The American Invitational Mathematics Examination-AIME is the second exam in the series of exams used to challenge bright students on the path toward choosing the team that represents the United States at the International Mathematics Olympiad (IMO). While most AIME participants are high school students, some bright middle school students also qualify each year.

This is a Example of a problem with a solution from this course.

**Question** Let the set S = {8,5,1,13,34,3,21,2}. Susan makes a list as follows: for each two-element subset of S, she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.

**Answer** Order the numbers in the set from greatest to least to reduce error: {34,21,13,8,5,3,2,1}. Each element of the set will appear in 77 two-element subsets, once with each other number.

34 will be the greater number in 77 subsets.

21 will be the greater number in 66 subsets.

1313 will be the greater number in 55 subsets.

88 will be the greater number in 44 subsets.

55 will be the greater number in 33 subsets.

33 will be the greater number in 22 subsets.

22 will be the greater number in 11 subsets.

11 will be the greater number in 00 subsets.

Therefore the desired sum is =484

**Sample** Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way).

**Solution** It is obvious that any configuration of one-way roads which contains a town whose roads all lead into it or lead out of it cannot satisfy the given. We claim that any configuration which does not have a town whose roads all lead into it or lead out of it does satisfy the given conditions. Now we show that a loop of 33 or more towns exist. Pick a town, then choose a neighboring town to travel to 55 times. Of these 66 towns visited, at least two of them must be the same; therefore there must exist a loop of 33 or more towns because a loop of 22 towns cannot exist. We want to show that the loop can be reached from any town, and any town can be reached from the loop.

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Official AIME Web Site : https://www.maa.org/math-competitions/aime

Level 1

- AIME I Exercise 1
- AIME I Exercise 2
- AIME I Exercise 3
- AIME I Exercise 4
- AIME I Exercise 5
- AIME I Exercise 6
- AIME I Exercise 7
- AIME I Exercise 8
- AIME I Exercise 9
- AIME I Exercise 10
- AIME I Exercise 11
- AIME I Exercise 12
- AIME I Exercise 13
- AIME I Exercise 14
- AIME I Exercise 15
- AIME I Exercise 16
- AIME I Exercise 17
- AIME I Exercise 18
- AIME I Exercise 19

Level 2

- AIME I Exercise 1
- AIME I Exercise 2
- AIME I Exercise 3
- AIME I Exercise 4
- AIME I Exercise 5
- AIME I Exercise 6
- AIME I Exercise 7
- AIME I Exercise 8
- AIME I Exercise 9
- AIME I Exercise 10
- AIME I Exercise 11
- AIME I Exercise 12
- AIME I Exercise 13
- AIME I Exercise 14
- AIME I Exercise 15
- AIME I Exercise 16
- AIME I Exercise 17
- AIME I Exercise 18
- AIME I Exercise 19

Level 3

- AIME I Exercise 1
- AIME I Exercise 2
- AIME I Exercise 3
- AIME I Exercise 4
- AIME I Exercise 5
- AIME I Exercise 6
- AIME I Exercise 7
- AIME I Exercise 8
- AIME I Exercise 9
- AIME I Exercise 10
- AIME I Exercise 11
- AIME I Exercise 12
- AIME I Exercise 13
- AIME I Exercise 14
- AIME I Exercise 15
- AIME I Exercise 16
- AIME I Exercise 17
- AIME I Exercise 18
- AIME I Exercise 19

Level 1

- AIME II Exam 1
- AIME II Exam 2
- AIME II Exam 3
- AIME II Exam 4
- AIME II Exam 5
- AIME II Exam 6
- AIME II Exam 7
- AIME II Exam 8
- AIME II Exam 9
- AIME II Exam 10
- AIME II Exam 11
- AIME II Exam 12
- AIME II Exam 13
- AIME II Exam 14
- AIME II Exam 15
- AIME II Exam 16
- AIME II Exam 17
- AIME II Exam 18
- AIME II Exam 19
- AIME II Exam 20

Level 2

- AIME II Exam 1
- AIME II Exam 2
- AIME II Exam 3
- AIME II Exam 4
- AIME II Exam 5
- AIME II Exam 6
- AIME II Exam 7
- AIME II Exam 8
- AIME II Exam 9
- AIME II Exam 10
- AIME II Exam 11
- AIME II Exam 12
- AIME II Exam 13
- AIME II Exam 14
- AIME II Exam 15
- AIME II Exam 16
- AIME II Exam 17
- AIME II Exam 18
- AIME II Exam 19
- AIME II Exam 20

Level 3

- AIME II Exam 1
- AIME II Exam 2
- AIME II Exam 3
- AIME II Exam 4
- AIME II Exam 5
- AIME II Exam 6
- AIME II Exam 7
- AIME II Exam 8
- AIME II Exam 9
- AIME II Exam 10
- AIME II Exam 11
- AIME II Exam 12
- AIME II Exam 13
- AIME II Exam 14
- AIME II Exam 15
- AIME II Exam 16
- AIME II Exam 17
- AIME II Exam 18
- AIME II Exam 19