2014 AMC 10B Problems
Problem 1
Leah has 13 coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth?
(A) 33 (B) 35 (C) 37 (D) 39 (E) 41
Problem 2
What is ?
(A) 16 (B) 24 (C) 32 (D) 48 (E) 64
Problem 3
Peter drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. In miles how long was Peter's trip?
Problem 4
Susie pays for 4 muffins and 3 bananas. Calvin spends twice as much paying for 2 muffins and 16 bananas. A muffin is how many times as expensive as a banana?
Problem 5
Camden constructs a square window using 8 equal-size panes of glass, as shown. The ratio of the height to width for each pane is 5 : 2, and the borders around and between the panes are 2 inches wide. In inches, what is the side length of the square window?
(A) 26 (B) 28 (C) 30 (D) 32 (E) 34
Problem 6
Orvin went to the store with just enough money to buy 30 balloons. When he arrived, he discovered that the store had a special sale on balloons: buy 1 balloon at the regular price and get a second at off the regular price. What is the greatest number of balloons Orvin could buy?
(A) 33 (B) 34 (C) 36 (D) 38 (E) 39
Problem 7
Suppose A > B > 0 and A is x% greater than B. What is x?
Problem 8
A truck travels feet every t seconds. There are 3 feet in a yard. How many yards does the truck travel in 3 minutes?
Problem 9
For real numbers w and z, . What is ?
Problem 10
In the addition shown below A, B, C and D are distinct digits. How many different values are possible for D?
(A) 2 (B) 4 (C) 87 (D) 8 (E) 9
Problem 11
For the consumer, a single discount of n% is more advantageous than any of the following discounts:
(1) two successive 15% discounts
(2) three successive 10% discounts
(3) a 25% discount followed by a 5% discount
What is the smallest possible positive integer value of n?
(A) 27 (B) 28 (C) 29 (D) 31 (E) 33
Problem 12
The largest divisor of 2,014,000,000 is itself. What is its fifth-largest divisor?
(A) 125,875,000 (B) 201,400,000 (C) 251,750,000
(D) 402,800,000 (E) 503,500,000
Problem 13
Six regular hexagons surround a regular hexagon of side length 1 as shown. What is the area of ∆ABC ?
Problem 14
Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, abc miles was displayed on the odometer, where abc is a 3-digit number with a ≥ 1 and a + b + c ≤ 7. At the end of the trip, the odometer showed cba miles. What is a2 + b2 + c2 ?
(A) 26 (B) 27 (C) 36 (D) 37 (E) 41
Problem 15
In rectangle ABCD, DC = 2CB and points E and F lie on so that and trisect ∠ADC as shown. What is the ratio of the area of ∆DEF to the area of rectangle ABCD?
Problem 16
Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value?
Problem 17
What is the greatest power of 2 that is a factor of 101002 -4501 ?
Problem 18
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 value of an integer in the list?
Problem 19
Two concentric circles have radii 1 and 2. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?
Problem 20
For how many integers x is the number x4 - 51x2 + 50 negative?
Problem 21
Trapezoid ABCD has parallel sides of length 33 and of length 21. The other two sides are of lengths 10 and 14. The angles at A and B are acute. What is the length of the shorter diagonal of ABCD?
Problem 22
Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?
Problem 23
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone?
Problem 24
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 1 to 15 one can find a subset of the numbers that appear consecutively on the circle that sum to n. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?
Problem 25
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 on pad N, 0 < N < 10, it will jump to pad N - 1 with probability and to pad N + 1 with probability . Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake?