Competition · AMC preparation · step 4 of 4

AMC 12 2010A: 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 2010A #1 grade 7+ arithmetic
What is (20-(2010-201))+(2010-(201-20))?
AMC 12 2010A #2 grade 4+ arithmetic
A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One…
AMC 12 2010A #3 grade 6+ geometry-2d
Rectangle ABCD, pictured below, shares 50% of its area with square EFGH. Square EFGH shares 20% of its area with rectang…
AMC 12 2010A #4 grade 8+ algebra
If x<0, then which of the following must be positive?
AMC 12 2010A #5 grade 7+ algebra
Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, wi…
AMC 12 2010A #6 grade 4+ number-theory
A palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers x and x+32 are…
AMC 12 2010A #7 grade 8+ geometry-3d
Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a…
AMC 12 2010A #8 grade 8+ geometry-2d
Triangle ABC has AB=2 · AC. Let D and E be on AB and BC, respectively, such that ∠ BAE = ∠ ACD. Let F be the intersectio…
AMC 12 2010A #9 grade 7+ geometry-3d
A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of…
AMC 12 2010A #10 grade 8+ algebra
The first four terms of an arithmetic sequence are p, 9, 3p-q, and 3p+q. What is the 2010^th term of this sequence?
AMC 12 2010A #11 grade 11+ algebra
The solution of the equation 7^(x+7) = 8^x can be expressed in the form x = log_b 7⁷. What is b?
AMC 12 2010A #12 grade 2+ logic
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whos…
AMC 12 2010A #13 grade 9+ algebra
For how many integer values of k do the graphs of x²+y²=k² and xy = k not intersect?
AMC 12 2010A #14 grade 7+ geometry-2d, number-theory
Nondegenerate △ ABC has integer side lengths, BD is an angle bisector, AD = 3, and DC=8. What is the smallest possible v…
AMC 12 2010A #15 grade 9+ probability
A coin is altered so that the probability that it lands on heads is less than 1/2 and when the coin is flipped four time…
AMC 12 2010A #16 grade 7+ probability
Bernardo randomly picks 3 distinct numbers from the set {1,2,3,...,7,8,9} and arranges them in descending order to form…
AMC 12 2010A #17 grade 8+ geometry-2d
Equiangular hexagon ABCDEF has side lengths AB=CD=EF=1 and BC=DE=FA=r. The area of △ ACE is 70% of the area of the hexag…
AMC 12 2010A #18 grade 8+ counting, geometry-2d
A 16-step path is to go from (-4,-4) to (4,4) with each step increasing either the x-coordinate or the y-coordinate by 1…
AMC 12 2010A #19 grade 7+ probability
Each of 2010 boxes in a line contains a single red marble, and for 1 ≤ k ≤ 2010, the box in the kth position also contai…
AMC 12 2010A #20 grade 6+ number-theory, algebra
Arithmetic sequences (a_n) and (b_n) have integer terms with a₁=b₁=1<a₂ ≤ b₂ and a_n b_n = 2010 for some n. What is the…
AMC 12 2010A #21 grade 11+ algebra
The graph of y=x⁶-10x⁵+29x⁴-4x³+ax² lies above the line y=bx+c except at three values of x, where the graph and the line…
AMC 12 2010A #22 grade 7+ algebra
What is the minimum value of f(x)=|x-1| + |2x-1| + |3x-1| + … + |119x - 1 |?
AMC 12 2010A #23 grade 8+ number-theory
The number obtained from the last two nonzero digits of 90! is equal to n. What is n?
AMC 12 2010A #24 grade 11+ algebra, counting
Let f(x) = log₁₀ (sin(π x) · sin(2 π x) · sin (3 π x) … sin(8 π x)). The intersection of the domain of f(x) with the int…
AMC 12 2010A #25 grade 11+ counting, geometry-2d
Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How ma…

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