Competition · AMC preparation · step 4 of 4

AMC 10 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 10 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 10 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 10 2012B #3 grade 5+ geometry-2d
The point in the xy-plane with coordinates (1000, 2012) is reflected across the line y=2000. What are the coordinates of…
AMC 10 2012B #4 grade 4+ arithmetic
When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with…
AMC 10 2012B #5 grade 7+ arithmetic
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the…
AMC 10 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 10 2012B #7 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 10 2012B #8 grade 7+ arithmetic
What is the sum of all integer solutions to 1<(x-2)²<25?
AMC 10 2012B #9 grade 2+ arithmetic
Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when tw…
AMC 10 2012B #10 grade 7+ algebra
How many ordered pairs of positive integers (M,N) satisfy the equation M/6=6/N?
AMC 10 2012B #11 grade 6+ arithmetic
A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, p…
AMC 10 2012B #12 grade 8+ geometry-2d
Point B is due east of point A. Point C is due north of point B. The distance between points A and C is 10√2, and ∠ BAC…
AMC 10 2012B #13 grade 6+ arithmetic
It takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escala…
AMC 10 2012B #14 grade 8+ geometry-2d
Two equilateral triangles are contained in square whose side length is 2√3. The bases of these triangles are the opposit…
AMC 10 2012B #15 grade 5+ arithmetic
In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one…
AMC 10 2012B #16 grade 8+ geometry-2d
Three circles with radius 2 are mutually tangent. What is the total area of the circles and the region bounded by them,…
AMC 10 2012B #17 grade 8+ geometry-3d
Jesse cuts a circular paper disk of radius 12 along two radii to form two sectors, the smaller having a central angle of…
AMC 10 2012B #18 grade 7+ probability
Suppose that one of every 500 people in a certain population has a particular disease, which displays no symptoms. A blo…
AMC 10 2012B #19 grade 8+ geometry-2d
In rectangle ABCD, AB=6, AD=30, and G is the midpoint of AD. Segment AB is extended 2 units beyond B to point E, and F i…
AMC 10 2012B #20 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 10 2012B #21 grade 8+ geometry-2d
Four distinct points are arranged on a plane so that the segments connecting them have lengths a, a, a, a, 2a, and b. Wh…
AMC 10 2012B #22 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 10 2012B #23 grade 8+ geometry-3d
A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one fa…
AMC 10 2012B #24 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 10 2012B #25 grade 7+ counting
A bug travels from A to B along the segments in the hexagonal lattice pictured below. The segments marked with an arrow…

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