How to Prepare for AMC 10: Topics, Plan and Tips

What Is The AMC 10, And Is It Multiple Choice?

Yes, the AMC 10 is entirely multiple choice: 25 questions in 75 minutes, written for students in grade 10 and below. It is the middle rung of the MAA's competition ladder, between the AMC 8 and the AMC 12.

Table 1: AMC 10 at a glance (sources named in each row, checked 11 October 2026).

The AMC 10 was introduced in 2000, according to the Wikipedia article on the American Mathematics Competitions, which also gives the scoring and calculator rules above. MAA's own policies page is the final word on rules for the current year, so check it before the competition.

Most children take the AMC 10 through their school. Ask the math department in September which date (A, B or both) the school offers, because registration closes well before November.

What Is The Difference Between AMC 8 And AMC 10?

The AMC 8 is a shorter middle-school contest; the AMC 10 is longer, harder, and opens the door to the next round, the AIME (American Invitational Mathematics Examination). The AMC 12 shares the AMC 10's format but covers math through grade 12.

Table 2: AMC 8, AMC 10 and AMC 12 compared.

AMC 10 vs AMC 12: which should your child take? If your child is in grade 10 or below and has not studied trigonometry and precalculus, the AMC 10 is the better fit, because the AMC 12 assumes topics they have not met. Many students take the AMC 10 until grade 10, then move up.

If your child has not yet sat the AMC 8, start there: our guide on how to prepare for AMC 8 covers that contest, and the wider math olympiad competitions overview shows where each contest leads.

Is AMC 10 Hard?

Yes, for almost everyone. The first questions on a paper are usually manageable for a strong algebra student, but the last few are hard enough that most students leave them blank, and that is by design. A child who answers the first 15 to 18 questions accurately is doing well.

The difficulty is not about advanced content. The AMC 10 stays within math through grade 10; the hardness comes from problems that need an idea rather than a procedure. A child who is top of the class in school algebra can still find the AMC 10 hard, because school tests reward speed on familiar steps and the AMC rewards finding a new route.

What Topics Are On The AMC 10?

The AMC 10 covers math through grade 10, and on past papers the problems fall into four areas. School covers the first two well; the last two are where most of the preparation time goes.

• Algebra. Linear and quadratic equations, systems, inequalities, sequences, functions and word problems. Your child should be fluent enough that algebra is a tool, not the obstacle.

• Geometry. Triangles, circles, areas, similar figures, coordinates and the Pythagoras theorem. Diagrams matter: drawing an accurate figure solves a surprising number of problems.

• Number theory. Divisibility, prime numbers, remainders, digits and the last digit of large powers. Most school curricula touch this only briefly.

• Counting and probability. Arrangements, choosing groups, and probability with equally likely outcomes. This is the area where a few weeks of focused study pays off fastest.

If school algebra itself is shaky, start there. A solid grip on algebra is the foundation for all four areas.

How Does AMC 10 Scoring Work, And When Should Your Child Guess?

Each correct answer earns 6 points, each blank earns 1.5, and each wrong answer earns 0, for a maximum of 150 (as reported by Wikipedia; confirm on MAA's policies page). Because a blank earns points and a wrong answer earns none, a blind guess usually costs your child points.

The break-even rule is simple. A guess is worth $6$ times the chance of being right; a blank is worth $1.5$. So a guess pays only when:

$$6p > 1.5$$

$$p > \frac{1}{4}$$

In plain words, guess only when your child is confident of being right more than one time in four, for example after ruling out most of the choices. Otherwise, leave it blank.

Here is what that means on a real paper. A child who answers 17 questions correctly and leaves 8 blank scores:

$$17 \times 6 + 8 \times 1.5 = 102 + 12 = 114$$

The same child who guesses wildly on those 8 and gets 1 right scores $18 \times 6 = 108$, which is lower.

What Is A Good AMC 10 Score?

A good AMC 10 score depends on your child's goal and grade. For the AIME, MAA listed a qualifying threshold of 100 on the AMC 10 (and 85 on the AMC 12) for the 2025-26 cycle, and the Wikipedia article (edited 6 October 2026) reports the AMC 10 cutoff as fixed at 100 from 2026-27. Thresholds are published by MAA, so check its announcement each year rather than relying on any single number.

For most children, especially a 7th, 8th or 9th grader taking it for the first time, a better question is: did the score go up from the last practice paper? A first-timer in grade 7 who scores in the 60s has done something real. The scoring section above shows the arithmetic: 100 points is roughly 15 correct answers with the rest left blank ($15 \times 6 + 10 \times 1.5 = 105$).

