- Intermediate Counting & Probabilty
- Intermediate Algebra
- PreCalculus
- Art of Problem Solving Volume 2
If you have qualified for AIME, you have probably studied the more elementary AopS books fully, and can expect to get 4-6 or less with just these basics. If you want to score 7+ on the AIME, you will need to study the more advanced AoPS books listed above, as well as past AIME problems and solutions.
The official solution sets generally provide a single solution to a given problem, demonstrating the feasibility of solution within the standard high school curriculum. However, AIME questions are often quite rich and have many possible alternate solutions, which also should also be studied for effective contest preparation. The largest collection of alternate solutions are available on the AoPS web sites' AIME Problems and Solutions page at https://artofproblemsolving.com/wiki/index.php/AIME_Problems_and_Solutions
| Past AIME Contest Booklets and Official Solution Sets | |||
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| Problems | Size | Official Solutions | Size |
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| 17.9 MB | 27.5 MB | ||
| Core Study and Review Books are the AoPS Series: | |
|---|---|
| AIME 4-6: (Basic AoPS Books) | AIME 7+: (Advanced AoPS Books) |
| AIME 7+ Topics (These topics will get you to 7+ on the AIME) | ||
|---|---|---|
| Topic | Additional Practice and Reference | |
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| AIME 10+ Topics (Add in these to get to 10+) | ||
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| Topic | Additional Advanced Practice and Reference | |
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| Recommended Problem and Review Books | ||
|---|---|---|
| Book | Size | |
| Challenging Problems in Algebra - Posamentier,Salkind 1970 (Dover) | ||
| Challenging Problems in Geometry - Posamentier,Salkind 1970 (Dover) | ||
| Challenging Problems with Elementary Solutions I - Yaglom, Yaglom, 1964 (Dover) | ||
| 7.3 MB | ||
| 3.8 MB | ||
| 9.2 MB | ||
| 14.5 MB | ||
| 3.8 MB | ||
| 3.2 MB | ||
| Elementary Number Theory: A Problem Oriented Approach - Roberts | ||
| Counting, 2nd Edition - Meng, Guan (2013) | ||
| Principles and Techniques in Combinatorics - Chen, Koh (1992) | ||
| More advanced ... | ||
| 21.4 MB | ||
| 6.7 MB | ||
Evan Chen《陳誼廷》
• CV
Teaching (otis), • otis excerpts, • mock aime, • for beginners, • for coaches, • problems, • mop, • elmo, • usemo, personal/hobbies, • puzzle hunts, • games, • photos, youtube/twitch, • discord, plz learn code, • filesys concepts, • learning path, • latex style, • asy guide, publications, • egmo book, • napkin (v1.5), • course notes, recommendations, • mentors, • quotes, • faqs, • rec letters.
Olympiad Problems and Solutions
This page contains problems and solutions to the International Math Olympiad and several USA contests, and a few others. Check the AoPS contest index for even more problems and solutions, including most of the ones below.
International Math Olympiad #
- IMO 1997 (PDF) (TeX)
- IMO 1998 (PDF) (TeX)
- IMO 1999 (PDF) (TeX)
- IMO 2000 (PDF) (TeX)
- IMO 2001 (PDF) (TeX)
- IMO 2002 (PDF) (TeX)
- IMO 2003 (PDF) (TeX)
- IMO 2004 (PDF) (TeX)
- IMO 2005 (PDF) (TeX)
- IMO 2006 (PDF) (TeX)
- IMO 2007 (PDF) (TeX)
- IMO 2008 (PDF) (TeX)
- IMO 2009 (PDF) (TeX)
- IMO 2010 (PDF) (TeX)
- IMO 2011 (PDF) (TeX)
- IMO 2012 (PDF) (TeX)
- IMO 2013 (PDF) (TeX)
- IMO 2014 (PDF) (TeX)
- IMO 2015 (PDF) (TeX)
- IMO 2016 (PDF) (TeX)
- IMO 2017 (PDF) (TeX)
- IMO 2018 (PDF) (TeX)
- IMO 2019 (PDF) (TeX)
- IMO 2020 (PDF) (TeX) (video)
- IMO 2021 (PDF) (TeX)
- IMO 2022 (PDF) (TeX)
- IMO 2023 (PDF) (TeX)
- IMO 2024 (PDF) (TeX)
Premier USA Contests
Usa math olympiad (usamo) #.
Despite being part of the USA team selection process, these are not the “official” solution files, rather my own personal notes. In particular, I tend to be more terse than other sources.
My understanding is that the internal problems and solutions, from the actual USA(J)MO committee, are copyrighted by MAA. To my knowledge they are not published anywhere. The Math Magazine has recently resumed publishing yet another version of the problems and solutions of the olympiad.
