 $AE?$