Competition · AMC preparation · step 4 of 4

AMC 10 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 10 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 10 2009B #2 grade 6+ arithmetic
Which of the following is equal to (1/3-1/4)/(1/2-1/3)?
AMC 10 2009B #3 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 10 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 10 2009B #5 grade 7+ rate-ratio
Twenty percent less than 60 is one-third more than what number?
AMC 10 2009B #6 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 10 2009B #7 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 10 2009B #8 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 10 2009B #9 grade 8+ geometry-2d
Segment BD and AE intersect at C, as shown, AB=BC=CD=CE, and ∠ A = 5/2 ∠ B. What is the degree measure of ∠ D?
AMC 10 2009B #10 grade 8+ geometry-2d
A flagpole is originally 5 meters tall. A hurricane snaps the flagpole at a point x meters above the ground so that the…
AMC 10 2009B #11 grade 7+ arithmetic
How many 7-digit palindromes (numbers that read the same backward as forward) can be formed using the digits 2, 2, 3, 3,…
AMC 10 2009B #12 grade 6+ geometry-2d
Distinct points A, B, C, and D lie on a line, with AB=BC=CD=1. Points E and F lie on a second line, parallel to the firs…
AMC 10 2009B #13 grade 5+ geometry-2d
As shown below, convex pentagon ABCDE has sides AB=3, BC=4, CD=6, DE=3, and EA=7. The pentagon is originally positioned…
AMC 10 2009B #14 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 10 2009B #15 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 10 2009B #16 grade 8+ geometry-2d
Points A and C lie on a circle centered at O, each of BA and BC are tangent to the circle, and △ ABC is equilateral. The…
AMC 10 2009B #17 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 10 2009B #18 grade 8+ geometry-2d
Rectangle ABCD has AB=8 and BC=6. Point M is the midpoint of diagonal AC, and E is on AB with ME⊥AC. What is the area of…
AMC 10 2009B #19 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 10 2009B #20 grade 8+ geometry-2d
Triangle ABC has a right angle at B, AB=1, and BC=2. The bisector of ∠ BAC meets BC at D. What is BD?
AMC 10 2009B #21 grade 4+ number-theory
What is the remainder when 3⁰ + 3¹ + 3² + … + 3²⁰⁰⁹ is divided by 8?
AMC 10 2009B #22 grade 8+ geometry-3d
A cubical cake with edge length 2 inches is iced on the sides and the top. It is cut vertically into three pieces as sho…
AMC 10 2009B #23 grade 7+ probability
Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert…
AMC 10 2009B #24 grade 8+ geometry-2d
The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together…
AMC 10 2009B #25 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…

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