art of problem solving 2021 amc 10

How to Prepare for the AMC 10 Math Contest

What is the AMC 10 Contest

The AMC 10 is a math contest organized by the Mathematical Association of America (MAA).  The AMC 10 contest contains 25 multiple choice questions that must be completed within 75 minutes. You get 1.5 points for every question that is left blank, so remember to not guess blindly. 

The top 2.5% (approximately) students on the AMC 10 are invited to take the American Invitational Mathematics Examination  (AIME) .

Who can tak e the AMC 10

Students in grade 10 or below and under 17.5 years of age on the day of the contest can take the AMC 10. Note that even students in elementary schools are eligible to take the contest.

Where can you take the AMC 10

Many middle/high schools host the contest in their school. If your school doesn't offer the contest, you can try requesting your math teachers or principal to consider becoming a hosting site. You can also search for nearby locations hosting the contest using this zip code search .

The Mathematical Association of America also requests its member institutions of higher learning to consider offering the AMC 10 to students whose schools will not be offering the contest. You can find a list of alternate locations in your area here.

If you are in a school which does not host the contest, you can also reach out to math teachers in nearby high schools and check if any of them will allow outside students to take the contest. 

How to score high on the AMC 10

So, how can you score well on the AMC 10? Here are four key things to focus on:

Understanding all the content : The problems go beyond the school curriculum, so if you're new to competition math you will need to learn a variety of concepts from Algebra, Number Theory, Combinatorics, and Geometry

Being able to apply concepts to tricky problems :  AMC 10 problems are not straightforward like in school math, so the problems require creative thinking and that's something that can only be achieved with significant practice

​ Solving the problems accurately : You need to simulate test conditions to learn how to avoid common mistakes like calculation errors, silly mistakes, misreading the questions, etc 

​ Working quickly : You need to finish 25 problems in 75 minutes, and you need to learn good time management so you can attempt most of the questions without getting stuck.

How to prepare for the AMC 10 contest

Step I: Reviewing the Basics

If you are new to competition math or taking the AMC 10 for the first or second time, you need to ensure that you have a good understanding of the fundamental mathematical concepts. You can use th e following resources to get started:

1.  Free Maste ring AMC 8  Book : If you are new to competition math, then this free book can help you get started. This is a 400+ page hybrid book with video lectures for every topic and hundreds of practice problems. 

2.  Free AMC 10/12 Fundamentals Course : A free course that covers the most basic concepts for the AMC 10/12 like Expected Value, Modular Arithmetic, Diophantine Equations, Power of a Point, etc.

3.  Volume 1 book by AoPS : The Volume 1 essentially provides a summary of the Introduction Series by AoPS, covering many important concepts.

Step II  (If time) : Build Str ong Fundamentals

If you are really serious about AMC 10 preparation, it is highly recommended that your work through the AoPS Introductory series books. They are quite comprehensive, but they take a long time to complete. If you are just starting your preparation less than 2 months before the contest, I recommend you skip to Step 3.  ​

4.  Introduction Series books by AoPS  (In order of importance)

    a.  Introduction to Counting & Probability

    b.  Introduction to Geometry

    c.  Introduction to Algebra

    d.  Introduction to Number Theory

You can also use Alcumus for extra practice along with these books. 

In addition, you can also check out paid classes from Alphastar Academy , Art of Problem Solving , and many other organizations

Step III : Maste ring AMC 10 Concepts

In the last 1-2 months before the contest, it is important to review and refresh all the concepts that are important for the AMC 10 test. Moreover the AoPS Introduction books do not cover some of the topics that have come up in recent AMC 10 tests, which need to be studied from other sources.

5.  Maste ring AMC 10/12 Ch apters 1 - 40: This free book will help you learn and practice advanced concepts useful for AMC 10 problems such as Linearity of Expectation, Markov Chains, Cyclic Quadrilaterals, etc. 

Step IV: Practice Contests & Mocks

In order to practice speed and accuracy, it's important to take full AMC 10 tests. 

Some tips w hen mocking AMC 10's:

1. Don't br ush off silly mistakes:  Rather than treating a silly as something that's not a big deal, analyze why you made that silly and what you will do not to make that mistake again. Here is a video with tips to avoid sillies .

2. Peek but not read: For problems that you are not able to solve, don't just "read the solution". Rather, try to look at a few lines of the solution and then solve the rest of the problem on your own.

3. Build speed gradually:  At first, try to work at a pace that you can achieve high accuracy with. Then, try to gradually increase your speed while still maintaining high accuracy. Never rush or work carelessly to increase speed as this will only reduce your accuracy significantly. 

