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AMC12 2014A Problem 1

What is

Solution

Problem 2

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

lt tickets and child tickets cost? child

Solution

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?

Solution

Problem 4

Suppose that cows give gallons of milk in days? days. At this rate, how many

gallons of milk will

cows give in

Solution

Problem 5

On an algebra quiz, scored scored points,

of the students scored points, scored points, and the rest

points. What is the difference between the mean and median score of

the students' scores on this quiz?

Solution

Problem 6

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

two digit number and its reverse?

Solution

Problem 7

The first three terms of a geometric progression are fourth term? , , and . What is the

Solution

Problem 8

A customer who intends to purchase an appliance has three coupons, only one of which may be used: Coupon 1: Coupon 2: Coupon 3: 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 offer a greater price reduction

For which of the following listed prices will coupon than either coupon or coupon ?

Solution

Problem 9

Five positive consecutive integers starting with average of consecutive integers that start with have average ? . What is the

Solution

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?

Solution

Problem 11

David drives from his home to the airport to catch a flight. He drives first hour, but realizes that he will be increases his speed by arrives miles in the

hour late if he continues at this speed. He

miles per hour for the rest of the way to the airport and

minutes early. How many miles is the airport from his home?

Solution

Problem 12

Two circles intersect at points circle and and . The minor arcs measure on one

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

the area of the smaller circle?

Solution

Problem 13

A fancy bed and breakfast inn has decor. One day rooms, each with a distinctive color-coded

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 friends per room. In how many ways can the innkeeper assign the guests to

the rooms?

Solution

Problem 14

Let and be three integers such that is an arithmetic progression ?

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

Solution

Problem 15

A five-digit palindrome is a positive integer with respective digits non-zero. Let digits of . , where is

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

Solution

Problem 16

The product , where the second factor has . What is ? digits, is an integer

whose digits have a sum of

Solution

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 ?

Solution

Problem 18

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

Solution

Problem 19

There are exactly distinct rational numbers such that . What is and ?

has at least one integer solution for

Solution

Problem 20

In on of , and , , and . Points and lie

respectively. What is the minimum possible value ?

Solution

Problem 21

For every real number let that and

, let

denote the greatest integer not exceeding The set of all numbers such

, and

is a union of disjoint intervals. What is the sum of

the lengths of those intervals?

Solution

Problem 22

The number integers is between are there such that and . How many pairs of and

Solution

Problem 23

The fraction

where

is the length of the period of the ?

repeating decimal expansion. What is the sum

Solution

Problem 24

Let For how many values of is , and for ? , let .

Problem 25

The parabola

has focus

and goes through the points with integer coefficients is it true

and

.

For how many points that ?

AMC 12 2013A Problem 1

Square of is has side length . What is ? . Point is on , and the area

Solution

Problem 2

A softball team played ten games, scoring , and runs. They

lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?

Solution

Problem 3

A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?

Solution

Problem 4

What is the value of

Solution

Problem 5

Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $ , Dorothy paid $ , and Sammy paid $ . In

order to share the costs equally, Tom gave Sammy dollars, and Dorothy gave Sammy dollars. What is ?

Solution

Problem 6

In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on her two-point shots. Shenille attempted of her three-point shots and of

shots. How many points did she score?

Solution

Problem 7

The sequence has the property that every term beginning with

the third is the sum of the previous two. That is, Suppose that and . What is ?

Solution

Problem 8

Given that what is ? and are distinct nonzero real numbers such that ,

Solution

Problem 9

In sides to , and , , and and . Points and and are on are parallel ?

, respectively, such that

, respectively. What is the perimeter of parallelogram

Solution

Problem 10

Let be the set of positive integers for which with ? and has the repeating decimal different digits. What is the

representation sum of the elements of

Solution

Problem 11

Triangle points and is equilateral with are on . Points and and and are on are parallel to and .

such that both and trapezoids ?

Furthermore, triangle same perimeter. What is

all have the

Solution

Problem 12

The angles in a particular triangle are in arithmetic progression, and the side lengths are and . The sum of the possible values of x equals are positive integers. What is ? where ,

Solution

Problem 13

Let points Quadrilateral

and

. .

is cut into equal area pieces by a line passing through

This line intersects What is ?

at point

, where these fractions are in lowest terms.

Solution

Problem 14

The sequence , , , , ?

is an arithmetic progression. What is

Solution

Problem 15

Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?

Solution

Problem 16

, , are three piles of rocks. The mean weight of the rocks in is is pounds,

the mean weight of the rocks in the combined piles combined piles and is is

pounds, the mean weight of the rocks in

pounds, and the mean weight of the rocks in the pounds. What is the greatest possible integer value and ?

and

for the mean in pounds of the rocks in the combined piles

Solution

Problem 17

A group of pirates agree to divide a treasure chest of gold coins among pirate to take a share takes of the coins that

themselves as follows. The

remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the pirate receive?

Solution

Problem 18

Six spheres of radius are positioned so that their centers are at the vertices of a . The six spheres are internally tangent to a larger

regular hexagon of side length

sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?

Solution

Problem 19

In radius , intersects , and at points ? . A circle with center and . Moreover and and have

integer lengths. What is

Solution

Problem 20

Let either of

be the set or

. For

, define

to mean that of elements

. How many ordered triples , , and ?

have the property that

Solution

Problem 21

Consider . Which of the following intervals contains ?

