##### Curriculum

There are four main subject areas covered in math competitions: Algebra, Combinatorics, Geometry, and Number Theory. Our goal at the AwesomeMath Summer Program is to build and hone students problem solving skills in these four fields. We offer 12 courses, each of which lasts for the duration of the camp and covers one of the four subjects mentioned above (though there will be some overlap between disciplines, many techniques are helpful for different kinds of problems.) Each course meets 5 times per week, with a lecture and a problem session taught by one or two instructors. The instructor(s) of each course will teach the 90-minute lecture, and supervise the 90-minute problem sessions following the lecture. Because communication plays an important role in developing problem solving skills, students will be asked to participate actively in the problem sessions, i.e., to ask questions and present and defend solutions. Each student will select two courses, one in the morning and one in the afternoon, based on the following criteria:

- Students personal choices and interests (please see below for course selection tips);
- Students mathematical background and ability as reflected in the personal application, recommendation letters, and achievements on AMC8, AMC10/12, AIME, ARML, USAMO, and AwesomeMath Summer Camp.

When you arrive, we will go over your selections with you to ensure that you are in the courses that suit you best. Switching between courses will be allowed only in exceptional cases and only during the first week. Permission to change will be granted based on the recommendations of the academic teams.

##### How Students Should Select Courses

Students will complete their course selections during the registration process. A camp registration email will be sent to all students (who have been accepted and paid) around the end of April. The registration process will explain to the students how to choose choose their courses. To keep the teaching quality high and ensure individual attention to each student, each class has a maximum size (popular courses might have multiple classes, but it is important for us to know this in advance to make such arrangements). We grant students course selections, following our guidelines, on a first-come-first-serve basis.

Visit the course descriptions page to view a list of course together with descriptions and sample problems. Check out our course selection tips to help you decide.

##### Course Selection Tips

- Select the areas in which the student is most interested, or in which the student needs the most work.
- Both the content and dificulty of each course are very important. Please read the course descriptions carefully.
- All of our courses are challenging and beyond the scope of regular/accelerated/honor classes in school settings. There is an entry level course in each area. Each of these entry level courses will be challenging for most able young minds because these are contest mathematics courses. In particular, Algebra 1.5 is harder than any algebra course (including Algebra 1 and 2) taught in high schools, and Math Counts with Proofs is beyond the requirements for the MathCounts competitions (at state and national levels).

##### Course Selection Guidelines

- If a student only has MathCounts state level experience, what courses should he/she take?

*Please choose***two level 1**courses. We recommend a combination of the student’s strong subject and his/her weak subject. - If a student’s (9th grader or below) AMC10/12 scores are below the qualifying line for AIME, what courses should he/she take?

*Please choose***two level 1**courses. We recommend a combination of the student’s strong subject and his/her weak subject. - If a student’s (10th grader or above) AMC10/12 scores are below the qualifying line for AIME, what course should he/she take?

*Please choose***one level 1**course, with this being the student’s weak subject, and one level 2 course, with this being his/her strong subject. - If a student has an AIME score between 1 and 3, what courses should he/she take?

*Please choose either***two level 1**courses. We recommend a combination of the student’s strong subject and his/her weak subject;**OR****one level 1 course**, with this being the student’s weak subject, and one level 2 course, with this being his/her strong subject.

- If a student has an AIME score between 4 and 7, what courses should he/she take?

*Please choose***two level 2**courses. We recommend a combination of the student’s strong subject and his/her weak subject. - If a student has an AIME score between 8 and 11, what courses should he/she take?

*Please either***two level 2**courses. We recommend a combination of the student’s strong subject and his/her weak subject;**OR****one level 2**course, with this being the student’s weak subject, and one level 3 course, with this being his/her strong subject.

- If a student has an AIME score of 12 or above, what courses should he/she take?

*Please choose***two level 3**courses. We recommend a combination of the student’s strong subject and his/her weak subject. - If a student’s background does not fall into any of the above, what courses should he/she take?