List of Mock Contests

1.  Official AMC 10 from previous years with solutions : Use these tests with a 75 minute timer to simulate test taking situations. The AMC's are increasing in difficulty every year, so only AMC 10 contests from 2021 Fall and after are representative of the difficulty of recent AMC 10.

2.  AoPS Created AMC 10 Mock Contest 2023 : A high quality mock contest created by Art of Problem Solving problem writers

3.  AoPS Created AMC 10 Mock Contest 2022 : A high quality mock contest created by Art of Problem Solving problem writers

4.  AoPS Created AMC 10  Mock Contest 2021 : Another high quality mock created by Art of Problem Solving problem writers

5.  AoPS User created AMC 10 Mock Contests : These mock contests are created by users on the AoPS forums and are of varying quality and difficulty levels. Still a good practice if you have completed all the previous year’s AMC 10 c ontests.

6.  AlphaStar AMC 10 Practice Tests Book : A book with some high quality mock contests created by the problem writing team at AlphaStar Academy.

7. CMC Mock Contests : Nearly all of the tests created by CMC (even the ones from 2017) are extremely high quality and close in difficulty to AMC's nowadays. 

8. DMC Mock Contests : Another site with high quality mock AMC 10s. 

9. OMC Mock Contest : A good AMC 10 mock from 2023.

Problem Trainers

For even more practice, there are some useful  websites which help you practice specific topics and improve problem solving speed.

1.  Alcu mus : Free resource by AoPS to practice problems from specific topics.  

2.  AMC Trivial : This site generates problems from past contests. You can even make custom mocks from older questions and it can give problems from specific topics or difficulty levels.

3.  MAATester : This site makes it easy to search for problems related to any topic from various contests.

4.  For the Win : Practice speed and accuracy by playing head to head against other players to answer questions quickly

By using these resources and practicing regularly, you can improve your skills and increase your chances of success on the AMC 10.

Crucial Tips for improving your score on the AMC 10

1. Bring colored pencils/pens . For Geo metry prob lems, sometimes it may help to use colors to mark different regions or draw similar triangles. Colors can help you visualize things better and make geometric observations.

2. Double check what the question is asking for. When you finish solving a problem, take a quick glance at the last sentence in the question to confirm that you are marking what the question is asking for. For example: You solve a complex Geometry problem and correctly calculate the side length of a square, mark that answer and move on…but the question was asking for the area of the square. You don’t want to ma ke that mistake. 

3. Double check the bubbled answers . Towards the end, don’t forget to double check the answer sheet to make sure that the answers you have bubbled match up to the answer you have marked in the question paper. 

One time I actually forgot that I h ad skipped a problem and then for all the subsequent problems the answers were shifted up by one so then all of those problems got marked wrong. Here is an important tip to check the answers - don't just check it as “A C D B”. Always try to associate the question number like “ 15 A, 16 C, 17 D, 18 D, etc.”

Also don’t leave this task to the last second, as you may be in the middle of solving a problem and won’t like to waste time in checking. Try to double check a few minutes before the end, maybe around the 65-70 minute mark. Most test centers will announce a time warning a few minutes before the end, so plan to check things at that time.

4. Don’t get stuck on any question . All questions have equal points, so your goal is to solve as many questions as possible. The questions are usually in increasing order of difficulty but some questions can be misplaced. Don’t get stuck on any question. If you are not making progress or the problem looks too bashy, then just mark it and skip it during the first pass. You should come back to it later. You should not have the mindset that oh it's the number five I should be able to solve it but I'm not able to solve it. 

5.  Don’t be intimidated by the questions at the end . There are 25 questions and 75 minutes to finish the test, so you have an average of 3 minutes per question which is not a lot of time. So make sure you budget your time wisely and leave enough time for the end questions which are typically harder. But some of them may be misplaced or from one of your stronger subjects, so depending on how much time you have left, try to solve the ones that are from your strong subjects or that look doable

6.  Sleep well, eat healthy, Stay c alm . This seems like common sense, but it is important to emphasize it again. Don’t try to cram till 4am on the day before the contest. A good night’s sleep and relaxed mind can be more helpful in making sure that you don’t make silly mistakes. 

One simple meditation trick that is worth trying just before the test: Just close your eyes and take 5 deep breaths and it will help to calm your nerves. 

Some people recommend eating dark chocolate before the test as it is considered a mood enhancer.

