Maths Competitions

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Last Updated: Sunday, 27 August 2023 04:24
Published: Tuesday, 11 April 2023 19:35
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Maths Olympiad and Competitions:

I've a separate section for Maths Competitions, as it deserves it's own section. There are many Maths competition organized all over the world. We'll look at few popular Maths competitions in the USA.

Mathworks Maths Contest (MMC):

This is an annual exam organized by Math Works, which is part of "Texas State College of Engineering and Science".This has nothing to do with "MathWorks" Company which developed MatLab. it is only for kids from 6th to 8th grade, so questions are relatively easy. t/here are 15 questions (no multiple choice). Test is administered in November. Top selected students participate in "Primary Math World Competition (PMWC)' in HongKong, with all expenses paid. Many students get prestigious "recognition certificates".

MMC: https://www.txst.edu/mathworks/MMC.html

Question papers from previous years => https://www.txst.edu/mathworks/mmc/previous-mmc-tests.html

 

 

Mathcounts Competition Series:

This is an annual exam series organized by Mathcounts, which is a non profit org. This exam is also only for middle schoolers. It's based on speed, and so questions are even easier than Mathworks exam. It's held in style similar to spelling bee. It has 4 levels, from school level to National level. Emphasis is more on speed, rather than complexity of problems. Many students get prestigious "recognition certificates".

Mathcounts.org: https://www.mathcounts.org/

Question papers from previous years => https://www.mathcounts.org/resources/past-competitions

 

 

Maths Olympiad

Maths Olympiad is one of the world's most prestigious Maths competition. The best thing about  Maths Olympiad is that it requires no prior knowledge except for basic Maths abilities (i.e basic algebra, geometry, etc). This makes it suitable for anyone to take this test irrespective of age (ideally >14 years) or grade (ideally > 8th grade).

IMO (International Maths Olympaid) is organized by an International organization every year. Each country gets to send 6 top maths students to participate in this. Anyone who is still in High School (i.e 12th grade or under0, but not in College yet, is qualified to participate. If you are selected to represent your country at the International Maths olympaid, you are among the top 0.001% of the population when it comes to Maths. That by itself is good enough to land you at worlds' top university, irrespective of your other achievements. When I say top universities, I mean MIT, Stanford and Berkeley !!!

Each country has it's own National Maths Olympiad competition, which they use to select top 500 or so candidates from all of the country. Then a second level exam is held to filter out top 30 candidates out of this, who go through a rigorous Maths Training for a couple of months. Finally 6 students are selected out of these top 30 to represent their country.

 

Indian National Maths Olympiad (INMO):

This is the Maths olympiad held every year in India conducted under the aegis of National Board of Higher Mathematics (NBHM). First a Regional Maths olympiad (RMO) is held at state level in which 30 top students from each state are invited to appear for the INMO. Top 30 students from INMO go thru a camp training and finally 6 are selected for IMO.

More details on wiki => https://en.wikipedia.org/wiki/Indian_National_Mathematical_Olympiad

Questions from prior years (1986 - 2012) in INMO => http://madanpur.in/inmo/inmo.htm

 

USA Maths Olympiad (USAMO):

This is the Maths olympiad held every year in USA conducted under the aegis of MAA (Mathematical Association of America). Any American or Candian citizen or resident is eligible to participate in these exams. It has been separated out in 2 parts for students in different levels. Students who are in college (i.e already graduated from high school) are not eligible to participate.

  • USAJMO (USA Junior MO) => This is for kids who are in 10th grade or lower or under 17.5 years of age.
  • USAMO (USA MO) => This is for kids in who are in 12th grade or lower or under 19.5 years of age.

More details on wiki => https://en.wikipedia.org/wiki/United_States_of_America_Mathematical_Olympiad

More details on USA MO is on MAA site: https://www.maa.org/math-competitions

Unfortunately none of the previous year olympiad papers are provided on MAA website for free. Instead they charge you $20 for a USB. Fortunately, there are other resources available which provide these papers for free.

