Competition · AMC preparation · step 4 of 4

AMC 12 2014B: 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 2014B #1 grade 3+ algebra
Leah has 13 coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would hav…
AMC 12 2014B #2 grade 6+ rate-ratio
Orvin went to the store with just enough money to buy 30 balloons. When he arrived, he discovered that the store had a s…
AMC 12 2014B #3 grade 6+ rate-ratio
Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on…
AMC 12 2014B #4 grade 8+ algebra, rate-ratio
Susie pays for 4 muffins and 3 bananas. Calvin spends twice as much paying for 2 muffins and 16 bananas. A muffin is how…
AMC 12 2014B #5 grade 8+ geometry-2d
Doug constructs a square window using 8 equal-size panes of glass, as shown. The ratio of the height to width for each p…
AMC 12 2014B #6 grade 8+ rate-ratio
Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is 50% mo…
AMC 12 2014B #7 grade 7+ number-theory
For how many positive integers n is n/(30-n) also a positive integer?
AMC 12 2014B #8 grade 6+ number-theory, logic
In the addition shown below A, B, C, and D are distinct digits. How many different values are possible for D? ABBCB + BC…
AMC 12 2014B #9 grade 8+ geometry-2d
Convex quadrilateral ABCD has AB=3, BC=4, CD=13, AD=12, and ∠ ABC=90°, as shown. What is the area of the quadrilateral?
AMC 12 2014B #10 grade 6+ number-theory
Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the tri…
AMC 12 2014B #11 grade 6+ arithmetic, logic
A list of 11 positive integers has a mean of 10, a median of 9, and a unique mode of 8. What is the largest possible val…
AMC 12 2014B #12 grade 7+ geometry-2d, counting
A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or si…
AMC 12 2014B #13 grade 9+ geometry-2d
Real numbers a and b are chosen with 1<a<b such that no triangle with positive area has side lengths 1, a, and b or 1/b,…
AMC 12 2014B #14 grade 9+ geometry-3d
A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. Wh…
AMC 12 2014B #15 grade 11+ number-theory
When p = sum_(k=1)⁶ k ln k, the number e^p is an integer. What is the largest power of 2 that is a factor of e^p?
AMC 12 2014B #16 grade 9+ algebra
Let P be a cubic polynomial with P(0) = k, P(1) = 2k, and P(-1) = 3k. What is P(2) + P(-2) ?
AMC 12 2014B #17 grade 9+ algebra
Let P be the parabola with equation y=x² and let Q = (20, 14). There are real numbers r and s such that the line through…
AMC 12 2014B #18 grade 7+ geometry-2d
The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is bad if it is not true that for every n from…
AMC 12 2014B #19 grade 8+ geometry-3d
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the…
AMC 12 2014B #20 grade 11+ algebra
For how many positive integers x is log₁₀(x-40) + log₁₀(60-x) < 2 ?
AMC 12 2014B #21 grade 10+ geometry-2d
In the figure, ABCD is a square of side length 1. The rectangles JKHG and EBCF are congruent. What is BE?
AMC 12 2014B #22 grade 8+ probability
In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is o…
AMC 12 2014B #23 grade 11+ number-theory
The number 2017 is prime. Let S = sum _(k=0)⁶² C(2014, k). What is the remainder when S is divided by 2017?
AMC 12 2014B #24 grade 11+ geometry-2d
Let ABCDE be a pentagon inscribed in a circle such that AB = CD = 3, BC = DE = 10, and AE= 14. The sum of the lengths of…
AMC 12 2014B #25 grade 11+ algebra
Find the sum of all the positive solutions of 2cos2x (cos2x - cos( 2014π²/x ) ) = cos4x - 1

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