USA-AMC

2014 AMC 12A Problems
Problem 1
What is

Problem 2
child tickets is

At the theater children get in for half price. The price for . How much would adult tickets and

child tickets cost?

Problem 3

Walking down Jane Street, Ralph passed four houses in a row, each painted a

different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?

Problem 4

Suppose that cows give in

cows give days?

gallons of milk in

days. At this rate, how many

gallons of milk will

Problem 5 On an algebra quiz,

of the students scored

points,

scored

points,

scored points, and the rest scored points. What is the difference between the mean and median score of the students' scores on this quiz?

Problem 6

The difference between a two-digit number and the number obtained by times the sum of the digits of either number. What is the sum of the two

reversing its digits is

digit number and its reverse?

Problem 7The first three terms of a geometric progression are
the fourth term?

,

, and

. What is

Problem 8
Coupon 1: Coupon 2: Coupon 3:

A customer who intends to purchase an appliance has three coupons, only one

of which may be used: off the listed price if the listed price is at least dollars off the listed price if the listed price is at least off the amount by which the listed price exceeds

For which of the following listed prices will coupon offer a greater price reduction than either coupon or coupon ?

Problem 9
average of

Five positive consecutive integers starting with ?

have average

. What is the

Problem 10

Three congruent isosceles triangles are constructed with their bases on the

sides of an equilateral triangle of side length . The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?

Problem 11
his speed by

David drives from his home to the airport to catch a flight. He drives

miles

in the first hour, but realizes that he will be hour late if he continues at this speed. He increases miles per hour for the rest of the way to the airport and arrives minutes early. How many miles is the airport from his home?

Problem 12
one circle and

Two circles intersect at points

and

. The minor arcs

measure

on

on the other circle. What is the ratio of the area of the larger circle to the area

of the smaller circle?

Problem 13

A fancy bed and breakfast inn has

rooms, each with a distinctive colorfriends per room.

coded decor. One day

friends arrive to spend the night. There are no other guests that night.

The friends can room in any combination they wish, but with no more than In how many ways can the innkeeper assign the guests to the rooms?

Problem 14
and

Let

be three integers such that

is an arithmetic progression ?

is a geometric progression. What is the smallest possible value of

Problem 15
where of .

A five-digit palindrome is a positive integer with respective digits

,

is non-zero. Let

be the sum of all five-digit palindromes. What is the sum of the digits

Problem 16 The product
whose digits have a sum of . What is ?

, where the second factor has

digits, is an integer

Problem 17

A

rectangular box contains a sphere of radius ?

and eight smaller

spheres of radius . The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is

Problem 18
interval of length

The domain of the function , where and are relatively prime positive integers. What is

is an ?

Problem 19

There are exactly

distinct rational numbers . What is ?

such that

and

has at least one integer solution for

Problem 20
and

In

,

,

, and

. Points

and ?

lie on

respectively. What is the minimum possible value of

Problem 21
, and let

For every real number

, let

denote the greatest integer not exceeding such that and

The set of all numbers

is a union of disjoint intervals. What is the sum of the lengths of those intervals?

Problem 22
are

The number such

is between that

and and

. How many pairs of integers

there

Problem 23 The fraction
of the repeating decimal expansion. What is the sum

where

is the length of the period ?

Problem

24

Let . For how many values of is

, ?

and

for

,

let

Problem 25
?

The parabola

has focus

and goes through the points

and

. For how many points

with integer coefficients is it true that

2.B 3.B

4.A

5.C 6.D 7.A 8.C 9.B 10.B 11.C 12.D 13.B 21.A 22.B 23.B 24.C 25.B

14.C 15.B

16.D 17.A 18.C 19.E

20.D

2014 AMC 12B Problems
Problem 1
Leah has 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?

Problem 2

Orvin went to the store with just enough money to buy

balloons. When he

arrived he discovered that the store had a special sale on balloons: buy balloon at the regular price and get a second at could buy? off the regular price. What is the greatest number of balloons Orvin

Problem 3

Randy drove the first third of his trip on a gravel road, the next

miles on

pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip?

Problem 4
for muffins and

Susie pays for

muffins and

bananas. Calvin spends twice as much paying

bananas. A muffin is how many times as expensive as a banana?

Problem 5
each pane is are window?

Doug constructs a square window using

equal-

size panes of glass, as shown. The ratio of the height to width for , and the borders around and between the panes inches wide. In inches, what is the side length of the square

Problem 6

Ed and Ann both have lemonade with their lunch. Ed orders the regular size.

Ann gets the large lemonade, which is 50% more than the regular. After both consume of their drinks, Ann gives Ed a third of what she has left, and 2 additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together?

Problem 7

For how many positive integers

is

also a positive integer?

Problem 8

In the addition shown below ?

,

,

, and

are distinct digits. How many

different values are possible for

Problem 9
,

has

, , as

shown. What is the area of the quadrilateral?

Problem 10

Danica drove her new car on a trip for a miles was . At the

whole number of hours, averaging 55 miles per hour. At the beginning of the trip, displayed on the odometer, where end of the trip, the odometer showed is a 3-digit number with miles. What is and .

Problem 11

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 12
can have?

A set

consists of triangles whose sides have integer lengths less than 5, are congruent or similar. What is the largest number of elements that

and no two elements of

Problem 13

Real numbers

and and

are chosen with or

such that no triangles with ?

positive area has side lengths

and . What is the smallest possible value of

Problem 14
interior diagonals?

A rectangular box has a total surface area of 94 square inches. The sum of

the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its

Problem 15

When ?

, the number

is an integer. What is the largest power

of 2 that is a factor of

Problem 16
What is

Let ?

be a cubic polynomial with

,

, and

.

Problem 17
real numbers if and

Let

be the parabola with equation with slope ?

and let does not intersect

. There are if and only

such that the line through

. What is

Problem 18
is

The numbers ,

,

,

,

, are to be arranged in a circle. An arrangement one can find a subset of the numbers that . Arrangements that differ only by a rotation or

if it is not true that for every

from to

appear consecutively on the circle that sum to

a reflection are considered the same. How many different bad arrangements are there?

Problem 19

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 20For how many positive integers

is

?

Problem 21
. The rectangles

In the figure, and

is a square of side length are congruent. What is ?

Problem 22

In a small pond there are eleven lily pads in a , , . Each

row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad with probability and to pad

with probability

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?

Problem 23 The number 2017 is prime. Let
is divided by 2017?

. What is the remainder when

Problem 24
to , where and

Let

be a pentagon inscribed in a circle such that . The sum of the lengths of all diagonals of

, is equal

, and

are relatively prime positive integers. What is

?

Problem 25

What is the sum of all positive real solutions

to the equation

Answer 1. C 2. C 3. E 4. B 5. A 6. D 7. D 8. C 9. B 10. D 11. E 12. B 13. C 14. D 15. C 16. E 17. E 18. B 19. E 20. B 21. C 22. C 23. C 24. D 25. D

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