2009 AMC 12A

All 25 problems from the 2009 AMC 12A. Take it as a timed mock test — 1:15:00 on the clock, scored the way the real contest is — or read through the problems and open any of them for hints and a worked solution.

25 problems · 6 points correct, 1.5 for blank, 0 for wrong

Problem 1
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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 $h + m?$
Problem 2
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Which of the following is equal to $1 + \dfrac{1}{1 + \dfrac{1}{1 + 1}}?$
Problem 3
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What number is one third of the way from $\dfrac{1}{4}$ to $\dfrac{3}{4}?$
Problem 4
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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?
Problem 5
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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?
Problem 6
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Suppose that $P = 2^m$ and $Q = 3^n.$ Which of the following is equal to $12^{mn}$ for every pair of integers $(m, n)?$
Problem 7
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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 $n?$
Problem 8
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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?
Problem 9
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Suppose that $f(x + 3) = 3x^2 + 7x + 4$ and $f(x) = ax^2 + bx + c.$ What is $a + b + c?$
Problem 10
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In quadrilateral $ABCD,$ $AB = 5,$ $BC = 17,$ $CD = 5,$ $DA = 9,$ and $BD$ is an integer. What is $BD?$
Problem 11
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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 $F_{20}?$
Problem 12
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How many positive integers less than $1000$ are $6$ times the sum of their digits?
Problem 13
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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 $AC^2?$
Problem 14
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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 $m?$
Problem 15
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For what value of $n$ is $\begin{aligned} &i + 2i^2 + 3i^3 + \cdots + ni^n \\ &= 48 + 49i? \end{aligned}$ Note: here $i = \sqrt{-1}.$
Problem 16
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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 $C?$
Problem 17
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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 $r_1 + r_2?$
Problem 18
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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 $N(k)?$
Problem 19
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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?
Problem 20
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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 $AE?$
Problem 21
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Let $p(x) = x^3 + ax^2 + bx + c,$ where $a,$ $b,$ and $c$ are complex numbers. Suppose that $\begin{gathered} p(2009 + 9002\pi i) \\ = p(2009) \\ = p(9002) = 0. \end{gathered}$ What is the number of nonreal zeros of $x^{12} + ax^8 + bx^4 + c?$
Problem 22
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A regular octahedron has side length $1.$ A plane parallel to two of its opposite faces cuts the octahedron into two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $\dfrac{a\sqrt{b}}{c},$ 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 $a + b + c?$
Problem 23
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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 $x_3 - x_2 = 150.$ The value of $x_4 - x_1$ is $m + n\sqrt{p},$ where $m,$ $n,$ and $p$ are positive integers, and $p$ is not divisible by the square of any prime. What is $m + n + p?$
Problem 24
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The tower function of twos is defined recursively as follows: $T(1) = 2$ and $T(n + 1) = 2^{T(n)}$ for $n \ge 1.$ Let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A.$ What is the largest integer $k$ such that $\underbrace{\log_2 \log_2 \log_2 \ldots \log_2}_{k} B$ is defined?
Problem 25
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The first two terms of a sequence are $a_1 = 1$ and $a_2 = \dfrac{1}{\sqrt{3}}.$ For $n \ge 1,$ $a_{n+2} = \dfrac{a_n + a_{n+1}}{1 - a_n a_{n+1}}.$ What is $|a_{2009}|?$

0 of 25 answered. Each problem left blank still scores 1.5 points.

Practise one problem at a time, with hints and solutions, on the practice page. Or work through the same ideas across every year by topic, or browse the full AMC 12 archive.

2009 AMC 12A

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25