7. Don’t guess blindly . Remember you get 1.5 points for any question left blank, so don’t guess blindly as the chances of getting it right is only 20%. But you can use meta solving techniques to improve your chances of  finding answers to problems without actually solving them.

8. Timed Test Practice . Since only a few days are remaining for the AMC 10, try to take at least one timed test - either previous year’s AMC 10 contest or a mock test, and remember to keep a 75 minute timer to simulate real test conditions. And try to apply the tips mentioned here during that practice test. This will make you become used to handling the stress of the test and help you with learning the time management that works best for you.

9. Review, Revise, Refresh . On the day before the test, try not to stress and don’t start learning new topics. Just review concepts and formulas for topics that you already know.   

Good Luck on the AMC 10.

What materials are allowed on the AM C 10 contest

According to MAA Guidelines , during the competition, students are only allowed:

Writing utensils ( Note: colored pens/pencils are allowed )

Blank scratch paper

Prohibited materials include:

Calculators

Smartwatches

Computing devices

Protractors

Graph paper

Calculate for all schools

Your chance of acceptance, your chancing factors, extracurriculars, competing in national math competitions.

I've always loved math and I'm interested in participating in national math competitions. Can someone give me an idea of what some of the major ones are? How do you prepare for these competitions? What benefits can they have on your college applications?

There are several major math competitions you can participate in, from regional to national levels. Some of the significant ones are:

1. American Mathematics Competitions (AMC): This includes the AMC 8, AMC 10, and AMC 12, each targeting a specific grade level. The AMC 10 and 12 act as qualifying exams for the American Invitational Mathematics Examination (AIME).

2. American Invitational Mathematics Examination (AIME): Students who perform well on the AMC 10 or AMC 12 qualify for this exam. A top AIME score can lead to participation in the USA Mathematical Olympiad (USAMO).

3. USA Mathematical Olympiad (USAMO) and Junior Mathematical Olympiad (USAJMO): These are very prestigious, invitation-only competitions for high school students. Success in these events can lead to joining the USA International Mathematical Olympiad team.

4. MathCounts: This is a national competition primarily for middle school students, featuring both individual and team rounds.

To prepare for these competitions, you can:

- Practice previous years' problems and solutions, which are often available online.

- Participate in math clubs or competition-focused extracurriculars at your school, or you can consider starting one if it doesn't exist.

- Explore online resources and forums (e.g., Art of Problem Solving) to further develop your problem-solving skills.

- Look for math summer camps or workshops that include competition preparation.

As for the benefits on your college applications:

- High achievements in these competitions can be an impressive addition to your resume. They showcase your mathematical prowess and dedication to the subject, making your application stand out.

- Such participation is considered a Tier 1 or Tier 2 extracurricular activity, which can positively impact your chances of admission.

- Strong performance in these competitions might even lead to scholarships or research opportunities in college.

Overall, competing in national math competitions demonstrates a commitment to pursuing a deeper understanding of math and can have a significant impact on your college applications, especially if you're planning on majoring in a STEM field. Good luck!

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The Moscow Metro Museum of Art: 10 Must-See Stations

There are few times one can claim having been on the subway all afternoon and loving it, but the Moscow Metro provides just that opportunity.  While many cities boast famous public transport systems—New York’s subway, London’s underground, San Salvador’s chicken buses—few warrant hours of exploration.  Moscow is different: Take one ride on the Metro, and you’ll find out that this network of railways can be so much more than point A to B drudgery.

The Metro began operating in 1935 with just thirteen stations, covering less than seven miles, but it has since grown into the world’s third busiest transit system ( Tokyo is first ), spanning about 200 miles and offering over 180 stops along the way.  The construction of the Metro began under Joseph Stalin’s command, and being one of the USSR’s most ambitious building projects, the iron-fisted leader instructed designers to create a place full of svet (radiance) and svetloe budushchee (a radiant future), a palace for the people and a tribute to the Mother nation.

Consequently, the Metro is among the most memorable attractions in Moscow.  The stations provide a unique collection of public art, comparable to anything the city’s galleries have to offer and providing a sense of the Soviet era, which is absent from the State National History Museum.  Even better, touring the Metro delivers palpable, experiential moments, which many of us don’t get standing in front of painting or a case of coins.

Though tours are available , discovering the Moscow Metro on your own provides a much more comprehensive, truer experience, something much less sterile than following a guide.  What better place is there to see the “real” Moscow than on mass transit: A few hours will expose you to characters and caricatures you’ll be hard-pressed to find dining near the Bolshoi Theater.  You become part of the attraction, hear it in the screech of the train, feel it as hurried commuters brush by: The Metro sucks you beneath the city and churns you into the mix.

