Competition · AMC preparation · step 4 of 4

AMC 12 2009B: 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 2009B #1 grade 4+ arithmetic
Each morning of her five-day workweek, Jane bought either a 50-cent muffin or a 75-cent bagel. Her total cost for the we…
AMC 12 2009B #2 grade 6+ rate-ratio
Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of…
AMC 12 2009B #3 grade 7+ rate-ratio
Twenty percent less than 60 is one-third more than what number?
AMC 12 2009B #4 grade 6+ geometry-2d
A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the ya…
AMC 12 2009B #5 grade 4+ number-theory
Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages?
AMC 12 2009B #6 grade 5+ algebra
By inserting parentheses, it is possible to give the expression 2×3 + 4×5 several values. How many different values can…
AMC 12 2009B #7 grade 7+ rate-ratio
In a certain year the price of gasoline rose by 20% during January, fell by 20% during February, rose by 25% during Marc…
AMC 12 2009B #8 grade 8+ algebra
When a bucket is two-thirds full of water, the bucket and water weigh a kilograms. When the bucket is one-half full of w…
AMC 12 2009B #9 grade 8+ geometry-2d
Triangle ABC has vertices A = (3,0), B = (0,3), and C, where C is on the line x + y = 7. What is the area of △ ABC?
AMC 12 2009B #10 grade 5+ counting
A particular 12-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to disp…
AMC 12 2009B #11 grade 6+ algebra
On Monday, Millie puts a quart of seeds, 25% of which are millet, into a bird feeder. On each successive day she adds an…
AMC 12 2009B #12 grade 8+ algebra
The fifth and eighth terms of a geometric sequence of real numbers are 7! and 8! respectively. What is the first term?
AMC 12 2009B #13 grade 8+ geometry-2d
Triangle ABC has AB = 13 and AC = 15, and the altitude to BC has length 12. What is the sum of the two possible values o…
AMC 12 2009B #14 grade 6+ geometry-2d
Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted l…
AMC 12 2009B #15 grade 11+ algebra
Assume 0 < r < 3. Below are five equations for x. Which equation has the largest solution x?
AMC 12 2009B #16 grade 10+ geometry-2d
Trapezoid ABCD has AD||BC, BD = 1, ∠ DBA = 23°, and ∠ BDC = 46°. The ratio BC: AD is 9: 5. What is CD?
AMC 12 2009B #17 grade 7+ probability, geometry-3d
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite ed…
AMC 12 2009B #18 grade 7+ probability
Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert…
AMC 12 2009B #19 grade 9+ number-theory
For each positive integer n, let f(n) = n⁴ - 360n² + 400. What is the sum of all values of f(n) that are prime numbers?
AMC 12 2009B #20 grade 7+ geometry-3d, counting
A convex polyhedron Q has vertices V₁,V₂,…,V_n, and 100 edges. The polyhedron is cut by planes P₁,P₂,…,P_n in such a way…
AMC 12 2009B #21 grade 7+ counting
Ten women sit in 10 seats in a line. All of the 10 get up and then reseat themselves using all 10 seats, each sitting in…
AMC 12 2009B #22 grade 10+ counting, number-theory
Parallelogram ABCD has area 1,000,000. Vertex A is at (0,0) and all other vertices are in the first quadrant. Vertices B…
AMC 12 2009B #23 grade 11+ probability
A region S in the complex plane is defined by S = {x + iy: - 1≤ x≤1, - 1≤ y≤1}. A complex number z = x + iy is chosen un…
AMC 12 2009B #24 grade 11+ algebra
For how many values of x in [0,π] is sin^(- 1)(sin 6x) = cos^(- 1)(cos x)? Note: The functions sin^(- 1) = arcsin and co…
AMC 12 2009B #25 grade 8+ counting, geometry-2d
The set G is defined by the points (x,y) with integer coordinates, 3≤|x|≤7, 3≤|y|≤7. How many squares of side at least 6…

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