There are 3 tier of Maths exams that students have to take depending on their grade level. Students who qualify in 1st tier (AMC 10/12), move to the 2nd tier exam (AIME), and finally to the 3rd tier (USAMO/USAJMO). There's also a 0th tier called AMC 8, which is to prepare young elementary school students for the tier 1 exam. About 300K students from USA/Canada sit in the below exams annually. Roughly 100K students appear for each of AMC 8, AMC 10 and AMC 12

  • AMC 8 (American Maths Competition): AMC 8 is for kids who are in 8th grade or lower. It's a 25-question, 75-minute, multiple-choice exam. Each question has only one correct answer and carries 1 mark, for a total of 25 marks. There's no -ve marking. It's a good stepping platform for kids to get their brains working in Maths and to prepare for higher level Maths Olympiad. About 100K students take AMC 8 exam. Scoring a perfect score here means you are in top 100 kids in Maths in all of USA (Top 0.1%), and most likely going to make it to AIME in high school. Even getting a score of 20+ gets you in top 5%.
  • AMC 10/12 (American Maths Competition): The AMC 10 and AMC 12 are both 25-question, 75-minute, multiple-choice examinations in high school mathematics. Question pattern is the same as AMC 8, but questions are harder. The AMC 10/12 being multiple choice questions are scored in a way that penalizes guesses. Correct answers are worth 6 points, incorrect questions are worth 0 points, and unanswered answers are worth 1.5 points (before 2006, unanswered questions were worth 2.5 points), to give a total score out of 150 points. Usually getting over 100 gets you close to qualifying for the next level exam (i.e AIME). The AMC 10 is for students in 10th grade and below (for USAJMO) and covers the high school curriculum up to 10th grade. The AMC 12 is for students in 12th grade or below (for USAMO) and covers the entire high school curriculum, including trigonometry, advanced algebra, and advanced geometry, but excluding calculus. Students may take the next level AIME after qualification by being in the (approximately) top 10% of scorers on AMC 10/12. Before 2020, it was lot tougher to qualify for AIME as only top 2.5% of scorers on the AMC 10 and the top 5% of scorers on the AMC 12.
  • AIME (American Invitational Maths Exam): AIME is a challenging competition offered for those who excelled on the AMC 10 and/or AMC 12. AMC 8 students are not considered for AIME. The AIME is a 15-question, 3-hour examination, in which each answer is an integer number between 0 to 999. Each question carries 10 marks for a total of 150 marks (In reality each question is only 1 mark for a total of 15 marks, but the total score later gets multiplied by 10 when considering for qualification). The questions on the AIME are much more difficult than those on the AMC 10 and AMC 12. Scores from both AMC 10 or AMC 12 and AIME are considered to qualify for next level. Total score from AMC 10/12 is 150 and from AIME is 150 for a total of 300. Scoring above 200 will usually qualify you for next level. To scroe 200+, it's better to score high in AMC10/12 as questions there are easier, and then try to solve may be 7-8 problems in AIME. Solving 10 or more problems in AIME is really tough, at least for me. Top-scoring participants on the AIME are invited to take the USAMO or USAJMO. About 250 students from AMC10 + AIME scores are invited for USJMO. Similarly 250 students from AMC12 + AIME scores are invited for USAMO. Here's the link for qualifying scores from all past years: https://www.momentumlearning.org/amc-cutoff-scores/
  • USAMO/USJMO: Once you have made it through the AMC 10/12 and AIME, you are invited to take the USAMO or USJMO exam. This is the toughest exam of all 3. Since 2002, the USAMO has been a six-question, nine-hour maths proof competition spread out over two days. (The IMO uses the same format.) On each day, four and a half hours are given for three questions. Each question is graded on a scale from 0 to 7, with a score of 7 representing a proof that is mathematically sound. Thus, a perfect score is 42 points, and is relatively rare (only 10 or so participants were able to get it for 2023)!! Top scorers in USAMO/USJMO (Total 60 from pool of 100K+ AMC10/12 participants) are invited to join the maths olympiad pgm (MOP), go thru an extensive training from which 6 students (6 selected from USAMO only. USJMO partcipants are eligible next year) are finally selected to represent USA in IMO.