With the recommendations of our born-and-bred Muscovite students, my wife Emma and I have just taken a self-guided tour of what some locals consider the top ten stations of the Moscow Metro. What most satisfied me about our Metro tour was the sense of adventure .  I loved following our route on the maps of the wagon walls as we circled the city, plotting out the course to the subsequent stops; having the weird sensation of being underground for nearly four hours; and discovering the next cavern of treasures, playing Indiana Jones for the afternoon, piecing together fragments of Russia’s mysterious history.  It’s the ultimate interactive museum.

Top Ten Stations (In order of appearance)

Kievskaya station.

Kievskaya Station went public in March of 1937, the rails between it and Park Kultury Station being the first to cross the Moscow River.  Kievskaya is full of mosaics depicting aristocratic scenes of Russian life, with great cameo appearances by Lenin, Trotsky, and Stalin.  Each work has a Cyrillic title/explanation etched in the marble beneath it; however, if your Russian is rusty, you can just appreciate seeing familiar revolutionary dates like 1905 ( the Russian Revolution ) and 1917 ( the October Revolution ).

Mayakovskaya Station

Mayakovskaya Station ranks in my top three most notable Metro stations. Mayakovskaya just feels right, done Art Deco but no sense of gaudiness or pretention.  The arches are adorned with rounded chrome piping and create feeling of being in a jukebox, but the roof’s expansive mosaics of the sky are the real showstopper.  Subjects cleverly range from looking up at a high jumper, workers atop a building, spires of Orthodox cathedrals, to nimble aircraft humming by, a fleet of prop planes spelling out CCCP in the bluest of skies.

Novoslobodskaya Station

Novoslobodskaya is the Metro’s unique stained glass station.  Each column has its own distinctive panels of colorful glass, most of them with a floral theme, some of them capturing the odd sailor, musician, artist, gardener, or stenographer in action.  The glass is framed in Art Deco metalwork, and there is the lovely aspect of discovering panels in the less frequented haunches of the hall (on the trackside, between the incoming staircases).  Novosblod is, I’ve been told, the favorite amongst out-of-town visitors.

Komsomolskaya Station

Komsomolskaya Station is one of palatial grandeur.  It seems both magnificent and obligatory, like the presidential palace of a colonial city.  The yellow ceiling has leafy, white concrete garland and a series of golden military mosaics accenting the tile mosaics of glorified Russian life.  Switching lines here, the hallway has an Alice-in-Wonderland feel, impossibly long with decorative tile walls, culminating in a very old station left in a remarkable state of disrepair, offering a really tangible glimpse behind the palace walls.

Dostoevskaya Station

Dostoevskaya is a tribute to the late, great hero of Russian literature .  The station at first glance seems bare and unimpressive, a stark marble platform without a whiff of reassembled chips of tile.  However, two columns have eerie stone inlay collages of scenes from Dostoevsky’s work, including The Idiot , The Brothers Karamazov , and Crime and Punishment.   Then, standing at the center of the platform, the marble creates a kaleidoscope of reflections.  At the entrance, there is a large, inlay portrait of the author.

Chkalovskaya Station

Chkalovskaya does space Art Deco style (yet again).  Chrome borders all.  Passageways with curvy overhangs create the illusion of walking through the belly of a chic, new-age spacecraft.  There are two (kos)mosaics, one at each end, with planetary subjects.  Transferring here brings you above ground, where some rather elaborate metalwork is on display.  By name similarity only, I’d expected Komsolskaya Station to deliver some kosmonaut décor; instead, it was Chkalovskaya that took us up to the space station.

Elektrozavodskaya Station

Elektrozavodskaya is full of marble reliefs of workers, men and women, laboring through the different stages of industry.  The superhuman figures are round with muscles, Hollywood fit, and seemingly undeterred by each Herculean task they respectively perform.  The station is chocked with brass, from hammer and sickle light fixtures to beautiful, angular framework up the innards of the columns.  The station’s art pieces are less clever or extravagant than others, but identifying the different stages of industry is entertaining.

Baumanskaya Statio

Baumanskaya Station is the only stop that wasn’t suggested by the students.  Pulling in, the network of statues was just too enticing: Out of half-circle depressions in the platform’s columns, the USSR’s proud and powerful labor force again flaunts its success.  Pilots, blacksmiths, politicians, and artists have all congregated, posing amongst more Art Deco framing.  At the far end, a massive Soviet flag dons the face of Lenin and banners for ’05, ’17, and ‘45.  Standing in front of the flag, you can play with the echoing roof.

