The Art of Problem Solving: AMC Strategies

So you’re taking the AMC. 25 problems, 75 minutes, and your brain feels like it’s going to explode. Been there.

The American Mathematics Competitions are brutal. They test everything: algebra, geometry, number theory, combinatorics, probability. And they do it in a way that makes you think, not just compute.

But here’s the thing: there are strategies. Ways to approach these problems that make them more manageable. Let’s talk about them.

The Format (So You Know What You’re Up Against)

There are three levels: AMC 8 (8th grade and below), AMC 10 (10th and below), and AMC 12 (12th and below). Each one has 25 multiple-choice questions in 75 minutes.

The scoring is interesting: 6 points for correct, 1.5 for blank, 0 for wrong. This means random guessing is a bad idea, but if you can eliminate even one answer, guessing becomes worth it.

How to Actually Solve Problems

Read. Then Read Again.

I can’t stress this enough. These problems are full of traps. They’ll say “exactly” when you think “at least.” They’ll say “distinct” when you forget to check. They’ll ask for something weird that you miss if you’re rushing.

Circle the important words. Underline constraints. Make sure you know what they’re actually asking for.

Use the Answer Choices

This is multiple choice. That’s a huge advantage. Don’t forget it.

For algebra problems, just plug in the answers. One of them will work. For geometry, estimate and eliminate the obviously wrong ones. If the answers are symmetric, your solution probably is too.

Example: Solving $x^2 + 5x + 6 = 0$? The choices are $-3, -2, 2, 3, 6$. Just test them. $-3$ works? Check. $-2$ works? Check. You’re done.

Look for Patterns

Math competitions love patterns. Powers cycle. Remainders repeat. Sequences have structure.

If you’re finding the last digit of $7^{2025}$, don’t compute the whole thing. Find the pattern: $7^1$ ends in 7, $7^2$ ends in 9, $7^3$ ends in 3, $7^4$ ends in 1, then it repeats. Since 2025 = 4 × 506 + 1, the answer is 7.

Break It Down

When counting gets messy, break it into cases. When direct counting is hard, count the complement. When you’re stuck, try the pigeonhole principle.

“How many 3-digit numbers have at least one 5?” That’s hard. But “how many have no 5s?” That’s easier. Total minus that gives you the answer: $900 - 8 \times 9 \times 9 = 252$.

Time Management (This Is Critical)

You have 75 minutes for 25 problems. That’s 3 minutes each, but the problems get harder. So don’t spend 3 minutes on each.

Problems 1-10: These should be quick. 1-2 minutes each. If you’re taking longer, move on.

Problems 11-15: A bit harder. 2-4 minutes each.

Problems 16-20: Now we’re getting serious. 3-5 minutes each.

Problems 21-25: These are the killers. 5-10 minutes each, and you might not solve them all.

The Two-Pass Strategy

Here’s what works for me:

First pass (30-40 minutes): Blast through problems 1-15. Solve the easy ones, mark the ones you skip. Don’t get stuck.

Second pass (35-45 minutes): Go back to the ones you skipped, then tackle the hard ones. You’ve already got points from the easy problems, so you can afford to spend time on the tough ones.

The key is: don’t spend 10 minutes on problem 5 when you could solve problems 6, 7, and 8 in that time.

Topic-Specific Tricks

Algebra

Factor everything. Look for differences of squares, sums of cubes, common factors. Sometimes a clever substitution helps—like $u = x + \frac{1}{x}$.

Vieta’s formulas are your friend for quadratics and cubics. And completing the square solves a lot of optimization problems.

Geometry

Draw the diagram. Actually draw it, don’t just imagine it. Label everything. Look for similar triangles—they’re everywhere.

If algebra is getting messy, try coordinate geometry. Sometimes it’s easier.

Remember the special triangles: 30-60-90, 45-45-90, 3-4-5, 5-12-13. They show up constantly.

Number Theory

Prime factorization is your best friend. GCD and LCM show up everywhere. Divisibility rules save time.

Modular arithmetic makes huge problems manageable. Finding $2^{100} \bmod 7$? Find the pattern, don’t compute $2^{100}$.

Combinatorics

Distinguish permutations (order matters) from combinations (order doesn’t). Use complementary counting when direct counting is hard.

Casework is your friend. Break the problem into mutually exclusive cases, solve each one, add them up.

Common Mistakes (Don’t Do These)

1. Arithmetic errors: Double-check your calculations, especially with negatives.

2. Misreading problems: “At least” vs “exactly” matters. Read carefully.

3. Overcomplicating: The simplest solution is usually right. If your solution is getting messy, there’s probably a simpler way.

4. Random guessing: Only guess if you can eliminate at least one answer.

5. Ignoring constraints: Check that your answer actually satisfies all the conditions.

Practice Strategy

Work through past AMC exams. The problems repeat themes, so you’ll start recognizing patterns.

Focus on your weak areas. If geometry kills you, do more geometry problems.

Time yourself. Practice under exam conditions. Get comfortable with the pressure.

Review solutions. There are often multiple ways to solve a problem, and seeing different approaches helps.

The Mental Game

Stay calm. If you’re stuck, move on. Come back later. Panic makes everything worse.

Use all your time. If you finish early, check your work. Review the problems you weren’t sure about.

Remember: you don’t need to solve everything. Getting 15-20 problems right is often enough to qualify for the next round.

Final Thoughts

The AMC is hard. That’s the point. But it’s also learnable. The more problems you do, the more patterns you see, the better you get.

Don’t just memorize formulas. Understand why they work. The problems test understanding, not just computation.

And most importantly: have fun with it. These are interesting problems. They’re puzzles. Enjoy the challenge.

Good luck. You’ve got this.