Here's the chart showing how this :

 

Here is the cutoff for all prior years: https://artofproblemsolving.com/wiki/index.php/AMC_historical_results

More on MAA for latest year: https://maa.edvistas.com/eduview/report.aspx?mode=6

  • AMC 8: You need to get at least 20 out of 25 questions correct in order to get in top 5% so that you get a certificate.
  • AMC 10: You need to get at least 15 out of 25 questions correct in order to get in top 10% so that you qualify for AIME. To get in top 1% for a "distinguished Honor Certificate", you need to get 21-22 questions correct.
  • AMC 12: You need to get at least 15 out of 25 questions correct in order to get in top 10% so that you qualify for AIME. To get in top 1% for a "distinguished Honor Certificate", you need to get 21-22 questions correct. So, it's same requirement as AMC12, but last 10 questions in AMC12 are usually harder than in AMC10, so getting same score in AMC12 is harder than in AMC10.
  • AIME: You need to get at least 10 out of 15 questions correct in order to comfortably get a total score of AMC10/AMC12 + AIME to be > 200. A combined score of 200+ is usually needed to qualify for USJMO, while 220+ is needed to qualify for USAMO. AMC questions are easier, so get 20 questions correct in AMC, and then shoot for 10 questions in AIME. There are no multiple choice in AIME, so getting 10 questions correct is very difficult.
  • USJMO/USAMO: From MAA website, looking at results for 2023, looks like USJMO participants need 4 out of 6 to qualify for selection, while USAMO need 5 out of 6. Since the questions are extremely tough here, less than 1/2 the participants are able to get 3 or more correct.

Sample Practice Paper for AMC 8:

There is a sample practice paper for starters from AMC10 2017 on MAA website (this is one of the easiest paper that you can possibly find in Maths olympiad, and I realized it's from AMC 8, even though it says it's AMC 10/12). If you are able to solve all 8 questions over here, then Maths Olympiad is definitely for you to try.

Link => https://www.maa.org/sites/default/files/2017_AMC1012_Practice_Questions.pdf

If you find that too easy, you may head to AIME papers which are significantly harder than AMC papers. One such paper from 2020 is here (local copy) => 2020 AIME Paper

Being able to solve more than 7 out of 15 questions here is already getting you as a strong participant for USAMO/USJMO.

Few more sample papers:

 

International Maths Olympiad (IMO):

This is the ultimate Maths Olympiad, where very few in the world get to participate and represent their country. After being selected in top 6 in your country, you get to participate in IMO. Participating in the IMO and winning a medal gets you to stand in the same league as the Noble Prize winners in Maths. That's something !!

Official website: https://www.imo-official.org

All question papers from 1959 to until now can be found on this link: https://www.imo-official.org/problems.aspx

 

 

Maths Olympiad topics:

There a lot of topics Maths Olympiad covers, but all of them are basic Maths (12th grade or lower with no AP/Pre-AP courses), and don't require formula. However, I've seen complex number problems, which seems unfair to kids who are not familiar with such topics. Below are topics that I gathered:

  • Probability: These are the most popular questions on AMC8/10/12 and AIME. Questions get harder with higher grade level exams, and most of the times you have to think of all possible permutation/combination. Some of the problems sre very hard, and it's easy to miss some possible combinations, getting ll your hard work in vain.
  • Geometry: Basic Geometry including traingles, circles, Pythagoras Theorem, Incircle, CircumCircle, Orthocenter, etc. Also basic area, volume for 2D/3D structures.
  • Algebra: Basic algebra using variables, reducing polynomials, solving equations, etc.
  • Logical: Basic logical questions which require minimal maths
  • Trignometry, Advanced Algebra and Advanced Geometry are for AMC12 and beyond.

Modulo questions:

A very interesting set of questions is finding "n mod m" where n and m are some integers. You have to apply some basic pattern finding skill to solve such problems.

ex: Find 2^101 mod 5 => Here we try to find repeating pattern. Let's try few values of n=1,2,3,....

n 2^n ones digit 2^n mod 5
1 2 2 2
2 4 4 4
3 8 8 3
4 16 6 1
5 32 2 2
6 64 4 4
7 128 8 3
8 256 6 1

From above table, we see that ones digit can only be 2,4,8,6. Since mod 5 will only depend on ones digit, we can ignore other digits as tens, hundreds, etc. The ones digit follows a pattern as follows:

For any whole number k=0,1,2,3...

when n=4k (i.e n=4,8,12,...) => ones digit=6, so 2^n mod 5 = 1 (exclude k=0)

when n=4k+1 (i.e n=1,5,9,...) => ones digit=2, so 2^n mod 5 = 2

when n=4k+2 (i.e n=2,6,10,...) => ones digit=4, so 2^n mod 5 = 4

when n=4k+3 (i.e n=3,7,11,...) => ones digit=8, so 2^n mod 5 = 3

Since 101 is of form 4k+1 => 2^101 mod 5 = 2