*Please contact AwesomeMath staff as soon as possible about the student’s background and interests, and we will make a recommendation promptly.* - If a student’s background falls into to the above categories, but he/she wants to choose the courses not following the guidelines, what should he/she do?

*Please contact AwesomeMath staff via email as soon as possible about the student’s background and interests, and we will make a recommendation promptly.*

**If there are still questions about our recommendation, the student might be required to take a placement test in the subjects for which he/she is not following our recommendation.**

Visit the course descriptions page for a list of courses and sample problems.

Subject | Beginning (Level 1) Courses |
Intermediate (Level 2) Courses |
Advanced (Level 3) Courses |
---|---|---|---|

Algebra | Algebra 1.5 | Algebra 2.5 | Algebra 3.5 |

Combinatorics | Math Counts with Proofs | Counting Strategies | Combinatorial Argument |

Geometry | Elements of Geometry | Computational Geometry | Geometry Proofs |

Number Theory | Number Sense | Modular Arithmetic | Number Theory |

These courses are computationally oriented with a touch on proofs. They are suited for most USA math competitions (MathCounts National level, AMC10, AMC12, ARML, and the entry level of AIME). | These courses are about half computational problems and half proofs. They are well suited for the hard end of AIME and the entry level of Math Olympiad contests. | These courses are proof oriented. They are well suited for students who can easily pass AIME and are seriously preparing for Math Olympiad contests. |

##### Algebra Courses

**Algebra 1.5**

Develops essential skills such as factoring, grouping, recognizing roots, telescoping sums/products, and rationalizing. Solving (systems of) equations/inequalities (linear, absolute value, quadratic, rational, radical) is the main theme of the course. Discriminants, Viete’s relations, and symmetric polynomials also play a central role. This is the entry level algebra course. It covers all AMC levels and easy end of AIME and ARML. This course is a good fit for students with MathCounts state level experience, AMC10/12 scores approaching AIME qualifying cuts, or an AIME score between 1 and 3.**Algebra 2.5**

Studies special systems of equations, discriminants, Viete’s relations, symmetric polynomials, functional properties. Introduces (weighted) AM–GM–HM and Cauchy–Schwarz inequalities. This is the intermediate level algebra course. It covers the hard end of AMC12, and the medium to hard end of ARML and AIME. A student with an AIME score between 4 and 7 should be a good fit for this course.**Algebra 3.5**

Discusses functional equations, classical inequalities such as AM-GM-HM, Cauchy-Schwarz, Power-mean, and Jensen’s inequalities, as well as Muirhead’s and Schur’s inequalities, and inequalities related to symmetric polynomials. This is the advanced level algebra course. It covers the hard end of AIME and all levels of USAMO. A student with a strong algebra background and an AIME score of 8 or above should consider this course.

##### Combinatorics Courses

**Math Counts with Proofs**

Studies the addition and multiplication principles, permutations and combinations, and probability. Teaches how to deal with over-counting and many useful properties of integer divisors. It also introduces mathematical proofs using pigeonhole principle, well-ordering, etc. This is the entry level combinatorics course. It covers MathCounts, all the AMC levels, and the easy end of AIME and ARML. This course is a good fit for students with MathCounts state level experience, AMC10/12 scores approaching AIME qualifying cuts, or AIME scores between 1 and 3.**Counting Strategies**

Discusses counting strategies such as the addition and multiplication principles, permutations and combinations, properties of the binomial coefficients, bijections, recursions, and the inclusion- exclusion principle. This is the intermediate level combinatorics course. It covers the hard end of AMC12, the medium to hard end of AIME and ARML, as well as the beginning USAMO level. A student with an AIME score between 4 and 7 should be a good fit for this course.**Combinatorial Arguments**

Introduces methods of mathematical proofs, including induction, proofs by contradiction, the Pigeonhole Principle, the well-ordering principle, colorings, assigning weights, bijections/mappings, recursion, calculating in two ways, and combinatorial constructions. Topics may include graph theory and combinatorial geometry. A focal point of the course is combinatorial number theory. This is the advanced level combinatorics course. It covers the hard end of AIME and the medium to hard end of USAMO. A student who is familiar with mathematics proofs and has an AIME score of 8 or above should consider this course.

