Competition · AMC preparation · step 4 of 4

AMC 12 2010B: 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 2010B #1 grade 6+ rate-ratio
Makarla attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took…
AMC 12 2010B #2 grade 4+ geometry-2d
A big L is formed as shown. What is its area?
AMC 12 2010B #3 grade 4+ number-theory
A ticket to a school play cost x dollars, where x is a whole number. A group of 9th graders buys tickets costing a total…
AMC 12 2010B #4 grade 4+ number-theory
A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the…
AMC 12 2010B #5 grade 7+ algebra
Lucky Larry's teacher asked him to substitute numbers for a, b, c, d, and e in the expression a-(b-(c-(d+e))) and evalua…
AMC 12 2010B #6 grade 6+ algebra
At the beginning of the school year, 50% of all students in Mr. Well's class answered "Yes" to the question "Do you love…
AMC 12 2010B #7 grade 7+ rate-ratio
Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining.…
AMC 12 2010B #8 grade 6+ logic
Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest rec…
AMC 12 2010B #9 grade 8+ number-theory
Let n be the smallest positive integer such that n is divisible by 20, n² is a perfect cube, and n³ is a perfect square.…
AMC 12 2010B #10 grade 8+ algebra
The average of the numbers 1, 2, 3,…, 98, 99, and x is 100x. What is x?
AMC 12 2010B #11 grade 7+ probability, number-theory
A palindrome between 1000 and 10,000 is chosen at random. What is the probability that it is divisible by 7?
AMC 12 2010B #12 grade 11+ algebra
For what value of x does log_√2√x+log₂x+log₄x²+log₈x³+log₁₆x⁴=40?
AMC 12 2010B #13 grade 11+ algebra, geometry-2d
In △ ABC, cos(2A-B)+sin(A+B)=2 and AB=4. What is BC?
AMC 12 2010B #14 grade 7+ algebra
Let a, b, c, d, and e be positive integers with a+b+c+d+e=2010 and let M be the largest of the sum a+b, b+c, c+d and d+e…
AMC 12 2010B #15 grade 11+ counting, algebra
For how many ordered triples (x,y,z) of nonnegative integers less than 20 are there exactly two distinct elements in the…
AMC 12 2010B #16 grade 7+ probability, number-theory
Positive integers a, b, and c are randomly and independently selected with replacement from the set {1, 2, 3,…, 2010}. W…
AMC 12 2010B #17 grade 7+ counting
The entries in a 3 × 3 array include all the digits from 1 through 9, arranged so that the entries in every row and colu…
AMC 12 2010B #18 grade 12+ probability
A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen independently at random. What is…
AMC 12 2010B #19 grade 8+ algebra
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of p…
AMC 12 2010B #20 grade 11+ algebra
A geometric sequence (a_n) has a₁=sin x, a₂=cos x, and a₃= tan x for some real number x. For what value of n does a_n=1+…
AMC 12 2010B #21 grade 7+ algebra, number-theory
Let a > 0, and let P(x) be a polynomial with integer coefficients such that P(1) = P(3) = P(5) = P(7) = a, and P(2) = P(…
AMC 12 2010B #22 grade 11+ geometry-2d, number-theory
Let ABCD be a cyclic quadrilateral. The side lengths of ABCD are distinct integers less than 15 such that BC· CD=AB· DA.…
AMC 12 2010B #23 grade 9+ algebra
Monic quadratic polynomial P(x) and Q(x) have the property that P(Q(x)) has zeros at x=-23, -21, -17, and -15, and Q(P(x…
AMC 12 2010B #24 grade 11+ algebra
The set of real numbers x for which 1/(x-2009)+1/(x-2010)+1/(x-2011)≥1 is the union of intervals of the form a<x≤ b. Wha…
AMC 12 2010B #25 grade 8+ number-theory
For every integer n≥2, let pow(n) be the largest power of the largest prime that divides n. For example pow(144)=pow(2⁴·…

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