2021 AMC 12A Problems/Problem 7
- 2 Solution 1 (Expand)
- 3 Solution 2 (Expand and then Factor)
- 4 Solution 3 (Beyond Overkill)
- 5 Video Solution (Simple & Quick)
- 6 Video Solution by Aaron He (Trivial Inequality)
- 7 Video Solution by North America Math Contest Go Go Go (Trivial Inequality, Simon's Favourite Packing Theorem)
- 8 Video Solution by Hawk Math
- 9 Video Solution by OmegaLearn (Trivial Inequality, Simon's Favorite Factoring)
- 10 Video Solution 6
- 11 Video Solution by TheBeautyofMath
- 12 Video Solution by The Learning Royal
- 13 Yet another Video Solution
- 14 See also
Solution 1 (Expand)
Solution 2 (Expand and then Factor)
~MRENTHUSIASM
Solution 3 (Beyond Overkill)
~ DBlack2021
Video Solution (Simple & Quick)
https://youtu.be/2CZ1u4J9yk4
~ Education, the Study of Everything
Video Solution by Aaron He (Trivial Inequality)
https://www.youtube.com/watch?v=xTGDKBthWsw&t=6m58s
Video Solution by North America Math Contest Go Go Go (Trivial Inequality, Simon's Favourite Packing Theorem)
https://www.youtube.com/watch?v=PbJK4KKfQjY&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=8
Video Solution by Hawk Math
https://www.youtube.com/watch?v=P5al76DxyHY
Video Solution by OmegaLearn (Trivial Inequality, Simon's Favorite Factoring)
https://youtu.be/DP0ppuQzFPE
~ pi_is_3.14
Video Solution 6
https://youtu.be/hmOGYmVmY1c
~savannahsolver
Video Solution by TheBeautyofMath
https://youtu.be/s6E4E06XhPU?t=640 (for AMC 10A)
https://youtu.be/cckGBU2x1zg?t=95 (for AMC 12A)
Video Solution by The Learning Royal
https://youtu.be/AWjOeBFyeb4
Yet another Video Solution
https://youtu.be/pUdeIBlp-y8
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Evan Chen《陳誼廷》
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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)
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)
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)
Other contests
Also listed on the USEMO page .
- (video 1) (video 2)
- USEMO 2022 (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 . ↩
IMAGES
VIDEO
COMMENTS
The test will be held on Thursday, February , . Please do not post the problems or the solutions until the contest is released. 2021 AMC 10A Problems. 2021 AMC 10A Answer Key. Problem 1.
Problem 5. Elmer the emu takes equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in equal leaps. The telephone poles are evenly spaced, and the st pole along this road is exactly one mile ( feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride?
Problem. Let be an isosceles trapezoid with and .Points and lie on diagonal with between and , as shown in the figure.Suppose , , , and .What is the area of . Solution 1. First realize that Thus, because we can say that and From the Pythagorean Theorem, we have and Because from the problem statement, we have that Solving, gives To find the area of the trapezoid, we can compute the area of and ...
AMC 8 Problems and Solutions - AoPS Wiki. Art of Problem Solving. AoPS Online. Math texts, online classes, and more. for students in grades 5-12. Visit AoPS Online ‚. Books for Grades 5-12 Online Courses. Beast Academy. Engaging math books and online learning.
Like solution 1, expand and simplify the original equation to and let . To find local extrema, find where . First, find the first partial derivative with respect to x and y and find where they are : Thus, there is a local extremum at . Because this is the only extremum, we can assume that this is a minimum because the problem asks for the ...
1961 IMO Problems/Problem 1. 1966 IMO Problems/Problem 5. 1968 IMO Problems/Problem 3. 1968 IMO Problems/Problem 5. 1970 Canadian MO Problems/Problem 1. 1970 IMO Problems/Problem 2. 1971 Canadian MO Problems/Problem 5. 1972 IMO Problems/Problem 5. 1972 USAMO Problems/Problem 4.
The center of the universe for students who love math.
ITMO/AITMO ( [Asian] International Teenagers Mathematical Olympiad) Directory of Problems w/ Solutions. Archive from 2005 to 2017. Every two years, except 2007. Mathematics. English. Taiwanese. IWYMIC (The International World Youth Mathematics Intercity Competition) All Problems Since 1999 w/ Solutions.
Art of Problem Solving. AoPS Online. Math texts, online classes, and morefor students in grades 5-12. Visit AoPS Online ‚. Books for Grades 5-12Online Courses. Beast Academy. Engaging math books and online learningfor students ages 6-13. Visit Beast Academy ‚.
Art of Problem Solving's Richard Rusczyk solves the 2020 AMC 10 A #21 / AMC 12 A #19.
USA Winter TST 2021 ; Because of the pandemic, there was no USA Winter TST for IMO 2022. ... 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 ...
The competitions start with the AMC 8, the AMC 10, and the AMC 12 exams, open to students in grade 8 or below, grade 10 or below, and grade 12 or below, respec-tively. Over 300,000 students take one of these multiple-choice exams each year. Afterward, based on their scores ontheAMC10or12,about5000studentsareinvitedto
Talk:2002 AMC 10B Problems/Problem 23. On solution 1, where did the value a1 go? Am I missing something? I am very confused. Art of Problem Solving is an. ACS WASC Accredited School.