Ploshchad Revolutsii Station

Novokuznetskaya Station

Novokuznetskaya Station finishes off this tour, more or less, where it started: beautiful mosaics.  This station recalls the skyward-facing pieces from Mayakovskaya (Station #2), only with a little larger pictures in a more cramped, very trafficked area.  Due to a line of street lamps in the center of the platform, it has the atmosphere of a bustling market.  The more inventive sky scenes include a man on a ladder, women picking fruit, and a tank-dozer being craned in.  The station’s also has a handsome black-and-white stone mural.

Here is a map and a brief description of our route:

Start at (1)Kievskaya on the “ring line” (look for the squares at the bottom of the platform signs to help you navigate—the ring line is #5, brown line) and go north to Belorusskaya, make a quick switch to the Dark Green/#2 line, and go south one stop to (2)Mayakovskaya.  Backtrack to the ring line—Brown/#5—and continue north, getting off at (3)Novosblodskaya and (4)Komsolskaya.  At Komsolskaya Station, transfer to the Red/#1 line, go south for two stops to Chistye Prudy, and get on the Light Green/#10 line going north.  Take a look at (5)Dostoevskaya Station on the northern segment of Light Green/#10 line then change directions and head south to (6)Chkalovskaya, which offers a transfer to the Dark Blue/#3 line, going west, away from the city center.  Have a look (7)Elektroskaya Station before backtracking into the center of Moscow, stopping off at (8)Baumskaya, getting off the Dark Blue/#3 line at (9)Ploschad Revolyutsii.  Change to the Dark Green/#2 line and go south one stop to see (10)Novokuznetskaya Station.

Check out our new Moscow Indie Travel Guide , book a flight to Moscow and read 10 Bars with Views Worth Blowing the Budget For

Jonathon Engels, formerly a patron saint of misadventure, has been stumbling his way across cultural borders since 2005 and is currently volunteering in the mountains outside of Antigua, Guatemala.  For more of his work, visit his website and blog .

Photo credits:   SergeyRod , all others courtesy of the author and may not be used without permission

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Title : Active carbons as nanoporous materials for solving of environmental problems

However, up to now, the main carriers of catalytic additives have been mineral sorbents: silica gels, alumogels. This is obviously due to the fact that they consist of pure homogeneous components SiO2 and Al2O3, respectively. It is generally known that impurities, especially the ash elements, are catalytic poisons that reduce the effectiveness of the catalyst. Therefore, carbon sorbents with 5-15% by weight of ash elements in their composition are not used in the above mentioned technologies. However, in such an important field as a gas-mask technique, carbon sorbents (active carbons) are carriers of catalytic additives, providing effective protection of a person against any types of potent poisonous substances (PPS). In ESPE “JSC "Neorganika" there has been developed the technology of unique ashless spherical carbon carrier-catalysts by the method of liquid forming of furfural copolymers with subsequent gas-vapor activation, brand PAC. Active carbons PAC have 100% qualitative characteristics of the three main properties of carbon sorbents: strength - 100%, the proportion of sorbing pores in the pore space – 100%, purity - 100% (ash content is close to zero). A particularly outstanding feature of active PAC carbons is their uniquely high mechanical compressive strength of 740 ± 40 MPa, which is 3-7 times larger than that of  such materials as granite, quartzite, electric coal, and is comparable to the value for cast iron - 400-1000 MPa. This allows the PAC to operate under severe conditions in moving and fluidized beds.  Obviously, it is time to actively develop catalysts based on PAC sorbents for oil refining, petrochemicals, gas processing and various technologies of organic synthesis.

Victor M. Mukhin was born in 1946 in the town of Orsk, Russia. In 1970 he graduated the Technological Institute in Leningrad. Victor M. Mukhin was directed to work to the scientific-industrial organization "Neorganika" (Elektrostal, Moscow region) where he is working during 47 years, at present as the head of the laboratory of carbon sorbents.     Victor M. Mukhin defended a Ph. D. thesis and a doctoral thesis at the Mendeleev University of Chemical Technology of Russia (in 1979 and 1997 accordingly). Professor of Mendeleev University of Chemical Technology of Russia. Scientific interests: production, investigation and application of active carbons, technological and ecological carbon-adsorptive processes, environmental protection, production of ecologically clean food.   

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