Competition · AMC preparation · step 4 of 4

AMC 12 2012B: 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 2012B #1 grade 4+ arithmetic
Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabb…
AMC 12 2012B #2 grade 7+ geometry-2d
A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1.…
AMC 12 2012B #3 grade 7+ arithmetic
For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in e…
AMC 12 2012B #4 grade 7+ rate-ratio
Suppose that the euro is worth 1.3 dollars. If Diana has 500 dollars and Etienne has 400 euros, by what percent is the v…
AMC 12 2012B #5 grade 2+ logic, number-theory
Two integers have a sum of 26. when two more integers are added to the first two, the sum is 41. Finally, when two more…
AMC 12 2012B #6 grade 7+ logic, algebra
In order to estimate the value of x-y where x and y are real numbers with x > y > 0, Xiaoli rounded x up by a small amou…
AMC 12 2012B #7 grade 5+ pattern, rate-ratio
Small lights are hung on a string 6 inches apart in the order red, red, green, green, green, red, red, green, green, gre…
AMC 12 2012B #8 grade 6+ counting
A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, p…
AMC 12 2012B #9 grade 6+ rate-ratio
It takes Clea 60 seconds to walk down an escalator when it is not moving, and 24 seconds when it is moving. How many sec…
AMC 12 2012B #10 grade 9+ geometry-2d
What is the area of the polygon whose vertices are the points of intersection of the curves x² + y² =25 and (x-4)² + 9y²…
AMC 12 2012B #11 grade 8+ algebra
In the equation below, A and B are consecutive positive integers, and A, B, and A+B represent number bases: 132_A+43_B=6…
AMC 12 2012B #12 grade 7+ counting
How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both?
AMC 12 2012B #13 grade 8+ probability
Two parabolas have equations y= x² + ax +b and y= x² + cx +d, where a, b, c, and d are integers, each chosen independent…
AMC 12 2012B #14 grade 7+ counting, probability
Bernardo and Silvia play the following game. An integer between 0 and 999 inclusive is selected and given to Bernardo. W…
AMC 12 2012B #15 grade 8+ geometry-3d
Jesse cuts a circular disk of radius 12, along 2 radii to form 2 sectors, one with a central angle of 120. He makes two…
AMC 12 2012B #16 grade 8+ counting, logic
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthe…
AMC 12 2012B #17 grade 10+ geometry-2d
Square PQRS lies in the first quadrant. Points (3,0), (5,0), (7,0), and (13,0) lie on lines SP, RQ, PQ, and SR, respecti…
AMC 12 2012B #18 grade 7+ counting
Let (a₁, a₂, ... a₁₀) be a list of the first 10 positive integers such that for each 2≤ i ≤10 either a_i + 1 or a_i-1 or…
AMC 12 2012B #19 grade 8+ geometry-3d
A unit cube has vertices P₁,P₂,P₃,P₄,P₁',P₂',P₃', and P₄'. Vertices P₂, P₃, and P₄ are adjacent to P₁, and for 1≤ i≤ 4,…
AMC 12 2012B #20 grade 9+ geometry-2d
A trapezoid has side lengths 3, 5, 7, and 11. The sum of all the possible areas of the trapezoid can be written in the f…
AMC 12 2012B #21 grade 10+ geometry-2d
Square AXYZ is inscribed in equiangular hexagon ABCDEF with X on BC, Y on DE, and Z on EF. Suppose that AB=40, and EF=41…
AMC 12 2012B #22 grade 7+ counting, pattern
A bug travels from A to B along the segments in the hexagonal lattice pictured below. The segments marked with an arrow…
AMC 12 2012B #23 grade 12+ algebra
Consider all polynomials of a complex variable, P(z)=4z⁴+az³+bz²+cz+d, where a,b,c, and d are integers, 0≤ d≤ c≤ b≤ a≤ 4…
AMC 12 2012B #24 grade 8+ number-theory
Define the function f₁ on the positive integers by setting f₁(1)=1 and if n=p₁^e₁p₂^e₂… p_k^(e_k) is the prime factoriza…
AMC 12 2012B #25 grade 10+ geometry-2d
Let S={(x,y) : x∈ {0,1,2,3,4}, y∈ {0,1,2,3,4,5}, and (x,y)≠ (0,0)}. Let T be the set of all right triangles whose vertic…

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