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Advanced Online Math and Language Arts Courses for Grades 2–12

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Since 1993, Art of Problem Solving has helped train the next generation of intellectual leaders. Hundreds of thousands of our students have gone on to attend prestigious universities, win global math competitions, and achieve success in highly competitive careers.

Why Students & Parents  AoPS Academy

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ABOUT AOPS ACADEMY

Rigorous Curriculum and an Unforgettable Experience

Since 1993, Art of Problem Solving has prepared hundreds of thousands of motivated students in grades 2–12 for college and career success. Through our innovative approach, students build a problem-solving foundation, an unparalleled skill set that helps them overcome obstacles in school and in life.

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We’re right in your neighborhood! Our campus location is at 788 Morris Turnpike, First Floor, Short Hills, NJ 07078.

A Problem-Solving Curriculum

Our curriculum is rigorous — it's built to be. If students never feel challenged, they’re not getting to the boundaries of what’s possible. By solving new and complex problems, AoPS students are inspired to expand to their fullest academic potential.

If you’re looking for a challenging, interactive environment where your student will build the skill stack to succeed in school and beyond, you'll find it at AoPS Academy Millburn!

Our family regularly talks about AoPS. We try to think about how our life would be different without you all. What if my daughter hadn't learned to love math? What if she never experienced being pushed to her limits? Overcoming failure? She wouldn't be who she is. And she is AWESOME! A very proud mom here. We are so fortunate that we found AoPS Academy. I credit AoPS for much of her confidence.

AoPS Academy parent to 16-year-old engineering major at UW and future cancer researcher

Problems Collection

This is a page where you can share the problems you made (try not to use past exams).

• 1 AMC styled
• 2 AIME styled
• 4.1 Problem 1
• 4.2 Solution 1
• 4.3 Problem 2
• 4.4 Solution 1 (Slow, probably official MAA)
• 4.5 Solution 2 (Fast)
• 4.6 Solution 3 (Faster)
• 4.7 Problem 3
• 4.8 Solution 1(Probably official MAA, lots of proofs)
• 4.9 Solution 2 (Fast, risky, no proofs)
• 4.10 Problem 4
• 4.11 Solution 1
• 4.12 Problem 5
• 4.13 Solution 1 (Euler's Totient Theorem)
• 4.14 Problem 6
• 4.15 Solution 1 (Recursion)
• 4.16 Problem 7
• 4.17 Solution 1 (Tedious Casework)
• 4.18 Solution 2 (Official)
• 4.19 Solution 3 (Official and Fastest)
• 4.20 Problem 8
• 4.21 Solution 1
• 4.22 Problem 9
• 4.23 Solution 1
• 4.24 Problem 10
• 4.25 Solution 1(Wordless endless bash)
• 4.26 Problem 11
• 4.27 Solution 1 (Analytic geo)
• 4.28.1 Solution 2a (Hard)
• 4.28.2 Solution 2b (Harder)

AIME styled

1. There is one and only one perfect square in the form

3.The fraction,

Someone mind making a diagram for this?

There is one and only one perfect square in the form

Solution 1 (Slow, probably official MAA)

Solution 2 (Fast)

Solution 3 (Faster)

~ (also) cxsmi

The fraction,

Solution 1(Probably official MAA, lots of proofs)

Proof: By the Triangle Inequality , we have

Add them together gives

Solution 2 (Fast, risky, no proofs)

To make things even simpler, let

Note: If you don't know Newton's Sums , you can also use Vieta's Formulas to bash.~ Ddk001

Solution 1 (Euler's Totient Theorem)

where the last step of all 3 congruences hold by the Euler's Totient Theorem . Hence,

(ii) No bigger rings are on top of smaller rings.

Solution 1 (Recursion)

Solution 1 (Tedious Casework)

In this case, we have

In this case, we have that

There is no apparent contradiction here, so we leave this as it is.

To simplify future calculations, note that

For this case, we must have

Solution 2 (Official)

Hence unit digit of RHS is 0,1,2, 6 or 4. 0,2,4 and 6 are rejected as follows:-

1 . 2 can't be the unit digit of a perfect square.

3 . If 0 is the unit digit of LHS then 50 60 70 are the only cases (as one of the digits is greater than or equal to 5) that don't satisfy

Solution 3 (Official and Fastest)

which is very easy to calculate and get 71 as the only possible solution to the problem and

Now, notice that

Similarly, we have

Now we state a few claims :

where the last equality holds by the Power of a Point Theorem .

With this in mind, we see that

Here, we state another claim :

Now, apply Ptolemy’s Theorem gives

Solution 1(Wordless endless bash)

Solution 1 (Analytic geo)

Now, we see that

Solution 2 (Hard vector bash)

Solution 2a (hard).

Solution 2b (Harder)

Now come the coordinates. Let

Here's the source for the problems:

1,2,3,4,5,6,8,9,10,11: Ddk001 , credits given to Ddk001

7: SANSKAR'S OG PROBLEMS , credits given to SANSGANKRSNGUPTA

• Note: Problem 6 is based on the Tower of Hanoi Problem

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About the Art of Problem Solving Initiative

Founded in 2004, the Art of Problem Solving Initiative, Inc. was created by people who love math and love teaching to help students access the study of advanced mathematics.

The Initiative began by running the USA Mathematical Talent Search , a nationwide math contest sponsored by the National Security Agency which continues to run to this day. In each round, this contest gives students a full month to work on five challenging proof-based problems. Students then get individual feedback on their work, including comments on their mathematical reasoning and on their proof-writing skills. By taking away the usual time pressures seen in many math contests, students have the opportunity to go more deeply and to explore.

For many years, the Initiative also ran the Local Programs Initiative which provided fiscal sponsorship to math circles and clubs across the country.

Now, the main project of the Initiative is Bridge to Enter Advanced Mathematics (BEAM) , a project to help underserved students find a realistic pathway towards becoming scientists, mathematicians, engineers, and programmers. During the summer after 7th grade, students are invited to participate in a free three-week residential program on a college campus where they gain the academic and social/emotional preparation to succeed in future programs for advanced study. They then receive academic advising throughout 8th grade and high school to help them and their families find the best opportunities for their educations.

The Art of Problem Solving Initiative receives support from Art of Problem Solving (AoPS) , which develops resources for high-performing middle and high school students including the largest online community of avid math students in the English-speaking world. The AoPS online school has trained many winners of major national mathematics competitions, including several gold medalists at the International Math Olympiad, Davidson Fellows, and winners of the Intel and Siemens Talent Search competitions.

The Art of Problem Solving Initiative is a 501(c)(3) charitable organization. It continues to be led by expert mathematicians and educators to this day. Please consider joining our contributors by donating to support advanced math education.

Financial Information

The Art of Problem Solving Initiative, Inc. makes its financial statements available for public inspection in the interest of greater transparency. Please see below to learn more about the organization.

• Tax return (Form 990)
• Audited financial statements
• Tax return (Form 990) (Note that due to a change in fiscal year these cover a shortened period.)