Competition · AMC preparation · step 4 of 4

AMC 12 2004A: 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 2004A #1 grade 6+ arithmetic
Alicia earns 20 dollars per hour, of which 1.45% is deducted to pay local taxes. How many cents per hour of Alicia's wag…
AMC 12 2004A #2 grade 7+ arithmetic
On the AMC 12, each correct answer is worth 6 points, each incorrect answer is worth 0 points, and each problem left una…
AMC 12 2004A #3 grade 7+ algebra
For how many ordered pairs of positive integers (x,y) is x + 2y = 100?
AMC 12 2004A #4 grade 4+ arithmetic
Bertha has 6 daughters and no sons. Some of her daughters have 6 daughters, and the rest have none. Bertha has a total o…
AMC 12 2004A #5 grade 8+ geometry-2d
The graph of the line y=mx+b is shown. Which of the following is true?
AMC 12 2004A #6 grade 8+ algebra
Let U=2· 2004²⁰⁰⁵, V=2004²⁰⁰⁵, W=2003· 2004²⁰⁰⁴, X=2· 2004²⁰⁰⁴, Y=2004²⁰⁰⁴ and Z=2004²⁰⁰³. Which of the following is the…
AMC 12 2004A #7 grade 4+ arithmetic
A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one t…
AMC 12 2004A #8 grade 6+ geometry-2d
In the overlapping triangles △ABC and △ABE sharing common side AB, ∠EAB and ∠ABC are right angles, AB=4, BC=6, AE=8, and…
AMC 12 2004A #9 grade 8+ geometry-3d
A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales…
AMC 12 2004A #10 grade 6+ arithmetic
The sum of 49 consecutive integers is 7⁵. What is their median?
AMC 12 2004A #11 grade 8+ arithmetic
The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is 20 cents. If she had one more qua…
AMC 12 2004A #12 grade 8+ geometry-2d
Let A = (0,9) and B = (0,12). Points A' and B' are on the line y = x, and AA' and BB' intersect at C = (2,8). What is th…
AMC 12 2004A #13 grade 8+ counting, geometry-2d
Let S be the set of points (a,b) in the coordinate plane, where each of a and b may be - 1, 0, or 1. How many distinct l…
AMC 12 2004A #14 grade 8+ arithmetic
A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second ter…
AMC 12 2004A #15 grade 7+ rate-ratio
Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first m…
AMC 12 2004A #16 grade 11+ algebra
The set of all real numbers x for which log₂₀₀₄(log₂₀₀₃(log₂₀₀₂(log₂₀₀₁x))) is defined is {x| x > c}. What is the value…
AMC 12 2004A #17 grade 8+ arithmetic
Let f be a function with the following properties: (i) f(1) = 1, and (ii) f(2n) = n · f(n) for any positive integer n. W…
AMC 12 2004A #18 grade 8+ geometry-2d
Square ABCD has side length 2. A semicircle with diameter AB is constructed inside the square, and the tangent to the se…
AMC 12 2004A #19 grade 8+ geometry-2d
Circles A, B and C are externally tangent to each other, and internally tangent to circle D. Circles B and C are congrue…
AMC 12 2004A #20 grade 7+ probability
Select numbers a and b between 0 and 1 independently and at random, and let c be their sum. Let A, B and C be the result…
AMC 12 2004A #21 grade 11+ algebra
If sum_(n = 0)^(∞)cos²ⁿθ = 5, what is the value of cos2θ?
AMC 12 2004A #22 grade 8+ geometry-3d
Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the d…
AMC 12 2004A #23 grade 11+ algebra
A polynomial P(x) = c₂₀₀₄x²⁰⁰⁴ + c₂₀₀₃x²⁰⁰³ + ... + c₁x + c₀ has real coefficients with c₂₀₀₄not = 0 and 2004 distinct c…
AMC 12 2004A #24 grade 10+ geometry-2d
A plane contains points A and B with AB = 1. Let S be the union of all disks of radius 1 in the plane that cover AB. Wha…
AMC 12 2004A #25 grade 9+ algebra
For each integer n≥ 4, let a_n denote the base-n number 0.133_n. The product a₄a₅… a₉₉ can be expressed as m/(n!), where…

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