Competition · AMC preparation · step 4 of 4

AMC 10 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 10 2010B #1 grade 7+ arithmetic
What is 100(100-3)-(100·100-3)?
AMC 10 2010B #2 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 10 2010B #3 grade 1+ counting
A drawer contains red, green, blue, and white socks with at least 2 of each color. What is the minimum number of socks t…
AMC 10 2010B #4 grade 6+ algebra
For a real number x, define ♡(x) to be the average of x and x². What is ♡(1)+♡(2)+♡(3)?
AMC 10 2010B #5 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 10 2010B #6 grade 8+ geometry-2d
A circle is centered at O, AB is a diameter and C is a point on the circle with ∠ COB = 50°. What is the degree measure…
AMC 10 2010B #7 grade 8+ geometry-2d
A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to the area of the triangle. What is…
AMC 10 2010B #8 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 10 2010B #9 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 10 2010B #10 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 10 2010B #11 grade 7+ algebra
A shopper plans to purchase an item that has a listed price greater than textdollar 100 and can use any one of the three…
AMC 10 2010B #12 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 10 2010B #13 grade 8+ algebra
What is the sum of all the solutions of x = |2x-|60-2x||?
AMC 10 2010B #14 grade 8+ algebra
The average of the numbers 1, 2, 3,…, 98, 99, and x is 100x. What is x?
AMC 10 2010B #15 grade 7+ number-theory
On a 50-question multiple choice math contest, students receive 4 points for a correct answer, 0 points for an answer le…
AMC 10 2010B #16 grade 8+ geometry-2d
A square of side length 1 and a circle of radius √3/3 share the same center. What is the area inside the circle, but out…
AMC 10 2010B #17 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 10 2010B #18 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 10 2010B #19 grade 8+ geometry-2d
A circle with center O has area 156π. Triangle ABC is equilateral, BC is a chord on the circle, OA = 4√3, and point O is…
AMC 10 2010B #20 grade 8+ geometry-2d
Two circles lie outside regular hexagon ABCDEF. The first is tangent to AB, and the second is tangent to DE. Both are ta…
AMC 10 2010B #21 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 10 2010B #22 grade 7+ counting
Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at…
AMC 10 2010B #23 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 10 2010B #24 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 10 2010B #25 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(…

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