Recent statistics for USAMO
Download statistics for 2015-present (PDF) .
Problems and solutions to USAMO
- USAMO 1996 (PDF) (TeX)
- USAMO 1997 (PDF) (TeX)
- USAMO 1998 (PDF) (TeX)
- USAMO 1999 (PDF) (TeX)
- USAMO 2000 (PDF) (TeX)
- USAMO 2001 (PDF) (TeX)
- USAMO 2002 (PDF) (TeX)
- USAMO 2003 (PDF) (TeX)
- USAMO 2004 (PDF) (TeX)
- USAMO 2005 (PDF) (TeX)
- USAMO 2006 (PDF) (TeX)
- USAMO 2007 (PDF) (TeX)
- USAMO 2008 (PDF) (TeX)
- USAMO 2009 (PDF) (TeX)
- USAMO 2010 (PDF) (TeX)
- USAMO 2011 (PDF) (TeX)
- USAMO 2012 (PDF) (TeX)
- USAMO 2013 (PDF) (TeX)
- USAMO 2014 (PDF) (TeX)
- USAMO 2015 (PDF) (TeX)
- USAMO 2016 (PDF) (TeX)
- USAMO 2017 (PDF) (TeX)
- USAMO 2018 (PDF) (TeX)
- USAMO 2019 (PDF) (TeX) (Math Jam)
- USAMOO 2020 (PDF) (TeX) (video)
- USAMO 2021 (PDF) (TeX) (video)
- USAMO 2022 (PDF) (TeX)
- USAMO 2023 (PDF) (TeX)
- USAMO 2024 (PDF) (TeX)
- JMO 2010 (PDF) (TeX)
- JMO 2011 (PDF) (TeX)
- JMO 2012 (PDF) (TeX)
- JMO 2013 (PDF) (TeX)
- JMO 2014 (PDF) (TeX)
- JMO 2015 (PDF) (TeX) , featuring Steve !
- JMO 2016 (PDF) (TeX)
- JMO 2017 (PDF) (TeX)
- JMO 2018 (PDF) (TeX)
- JMO 2019 (PDF) (TeX) (Math Jam)
- JMOO 2020 (PDF) (TeX) (video)
- JMO 2021 (PDF) (TeX) (video)
- JMO 2022 (PDF) (TeX)
- JMO 2023 (PDF) (TeX)
- JMO 2024 (PDF) (TeX)
USA TST Selection Test (TSTST) #
For an explanation of the name, see the FAQ on the USA IMO team selection .
- TSTST 2011 (probs) (sols) (TeX)
- TSTST 2012 (probs) (sols) (TeX)
- TSTST 2013 (probs) (sols) (TeX)
- TSTST 2014 (probs) (sols) (TeX)
- TSTST 2015 (probs) (sols) (TeX)
- TSTST 2016 (probs) (sols) (TeX)
- TSTST 2017 (probs) (sols) (TeX)
- TSTST 2018 (probs) (sols) (TeX) (stats)
- TSTST 2019 (probs) (sols) (TeX) (stats)
- (video 1) (video 2) (video 3)
- TSTST 2021 (probs) (sols) (TeX) (stats)
- TSTST 2022 (probs) (sols) (TeX) (stats)
- TSTST 2023 (probs) (sols) (TeX) (stats)
- TSTST 2024 (probs) (sols) (TeX) (stats)
USA Team Selection Test (TST) #
These exams are used in the final part of the selection process for the USA IMO team.
- USA Team Selection Test 2000 (probs)
- USA Team Selection Test 2001 (probs)
- USA Team Selection Test 2002 (probs)
- USA Winter TST 2012 (probs)
- USA Winter TST 2013 (probs)
- USA Winter TST 2014 (probs) (sols) (TeX)
- USA Winter TST 2015 (probs) (sols) (TeX)
- USA Winter TST 2016 (probs) (sols) (TeX)
- USA Winter TST 2017 (probs) (sols) (TeX)
- USA Winter TST 2018 (probs) (sols) (TeX) (stats)
- USA Winter TST 2019 (probs) (sols) (TeX) (stats)
- USA Winter TST 2020 (probs) (sols) (TeX) (stats)
- USA Winter TST 2021 (probs) (sols) (TeX) (stats) (video)
- Because of the pandemic, there was no USA Winter TST for IMO 2022.
- USA Winter TST 2023 (probs) (sols) (TeX) (stats)
- USA Winter TST 2024 (probs) (sols) (TeX) (stats)
Other contests
Also listed on the USEMO page .
- (video 1) (video 2)
- USEMO 2022 (problems) (solutions+results)
- USEMO 2023 (problems) (solutions+results)
See also general ELMO information .