##### Geometry Courses

**Elements of Geometry**

Deals with computational geometry in two dimensions using Euclidean methods, including manipulation of angles and lengths, as well as the basic properties of polygons, circles, and the relations between figures. Analytic geometry is also a focal point. This is the entry level geometry course. It covers MathCounts, all AMC levels, and the easy end of AIME and ARML. This course is a good fit for students with MathCounts state level experience, AMC10/12 scores approaching AIME qualifying cuts, or AIME scores between 1 and 3.**Computational Geometry**

Studies non-synthetic techniques in solving geometry problems: coordinate geometry, vectors (2- and 3- dimensional), planes, spheres, trigonometry, and complex numbers. Features many important geometric themes: The Law of Sines and the Law of Cosines, Ptolemys theorem, Cevas theorem, Menelauss theorem, Stewarts theorem, Herons and Brahmaguptas formulas, Brocard points, dot product and the vector form of the Law of Cosines, the Cauchy-Schwarz inequality, 3-dimensional coordinate systems, as well as linear representation and traveling on the earth (sphere). This is the intermediate level geometry course. It covers the hard end of AMC12, the medium to hard end of AIME and ARML. A student with an AIME score between 4 and 7 should consider this course.**Geometric Proofs**

Focuses on classical topics such as concurrency, col-linearity, cyclic quadrilaterals, special centers/points of triangles, and geometric constructions. Introduces important transformations translation, reflections, and spiral similarities, with a touch on projective and inversive geometry. This is the advanced level geometry course. It covers the hard end of AIME and the medium to hard end of USAMO. A student with a strong background in geometry and an AIME score of 8 or above should consider this course.

##### Number Theory Courses

**Number Sense**

Studies divisibility, factoring, numerical systems, divisors and arithmetic functions of divisors. Setting-up and solving linear Diophantine equations is also a focal point of the course. This is the entry level number theory course. It covers MathCounts, all AMC levels, and the easy end of AIME and ARML. This course is a good fit for students with MathCounts state level experience, AMC10/12 scores approaching AIME qualifying cuts, or AIME scores between 1 and 3.**Modular Arithmetic**

Develops essential skills in number theory: divisibility, the division algorithm, prime numbers, the Fun- damental Theorem of Arithmetic, GCD, LCM, Bezouts identity, the Euclidean algorithm, modular arithmetic, and divisibility criteria in the decimal system. Studies numerical functions such as the number of divisors or the sum of divisors of integers. This is the intermediate level number theory course. It covers the hard end of AMC12 and the medium to hard end of AIME and ARML. A student qualified for AIME with a score between 4 and 7 should be a good fit for this course.**Number Theory**

Focuses on in-depth discussions of Diophantine equations, residue classes, quadratic reciprocity, Fermats little theorem, Eulers theorem, primitive roots, and Eulers totient function, etc. This is the advanced level number theory course. It covers the hard end of AIME and the medium to hard end of USAMO. A student with a strong background in number theory and an AIME score of 8 or above should consider this course.

Download sample problems. Check out course selection tips.

### Project 1:

Location: | The University of Texas at Dallas |

Project Title: | Perfect Powers and Associated Arithmetic Sequences |

Project Specific Prerequisites: | Very good understanding of NT2 material. Familiarity with Groups and Rings is desirable, but not mandatory |

Instructor: | Vlad Crisan |

More Details: | Research Program – Project Information |

### Project 2:

Location: | University of Puget Sound |

Project Title: | Perfect Numbers and Diophantine Equations |

Project Specific Prerequisites: | Very solid understanding of NT2 and parts of NT3 |

Instructor: | Oleksiy Klurman |

More Details: | Research Program – Project Information |