Solution

Problem 22

A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome chosen uniformly at random. What is the probability that is

is also a palindrome?

Solution

Problem 23

is a square of side length The square region bounded by center . Point is rotated is on such that .

counterclockwise with , where ? , , and are

, sweeping out a region whose area is . What is

positive integers and

Solution

Problem 24

Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?

Solution

Problem 25

Let numbers parts of be defined by are there such that . How many complex and both the real and the imaginary ?

are integers with absolute value at most

AMC12 2012A Problem 1

A bug crawls along a number line, starting at around and crawls to . It crawls to , then turns

. How many units does the bug crawl altogether?

Solution

Problem 2

Cagney can frost a cupcake every every in seconds and Lacey can frost a cupcake

seconds. Working together, how many cupcakes can they frost

minutes?

Solution

Problem 3

A box hold centimeters high, centimeters wide, and centimeters long can

grams of clay. A second box with twice the height, three times the width, grams of clay. What is ?

and the same length as the first box can hold

Solution

Problem 4

In a bag of marbles, of the marbles are blue and the rest are red. If the number

of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?

Solution

Problem 5

A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of pieces of fruit. There are twice as many raspberries as

blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad?

Solution

Problem 6

The sums of three whole numbers taken in pairs are middle number? , , and . What is the

Solution

Problem 7

Mary divides a circle into sectors. The central angles of these sectors, measured

in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?

Solution

Problem 8

An iterative average of the numbers , , , , and is computed in the following

way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?

Solution

Problem 9

A year is a leap year if and only if the year number is divisible by or is divisible by but not by (such as ). The (such as )

anniversary of the , , a Tuesday.

birth of novelist Charles Dickens was celebrated on February On what day of the week was Dickens born?

Solution

Problem 10

A triangle has area length is ? . Let , one side of length , and the median to that side of

be the acute angle formed by that side and the median. What

Solution

Problem 11

Alex, Mel, and Chelsea play a game that has rounds. In each round there is a

single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is , and Mel is twice as likely to win as Chelsea. What is

the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round?

Solution

Problem 12

A square region equation

is externally tangent to the circle with at the point on the side . Vertices and are

on the circle with equation

. What is the side length of this square?

Solution

Problem 13

Paula the painter and her two helpers each paint at constant, but different, rates. They always start at , and all three always take the same amount of time of a house, quitting

to eat lunch. On Monday the three of them painted at only

. On Tuesday, when Paula wasn't there, the two helpers painted of the house and quit at . On Wednesday Paula worked by . How long, in minutes,

herself and finished the house by working until was each day's lunch break?

Solution

Problem 14

The closed curve in the figure is made up of length congruent circular arcs each of

, where each of the centers of the corresponding circles is among the . What is the area enclosed by the curve?

vertices of a regular hexagon of side

Solution

Problem 15

A square is partitioned into unit squares. Each unit square is painted either

white or black with each color being equally likely, chosen independently and at random. The square is the rotated clockwise about its center, and every white

square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black?

Solution

Problem 16

Circle Point and has its center in the exterior of lying on circle lies on circle ? . The two circles meet at and , , and .

. What is the radius of circle

Solution

Problem 17

Let be a subset of has a sum divisible by with the property that no pair of distinct . What is the largest possible size of ?

elements in

Solution

Problem 18

Triangle has , , and . Let . What is denote the ?

intersection of the internal angle bisectors of

Solution

Problem 19

Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?

Solution

Problem 20

Consider the polynomial

The coefficient of

is equal to

. What is

?

Solution

Problem 21

Let , , and be positive integers with such that

What is

?

Solution

Problem 22

Distinct planes

intersect the interior of a cube

. Let

be the union

of the faces of

and let

. The intersection of

and

consists of the

union of all segments joining the midpoints of every pair of edges belonging to the same face of . What is the difference between the maximum and minimum ?

possible values of

Solution

Problem 23

Let be the square one of whose diagonals has and . A point and . Let such be a translated copy is chosen uniformly at

endpoints

random over all pairs of real numbers that of by centered at and

. What is the probability that the square region determined

contains exactly two points with integer coefficients in its interior?

Solution

Problem 24

Let by general, be the sequence of real numbers defined , and in

Rearranging the numbers in the sequence new sequence that

in decreasing order produces a , , such

. What is the sum of all integers

Solution

Problem 25

Let number where denotes the fractional part of . The

is the smallest positive integer such that the equation real solutions. What is such that ? Note: the fractional part of is an integer. is a real

has at least number

and

2014A 1.C 2.B 3.B 4.A 5.C 6.D 7.A 8.C 9.B 10.B 11.C 12.D 13.B 14.C 15.B 16.D 17.A 18.C 19.E 20.D 21.A 22.B 23.B 24.C 25.B

2013A 1. E 2. C 3. E 4. C 5. B 6. B 7. C 8. D 9. C 10. D 11. C 12. A 13. B 14. B 15. D 16. E 17. D 18. B 19. D 20. B 21. A 22. E 23. C 24. E 25. A

2012A 1. E 2. D 3. D 4. C 5. D 6. D 7. C 8. C 9. A 10. D 11. B 12. D 13. D 14. E 15. A 16. E 17. B 18. A 19. B 20. B 21. E 22. C 23. C 24. C 25. C

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