- ELMO 2010 (problems) (solutions)
- ELMO 2011 (problems) (solutions)
- ELMO 2012 (problems)
- ELMO 2013 (problems) (solutions) (shortlist) (中文)
- ELMO 2014 (problems) (solutions) (shortlist)
- ELMO 2016 (problems) (solutions) (ELSMO)
- ELMO 2017 (problems) (shortlist) (ELSMO) (ELSSMO)
- ELMO 2018 (problems) (shortlist) (ELSMO)
- ELMO 2019 (problems) (shortlist) (ELSMO)
- ELMO 2020 (problems) (ELSMO)
- ELMO 2021 (problems) (ELSMO)
- ELMO 2022 (problems) (ELSMO)
Taiwan Team Selection Test #
These are the problems I worked on in high school when competing for a spot on the Taiwanese IMO team. These problems are in Chinese; English versions here .
- Taiwan TST 2014 Round 1 (problems)
- Taiwan TST 2014 Round 2 (problems)
- Taiwan TST 2014 Round 3 (problems)
NIMO / OMO #
In high school, I and some others ran two online contests called NIMO (National Internet Math Olympiad) and OMO (Online Math Open). Neither contest is active at the time of writing (April 2021) but I collected all the materials and put them in a Google Drive link since the websites for those contests is not currently online. Most of the problems are short-answer problems.
Hardness scale #
Here is an index of many problems by my opinions on their difficulty and subject. The difficulties are rated from 0 to 50 in increments of 5, using a scale I devised called MOHS . 1
In 2020, Rustam Turdibaev and Olimjon Olimov, compiled a 336-problem index of recent problems by subject and MOHS rating . In addition, the linked file also contains a hyperlink to each of the corresponding solution threads on Art of Problem-Solving.
This document will probably see a lot of updates. Anyway, I cannot repeat enough the disclaimer that the ratings (and even philosophy) are my own personal opinion, rather than some sort of indisputable truth.
The acronym stands from “math olympiad hardness scale”, pun fully intended . ↩
2020 AIME I Problems/Problem 10
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3 (Official MAA)
- 5 Video Solution
- 6 Video Solution
Solution 3 (Official MAA)
Video Solution
https://youtu.be/Z47NRwNB-D0
https://www.youtube.com/watch?v=FQSiQChGjpI&list=PLLCzevlMcsWN9y8YI4KNPZlhdsjPTlRrb&index=7 ~ MathEx
| ( • • ) | ||
| Preceded by | Followed by | |
- Intermediate Number Theory Problems
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Barnaul, Altai Krai (Siberia)
Barnaul is the administrative center of the Altai Krai [Altai Territory] which had been created in 1937. In the years following the 1941 Deportation of the Volga Germans, many Volga German families moved from the rural settlements in the Altai Krai into the city of Barnaul.
During World War II, more than half of the light ammunition used by the Soviet Army was manufactured in Barnaul in factories that had been relocated there from Moscow, Leningrad (St. Petersburg), Odessa, and Kharkov following the invasion of the Nazis.
- Barnaul (Wikipedia)
Barnaul street scene. Source: Marina Webber.
Panorama of Barnaul (2007). Source: Russian Wikipedia.
Pre-Volga Origin
Altai Krai State Fine Arts Museum
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Altai Krai State Fine Arts Museum - All You Need to Know BEFORE You Go (2024)
IMAGES
COMMENTS
2020 AIME I. 2020 AIME I problems and solutions. The test was held on Wednesday, March 11, 2020. The first link contains the full set of test problems. The rest contain each individual problem and its solution. Entire Test. Answer Key. Problem 1. Problem 2.
Problem 4. Let be the set of positive integers with the property that the last four digits of are and when the last four digits are removed, the result is a divisor of For example, is in because is a divisor of Find the sum of all the digits of all the numbers in For example, the number contributes to this total. Solution.
Je rey Chen and Dylan Yu (May 13, 2020) Recursion in the AMC and AIME §0Acknowledgements This was made for the Art of Problem Solving Community out there! We would like to thank Evan Chen for his evan.sty code. In addition, all problems in the handout were either copied from the Art of Problem Solving Wiki or made by ourselves.
Art of Problem Solving Volume 2 If you have qualified for AIME, you have probably studied the more elementary AopS books fully, and can expect to get 4-6 or less with just these basics. If you want to score 7+ on the AIME, you will need to study the more advanced AoPS books listed above, as well as past AIME problems and solutions.
