Challenging Recreational Mathematics: ▪ USA AIME 2012

Σάββατο 31 Μαρτίου 2012

▪ USA AIME 2012

Part Ι

15 March 2012

1. Find the number of positive integers with three not necessarily distinct digits,

, with

such that both

and

are divisible by 4.

2. The terms of an arithmetic sequence add to

. The first term of the sequence is increased by

, the second term is increased by

, the third term is increased by

, and in general, the

th term is increased by the

th odd positive integer. The terms of the new sequence add to

. Find the sum of the first, last, and middle terms of the original sequence.

3. Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people such that exactly one person receives the type of meal ordered by that person.

4. Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins walking as Sundance rides. When Sundance reaches the first of their hitching posts that are conveniently located at one-mile intervals along their route, he ties Sparky to the post and begins walking. When Butch reaches Sparky, he rides until he passes Sundance, then leaves Sparky at the next hitching post and resumes walking, and they continue in this manner. Sparky, Butch, and Sundance walk at 6, 4, and 2.5 miles per hour, respectively. The first time Butch and Sundance meet at a milepost, they are

miles from Dodge, and have been traveling for

minutes. Find

5. Let

be the set of all binary integers that can be written using exactly 5 zeros and 8 ones where leading zeros are allowed. If all possible subtractions are performed in which one element of

is subtracted from another, find the number of times the answer 1 is obtained.

6. The complex numbers

and

satisfy

, and the imaginary part of

for relatively prime positive integers

and

with

. Find

7. At each of the sixteen circles in the network below stands a student. A total of 3360 coins are distributed among the sixteen students. All at once, all students give away all their coins by passing an equal number of coins to each of their neighbors in the network. After the trade, all students have the same number of coins as they started with. Find the number of coins the student standing at the center circle had originally.

8. Cube

, labeled as shown below, has edge length

and is cut by a plane passing through vertex

and the midpoints

and

respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form

, where

and

are relatively prime positive integers. Find

9. Let

, and

be positive real numbers that satisfy

The value of

can be expressed in the form

, where

and

are relatively prime integers. Find

10. Let

be the set of all perfect squares whose rightmost three digits in base

are

. Let

be the set of all numbers of the form

, where

is in

. In other words,

is the set of numbers that result when the last three digits of each number in

are truncated. Find the remainder when the tenth smallest element of

is divided by

11. A frog begins at

and makes a sequence of jumps according to the following rule: from

, the frog jumps to

, which may be any of the points

, or

. There are

points

with

that can be reached by a sequence of such jumps. Find the remainder when

is divided by

12. Let

be a right triangle with right angle at

. Let

and

be points on

with

betwen

and

such that

and

trisect

. If

, then

can be written as

, where

and

are relatively prime positive integers, and

is a positive integer not divisible by the square of any prime. Find

13. Three concentric circles have radii

, and

. An equilateral triangle with one vertex on each circle has side length

. The largest possible area of the triangle can be written as

, where

and

are positive integers,

and

are relatively prime, and

is not divisible by the square of any prime. Find

14. Complex numbers

and

are the zeros of a polynomial

, and

. The points corresponding to

, and

in the complex plane are the vertices of a right triangle with hypotenuse

. Find

15. There are

mathematicians seated around a circular table with

seats numbered

in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer

such that

(1) for each

, the mathematician who was seated in seat

before the break is seated in seat

after the break (where seat

is seat

);

(2) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break.

Find the number of possible values of

with

Part II

28 March 2012

1. Find the number of ordered pairs of positive integer solutions (m,n) to the equation

2. Two geometric sequences

and

have the same common ratio, with

, and

. Find

3. At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible comittees that can be formed subject to these requirements.

4. Ana, Bob, and Cao bike at constant rates of

meters per second,

meters per second, and

meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the

field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point D on the south edge of the field. Cao arrives at point D at the same time that Ana and Bob arrive at D for the first time. The ratio of the field's length to the field's width to the distance from point D to the southeast corner of the field can be represented as

, where

, and

are positive integers with p and q relatively prime. Find

5. In the accompanying figure, the outer square has side length 40. A second square S' of side length 15 is constructed inside S with the same center as S and with sides parallel to those of S. From each midpoint of a side of S, segments are drawn to the two closest vertices of S'. The result is a four-pointed starlike figure inscribed in S. The star figure is cut out and then folded to form a pyramid with base S'. Find the volume of this pyramid.

6. Let

be the complex number with

and

such that

the distance between

and

is maximized, and let

Find

7. Let

be the increasing sequence of positive integers whose binary representation has exactly

ones. Let

be the

number in

. Find the remainder when

is divided by

8. The complex numbers

and

satisfy the system

Find the smallest possible value of

9. Let

and

be real numbers such that

and

. The value of

can be expressed in the form

, where

and

are relatively prime positive integers. Find

10. Find the number of positive integers

less than

for which there exists a positive real number

such that

Note:

is the greatest integer less than or equal to

11. Let

, and for

, define

. The value of x that satisfies

can be expressed in the form

, where

and

are relatively prime positive integers. Find

12. For a positive integer

, define the positive integer

to be

-safe if

differs in absolute value by more than

from all multiples of

. For example, the set of

-safe numbers is {}. Find the number of positive integers less than or equal to

which are simultaneously

-safe,

-safe, and

-safe.·

13. Equilateral

has side length

. There are four distinct triangles

, and

, each congruent to

, with

. Find

14. In a group of nine people each person shakes hands with exactly two of the other people from the group. Let N be the number of ways this handshaking can occur. Consider two handshaking arrangements different

if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when N is divided by 1000.

15. Triangle