Competition · AMC preparation · step 4 of 4

AMC 10 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 10 2009A #1 grade 4+ arithmetic
One can holds 12 ounces of soda, what is the minimum number of cans needed to provide a gallon (128 ounces) of soda?
AMC 10 2009A #2 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 10 2009A #3 grade 6+ arithmetic
Which of the following is equal to 1 + 1/(1 + 1/(1 + 1))?
AMC 10 2009A #4 grade 6+ rate-ratio
Eric plans to compete in a triathlon. He can average 2 miles per hour in the 1/4-mile swim and 6 miles per hour in the 3…
AMC 10 2009A #5 grade 5+ number-theory
What is the sum of the digits of the square of 111111111?
AMC 10 2009A #6 grade 7+ geometry-2d
A circle of radius 2 is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is sh…
AMC 10 2009A #7 grade 7+ arithmetic
A carton contains milk that is 2% fat, an amount that is 40% less fat than the amount contained in a carton of whole mil…
AMC 10 2009A #8 grade 6+ arithmetic
Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest g…
AMC 10 2009A #9 grade 6+ number-theory
Positive integers a, b, and 2009, with a<b<2009, form a geometric sequence with an integer ratio. What is a?
AMC 10 2009A #10 grade 8+ geometry-2d
Triangle ABC has a right angle at B. Point D is the foot of the altitude from B, AD=3, and DC=4. What is the area of △ A…
AMC 10 2009A #11 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 10 2009A #12 grade 7+ geometry-2d
In quadrilateral ABCD, AB = 5, BC = 17, CD = 5, DA = 9, and BD is an integer. What is BD?
AMC 10 2009A #13 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 10 2009A #14 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 10 2009A #15 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 10 2009A #16 grade 7+ arithmetic
Let a, b, c, and d be real numbers with |a-b|=2, |b-c|=3, and |c-d|=4. What is the sum of all possible values of |a-d|?
AMC 10 2009A #17 grade 8+ geometry-2d
Rectangle ABCD has AB=4 and BC=3. Segment EF is constructed through B so that EF is perpendicular to DB, and A and C lie…
AMC 10 2009A #18 grade 7+ rate-ratio
At Jefferson Summer Camp, 60% of the children play soccer, 30% of the children swim, and 40% of the soccer players swim.…
AMC 10 2009A #19 grade 7+ geometry-2d
Circle A has radius 100. Circle B has an integer radius r<100 and remains internally tangent to circle A as it rolls onc…
AMC 10 2009A #20 grade 6+ rate-ratio
Andrea and Lauren are 20 kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lau…
AMC 10 2009A #21 grade 8+ geometry-2d
Many Gothic cathedrals have windows with portions containing a ring of congruent circles that are circumscribed by a lar…
AMC 10 2009A #22 grade 7+ probability
Two cubical dice each have removable numbers 1 through 6. The twelve numbers on the two dice are removed, put into a bag…
AMC 10 2009A #23 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 10 2009A #24 grade 7+ geometry-3d
Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three…
AMC 10 2009A #25 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…

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