Math competitions aren't just for prodigies. They're for anyone who enjoys problem-solving and wants to push their mathematical thinking beyond the classroom. Here's how to start.

The Competition Landscape

Middle School (Ages 11-14)

• MATHCOUNTS - Team and individual rounds, school→chapter→state→national progression
• AMC 8 - 25 questions, 40 minutes, multiple choice
• Math Kangaroo - International, puzzle-focused, accessible entry point

High School (Ages 14-18)

• AMC 10/12 - The main entry point to the US math competition pipeline
• AIME - Invitation-only based on AMC scores, 15 questions, 3 hours
• USAMO/USA(J)MO - Proof-based olympiad for top AIME scorers
• IMO - International Mathematical Olympiad, the pinnacle

Open/Online

• Advent of Math - Daily problems in December
• Project Euler - Computational math problems (programming helpful)
• Math Gym Arena - Online timed practice with real-time study buddies

What Competition Math Actually Tests

Competition math is fundamentally different from school math. School tests whether you can apply learned procedures. Competitions test whether you can solve problems you've never seen before.

The four pillars:

1. Number Theory - Primes, divisibility, modular arithmetic, GCD/LCM
2. Algebra - Equations, inequalities, polynomials, sequences
3. Geometry - Euclidean geometry, coordinate geometry, transformations
4. Combinatorics - Counting, probability, graph theory, recursion

The 6-Month Preparation Plan

Months 1-2: Foundation

• Master arithmetic speed (multiplication, division, fractions, percentages)
• Learn the basic theorems: Pythagorean theorem, triangle inequality, pigeonhole principle
• Solve 5 problems per day from past AMC 8 or easy AMC 10 papers

Months 3-4: Technique Building

• Study one topic deeply per week (number theory, then combinatorics, etc.)
• Solve 5-10 AMC 10 problems daily
• Start a problem journal: for every problem you can't solve, write the solution AND the technique used

Months 5-6: Competition Simulation

• Take full-length practice tests under timed conditions
• Review every mistake thoroughly
• Solve 2-3 AIME problems per week (even if you can't finish them - struggling builds skill)

Strategy on Competition Day

Time Management

• AMC 10: 30 questions in 75 minutes = 2.5 minutes per question
• But not all questions are equal. The first 15 are significantly easier
• Strategy: Blitz questions 1-15 (60-90 seconds each), spend remaining time on 16-25

Guessing Strategy (AMC)

• Right answer: +6 points
• Wrong answer: +0 points (no penalty on recent AMCs)
• Blank: +1.5 points
• If you can eliminate 2+ choices, guessing has positive expected value

Common Traps

• Answer choices designed to match common mistakes - if your answer appears, double-check
• Problems that seem to require heavy computation usually have a shortcut - you're missing it
• Read the question twice. "How many integers..." vs "How many positive integers..." are different questions

The Real Value

Score aside, math competitions build problem-solving instincts that transfer everywhere: programming, science, finance, engineering. The skills are the reward. The trophies are a bonus.

Many of the world's top engineers, scientists, and entrepreneurs were math competitors. Not because competition math is directly useful in their careers, but because the thinking patterns it builds are universally valuable.

Start This Week

Pick one past competition test at your level. Set a timer. See how you do. Don't worry about the score - worry about whether you enjoyed the challenge. If you did, you're in the right place.