Competition · AMC preparation · step 4 of 4

AMC 10 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 10 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 10 2014B #2 grade 8+ arithmetic
What is (2³ + 2³)/(2⁻³ + 2⁻³)?
AMC 10 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 10 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 10 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 10 2014B #6 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 10 2014B #7 grade 7+ arithmetic
Suppose A>B>0 and A is x% greater than B. What is x?
AMC 10 2014B #8 grade 6+ arithmetic
A truck travels b/6 feet every t seconds. There are 3 feet in a yard. How many yards does the truck travel in 3 minutes?
AMC 10 2014B #9 grade 7+ arithmetic
For real numbers w and z, (1/w + 1/z)/(1/w - 1/z) = 2014. What is (w+z)/(w-z)?
AMC 10 2014B #10 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 10 2014B #11 grade 7+ arithmetic
For the consumer, a single discount of n% is more advantageous than any of the following discounts: (1) two successive 1…
AMC 10 2014B #12 grade 6+ number-theory
The largest divisor of 2,014,000,000 is itself. What is its fifth-largest divisor?
AMC 10 2014B #13 grade 8+ geometry-2d
Six regular hexagons surround a regular hexagon of side length 1 as shown. What is the area of △ABC?
AMC 10 2014B #14 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 10 2014B #15 grade 8+ geometry-2d
In rectangle ABCD, DC = 2 · CB and points E and F lie on AB so that ED and FD trisect ∠ ADC as shown. What is the ratio…
AMC 10 2014B #16 grade 7+ probability
Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?
AMC 10 2014B #17 grade 8+ number-theory
What is the greatest power of 2 that is a factor of 10¹⁰⁰² - 4⁵⁰¹?
AMC 10 2014B #18 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 10 2014B #19 grade 8+ geometry-2d
Two concentric circles have radii 1 and 2. Two points on the outer circle are chosen independently and uniformly at rand…
AMC 10 2014B #20 grade 8+ arithmetic
For how many integers x is the number x⁴-51x²+50 negative?
AMC 10 2014B #21 grade 8+ geometry-2d
Trapezoid ABCD has parallel sides AB of length 33 and CD of length 21. The other two sides are of lengths 10 and 14. The…
AMC 10 2014B #22 grade 8+ geometry-2d
Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to a…
AMC 10 2014B #23 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 10 2014B #24 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 10 2014B #25 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…

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