How Long Does It Take To Prepare For AMC 10?

Three to six months is realistic for a child who already handles school algebra comfortably and enjoys problem solving. A child in grade 7 or 8 who is newer to algebra should plan on a year or more, usually by working through the AMC 8 first.

The time is not mainly about learning new content. It is about building three habits: recognising which topic a problem belongs to, starting a problem without knowing the full path, and pacing 25 questions across 75 minutes.

How To Prepare For AMC 10 In Three To Six Months?

Work in three phases: fill topic gaps, practise problems by topic, then sit full timed papers. The six-month plan below compresses to three months by halving each phase.

Table 3: A six-month AMC 10 preparation plan (about 4 to 5 hours a week).

Three rules make the plan work:

1. Review beats volume. For every missed problem, your child writes one sentence: what idea they did not see. Reread the log every Sunday.

2. Protect the first 15 questions. Accuracy on the early questions matters more than reaching question 25.

3. Practise without a calculator from day one, since none is allowed on the day.

Which Books And Resources Help Most?

Past AMC 10 papers are the single most useful resource, because nothing else matches the style. After that, a few topic books fill the gaps school leaves.

• Introduction to Number Theory by Mathew Crawford (Art of Problem Solving, 2008).

• Introduction to Counting and Probability by David Patrick (Art of Problem Solving, second edition 2007).

• Introduction to Algebra and Introduction to Geometry by Richard Rusczyk (Art of Problem Solving, 2007 and 2006), for a child whose school algebra or geometry needs strengthening.

• The Art of Problem Solving, Volume 1: The Basics by Sandor Lehoczky and Richard Rusczyk (2006), which mixes all four areas in contest style.

These titles and dates were checked against library catalogue records on 11 October 2026; we name them because parents ask, and they are not affiliated with us. Start with the number theory or counting book, whichever area is weaker, rather than buying the full set. For the longer olympiad path beyond the AMC, see how to prepare for a math olympiad.

Examples Of AMC 10 Practice Problems

These six original problems match AMC 10 style across the four areas. Every answer was checked by a second method.

Example 1

If $3x + 2y = 18$ and $x - y = 1$, what is $x + y$?

From the second equation, $x = y + 1$. Substitute into the first:

$$3(y + 1) + 2y = 18$$

$$5y + 3 = 18$$

$$y = 3, \quad x = 4$$

Check: $3 \times 4 + 2 \times 3 = 18$ and $4 - 3 = 1$.

Final answer: $x + y = 7$.

Example 2

A price rises by 25 percent and then falls by 20 percent. What is the overall change?

The tempting answer: $25 - 20 = 5$, so a 5 percent rise.

Test it with a real price. Start at 100. A 25 percent rise gives 125. A 20 percent fall takes away 20 percent of 125, which is 25, not 20:

$$125 - 25 = 100$$

The two percentages are taken of different amounts, so they cannot simply be subtracted. As multipliers:

$$1.25 \times 0.8 = 1$$

Final answer: no overall change.

Example 3

What is the units digit of $7^{2026}$?

The units digits of powers of 7 repeat in a cycle of four: 7, 9, 3, 1.

$$2026 = 4 \times 506 + 2$$

So $7^{2026}$ has the same units digit as $7^{2}$, which is 49.

Final answer: 9.

Example 4

How many diagonals does a 12-sided polygon have?

Each vertex connects by a diagonal to every vertex except itself and its two neighbours, so to $12 - 3 = 9$ others. Each diagonal is counted from both ends:

$$\frac{12 \times 9}{2} = 54$$

Check with a square: $\frac{4 \times 1}{2} = 2$ diagonals, which is right.

Final answer: 54.

Example 5

A right triangle has legs 5 and 12. What is the length of the altitude to the hypotenuse?

The hypotenuse is $\sqrt{5^{2} + 12^{2}} = \sqrt{169} = 13$. Find the area two ways:

$$\frac{1}{2} \times 5 \times 12 = 30$$

$$\frac{1}{2} \times 13 \times h = 30$$

$$h = \frac{60}{13}$$

Final answer: $\frac{60}{13}$.

Example 6

Two different numbers are chosen at random from 1, 2, 3, 4, 5, 6. What is the probability that their product is even?

Count the opposite case. The product is odd only when both numbers are odd, choosing 2 of the 3 odd numbers. There are 15 pairs in total and 3 odd-odd pairs:

$$1 - \frac{3}{15} = \frac{4}{5}$$

Check by direct count: 12 of the 15 pairs contain at least one even number, and $\frac{12}{15} = \frac{4}{5}$.