Problem. Find the number of ordered pairs of positive integers such that .. Solution. In this problem, we want to find the number of ordered pairs such that .Let .Therefore, we want two numbers, and , such that their product is and is a perfect square. Note that there is exactly one valid for a unique , which is .This reduces the problem to finding the number of unique perfect square factors of .
naman12 and freeman66 (May 26, 2020) Trigonometry in the AIME and the USA(J)MO §0Acknowledgements This was made for the Art of Problem Solving Community out there! We would like to thank Evan Chen for his evan.sty code. In addition, all problems in the handout were either copied from the Art of Problem Solving Wiki or made by ourselves.
2020 AIME II problems and solutions. The test was offered on June 6, 2020 for students who have took 2020 AIME I and students who were planning to take the cancelled 2020 AIME II on March 19, 2020. The first link will contain the full set of test problems. The rest will contain each individual problem and its solution. Entire Test.
Solution 4 (Official MAA) Analyze each prime power separately. Start with the case of . By the Binomial Theorem, Because is divisible by , all terms after the first two are divisible by , and the exponent of in the first term is less than that in the second term. Hence it is necessary and sufficient that , that is, .
This is a list of all AIME exams in the AoPSWiki. Many of these problems and solutions are also available in the AoPS Resources section. If you find problems that are in the Resources section which are not in the AoPSWiki, please consider adding them. Also, if you notice that a problem in the Wiki differs from the original wording, feel free to ...
Problem 7. Two congruent right circular cones each with base radius and height have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance from the base of each cone. A sphere with radius lies within both cones. The maximum possible value of is , where and are relatively prime positive integers.
freeman66 (May 13, 2020) Modular Arithmetic in the AMC and AIME §0Acknowledgements This was made for the Art of Problem Solving Community out there! I would like to thank Evan Chen for his evan.sty code. In addition, all problems in the handout were either copied from the Art of Problem Solving Wiki or made by myself. Art of Problem Solving ...
In 2020, Rustam Turdibaev and Olimjon Olimov, compiled a 336-problem index of recent problems by subject and MOHS rating. In addition, the linked file also contains a hyperlink to each of the corresponding solution threads on Art of Problem-Solving. This document will probably see a lot of updates.
Since 2020, the AIME floor has been set to a higher percentage of scores, likely to ensure that a consistent number of students qualify for AIME each year, rather than a fixed percentage. Usually, 6000-7000 competitors from the AMC 10 and 12 qualify for the AIME. Honor Roll (also known as Distinction since 2020): Awarded to top 5% of scorers on ...
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Solution 4. Using the formula for , Since divided by has a remainder of , Using the rules of modular arithmetic, Expanding the left hand side, This means that is divisible by . Note that and (because the remainder when dividing by is , so must be greater than ), so all options can be eliminated. Checking all 3 cases, and work; fails.
Solution 2. Assume for the sake of contradiction that is a multiple of a single digit prime number, then must also be a multiple of that single digit prime number to accommodate for . However that means that is divisible by that single digit prime number, which violates , so contradiction. is also not 1 because then would be a multiple of it.
Barnaul is the administrative center of the Altai Krai [Altai Territory] which had been created in 1937. In the years following the 1941 Deportation of the Volga Germans, many Volga German families moved from the rural settlements in the Altai Krai into the city of Barnaul.
AIME 2020(MOCK) Problems. Contents. 1 Problem 1; 2 Problem 2; 3 Problem 3; 4 Problem 4; Problem 1. Let be . What is the remainder when is divided by ? ... Art of Problem Solving is an ACS WASC Accredited School. aops programs. AoPS Online. Beast Academy. AoPS Academy. About. About AoPS. Our Team. Our History. Jobs. AoPS Blog. Site Info. Terms.
Jul 2023. This attractive building was built in 1937 for a high school. Since 1993, part of the premises in the building has been occupied by the Altai Museum of Fine Arts, and since 2012 the building has been reconstructed to adapt it to the needs of the museum. The building is an example of Soviet architecture of the transition period - from ...
2021 AIME II. 2021 AIME II problems and solutions. The test was held on Thursday, March 18, 2021. The first link will contain the full set of test problems. The rest will contain each individual problem and its solution. Entire Test. Answer Key.
Problem. In acute , is the orthocenter, is the centroid, and is the midpoint of .It is obvious that , but does not always hold. If , , then the value of which produces the smallest value of such that can be expressed in the form , for squarefree. Compute .. Solution. Because and , we know that the height from to must be .Thus, because the perpendicular is the shortest segment from a line to a ...
Mock AIME template. Please see my changes to Mock AIME 5 2005-2006/Problems for the standard template for AIME problems articles. --JBL 10:01, 25 February 2007 ... Art of Problem Solving is an ACS WASC Accredited School. aops programs. AoPS Online. Beast Academy. AoPS Academy. About. About AoPS. Our Team. Our History. Jobs. AoPS Blog. Site Info ...