Competition · AMC preparation · step 4 of 4

AMC 12 2009A: all 25 problems

Each problem has a solution worked from first principles and the first grade (by CCSS standards) that can solve it.

AMC 12 2009A #1 grade 4+ rate-ratio
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…
AMC 12 2009A #2 grade 6+ arithmetic
Which of the following is equal to 1 + 1/(1 + 1/(1 + 1))?
AMC 12 2009A #3 grade 4+ arithmetic
What number is one third of the way from 1/4 to 3/4?
AMC 12 2009A #4 grade 2+ number-theory
Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of…
AMC 12 2009A #5 grade 7+ geometry-3d
One dimension of a cube is increased by 1, another is decreased by 1, and the third is left unchanged. The volume of the…
AMC 12 2009A #6 grade 8+ algebra
Suppose that P = 2^m and Q = 3ⁿ. Which of the following is equal to 12^mn for every pair of integers (m,n)?
AMC 12 2009A #7 grade 8+ algebra
The first three terms of an arithmetic sequence are 2x - 3, 5x - 11, and 3x + 1 respectively. The nth term of the sequen…
AMC 12 2009A #8 grade 8+ geometry-2d
Four congruent rectangles are placed as shown. The area of the outer square is 4 times that of the inner square. What is…
AMC 12 2009A #9 grade 9+ algebra
Suppose that f(x+3)=3x² + 7x + 4 and f(x)=ax² + bx + c. What is a+b+c?
AMC 12 2009A #10 grade 7+ geometry-2d
In quadrilateral ABCD, AB = 5, BC = 17, CD = 5, DA = 9, and BD is an integer. What is BD?
AMC 12 2009A #11 grade 6+ pattern, geometry-2d
The figures F₁, F₂, F₃, and F₄ shown are the first in a sequence of figures. For n≥3, F_n is constructed from F_(n - 1)…
AMC 12 2009A #12 grade 7+ number-theory
How many positive integers less than 1000 are 6 times the sum of their digits?
AMC 12 2009A #13 grade 10+ geometry-2d
A ship sails 10 miles in a straight line from A to B, turns through an angle between 45° and 60°, and then sails another…
AMC 12 2009A #14 grade 10+ geometry-2d
A triangle has vertices (0,0), (1,1), and (6m,0), and the line y = mx divides the triangle into two triangles of equal a…
AMC 12 2009A #15 grade 11+ algebra
For what value of n is i + 2i² + 3i³ + … + niⁿ = 48 + 49i? Note: here i = √(- 1).
AMC 12 2009A #16 grade 10+ geometry-2d
A circle with center C is tangent to the positive x and y-axes and externally tangent to the circle centered at (3,0) wi…
AMC 12 2009A #17 grade 11+ algebra
Let a + ar₁ + ar₁² + ar₁³ + … and a + ar₂ + ar₂² + ar₂³ + … be two different infinite geometric series of positive numbe…
AMC 12 2009A #18 grade 8+ number-theory
For k > 0, let I_k = 10… 064, where there are k zeros between the 1 and the 6. Let N(k) be the number of factors of 2 in…
AMC 12 2009A #19 grade 8+ geometry-2d
Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area…
AMC 12 2009A #20 grade 8+ geometry-2d
Convex quadrilateral ABCD has AB = 9 and CD = 12. Diagonals AC and BD intersect at E, AC = 14, and △ AED and △ BEC have…
AMC 12 2009A #21 grade 11+ algebra
Let p(x) = x³ + ax² + bx + c, where a, b, and c are complex numbers. Suppose that p(2009 + 9002π i) = p(2009) = p(9002)…
AMC 12 2009A #22 grade 10+ geometry-3d
A regular octahedron has side length 1. A plane parallel to two of its opposite faces cuts the octahedron into the two c…
AMC 12 2009A #23 grade 9+ algebra
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…
AMC 12 2009A #24 grade 11+ algebra
The tower function of twos is defined recursively as follows: T(1) = 2 and T(n + 1) = 2^(T(n)) for n≥1. Let A = (T(2009)…
AMC 12 2009A #25 grade 11+ algebra, pattern
The first two terms of a sequence are a₁ = 1 and a₂ = 1/√3. For n≥1, a_(n + 2) = (a_n + a_(n + 1))/(1 - a_na_(n + 1)). W…

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