Final answer: $\frac{4}{5}$.

Is Your Child Ready For The AMC 10?

Your child is ready to start AMC 10 preparation when school algebra feels easy and they enjoy a problem that does not have an obvious first step. Signs to look for:

• They have done well on the AMC 8 or found its later questions interesting rather than impossible.

• They solve multi-step algebra word problems without needing the method shown first.

• They come back to a problem after leaving it, instead of asking for the answer.

• They are curious about why, not just how; the second-guesser who keeps checking a correct answer will need practice trusting their work under time.

If your child is in grade 6 or 7 and shows these signs, the AMC 10 can be a stretch goal for the following year, with the AMC 8 as this year's target.

Where Does AMC 10 Preparation Go Wrong?

Most lost points come from habits, not from missing knowledge. These four mistakes cost students the most.

Mistake 1: Racing To Answer All 25

Where it slips in: The rusher treats the AMC 10 like a school test and tries to answer every question.

Don't do this: Do not aim to finish the paper.

The correct way: Aim for accuracy on the first 15 to 18 questions, then attempt later questions only where an idea appears. The student who protects the early questions usually beats the one who reaches question 25 with careless errors.

Mistake 2: Blind Guessing

Where it slips in: In the last ten minutes, a child fills every remaining bubble.

Don't do this: Do not guess when the chance of being right is one in four or less.

The correct way: Leave it blank unless most choices have been ruled out. The real-world version is in the scoring section: eight blind guesses that land one correct answer turn a 114 into a 108.

Mistake 3: Studying Only School Topics

Where it slips in: A strong algebra student spends all their time on harder algebra.

Don't do this: Do not skip number theory and counting because school does not test them.

The correct way: Spend the first month on whichever of the two is weaker. These areas show up on every paper, and a few weeks of focused study moves the score fastest.

Mistake 4: Doing Papers Without Reviewing Them

Where it slips in: A child sits a practice paper every weekend, checks the score, and moves on.

Don't do this: Do not measure progress by the number of papers done.

The correct way: Review every miss the next day and log the missing idea. The first instinct on Example 2 (subtracting the percentages) is a good test: children who log it once rarely repeat it, while children who only check the score make the same slip on the next paper.

When Should You Get Outside Help?

Many students prepare well with past papers and a topic book. Outside help is worth considering when:

• Your child's practice scores have been flat for six or more weeks despite regular work.

• The problem log keeps showing the same gap, usually in number theory or counting.

• School algebra itself is slowing them down on early questions.

• Practice papers are causing stress or avoidance, which can point to math test anxiety rather than a math gap.

In the last case, reduce the load before adding more. A child who is stuck often needs time with harder problems, which is the idea behind productive struggle in math, not more pressure.

How Does Bhanzu Help Students Preparing For Contests Like The AMC 10?

Bhanzu teaches live online classes from UKG to Grade 10, which covers the full AMC 10 age range. Every topic starts with why a method works, so a student can rebuild an idea under time pressure instead of hoping to remember a procedure, which is exactly what AMC problems demand.

Trainers begin with a Level 0 diagnostic that finds the real gaps, often in algebra fluency or in topics like ratios, that cost points on the early questions. Classes build strong foundations and reasoning; they are not an AMC cram course and do not work through past papers question by question.

Fit signal: this fits a student who wants deeper understanding of math alongside contest practice. It is not the right fit if you want a dedicated AMC coach focused only on past papers in the weeks before November.

Where Can Your Child Get Extra Help With Contest Math?

If your child wants more challenge than school provides, these programs match different stages:

• Advanced math classes for kids online for students working beyond their grade.

• High school math tutor support for grade 9 and 10 AMC 10 candidates.

• Middle school math tutor support for grade 7 and 8 students moving up from the AMC 8.

• Math enrichment programs for problem solving beyond the school syllabus.

Key Takeaways

• The AMC 10 is a 25-question, 75-minute multiple-choice paper for grade 10 and below, in A and B versions each November.

• How to prepare for AMC 10: fill gaps in number theory and counting, practise by topic, then sit full timed papers.

• A blank earns 1.5 points, so guess only when your child is right more than one time in four.

• The AIME threshold was 100 on the AMC 10 for 2025-26; check MAA's announcement each year.

• Review every miss in a problem log; review beats volume.

What Should You Do This Week?

Ask your child's school which AMC 10 date it offers and when registration closes. Then sit one past paper's first 15 questions together, with no calculator and a 45-minute timer, and note which of the four areas the misses came from: that area is month one of the plan. If you would like a trainer to find the gaps first, book a free demo class with a Bhanzu trainer.

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