The AMC 10 and the AIME are the first two steps of the main math competition path for U.S. high school students. They reward a kind of thinking that school math doesn't always practice: noticing structure, trying small cases, and finding a short path through a problem that looks long. The good news is that this kind of thinking improves with steady practice on real problems. This guide explains how the contests work and how to build a practice routine that actually helps.
How the contests work
| AMC 10 | AIME | |
|---|---|---|
| Who takes it | Students in grade 10 or below | Students who score high enough on the AMC 10 or AMC 12 |
| Problems | 25 multiple choice (A to E) | 15, each with a whole-number answer from 0 to 999 |
| Time | 75 minutes | 3 hours |
| Scoring | 6 points for a right answer, 1.5 for a blank, 0 for a wrong answer (150 max) | 1 point per right answer, no penalty for guessing |
The AIME cutoff changes every year, so aim to improve your score rather than to hit a particular number. Rules and dates can change too, so check the official Mathematical Association of America page for your year before the contest.
The blank-or-guess rule
On the AMC 10 a blank is worth 1.5 points, and a pure guess between five choices is worth 6 × 1/5 = 1.2 points on average. So a blind guess is slightly worse than leaving it blank. But if you can rule out two choices, a guess between the other three is worth 6 × 1/3 = 2 points on average, which is better than a blank. In short: don't guess blind, but do guess once you've eliminated two answers.
On the AIME there's no penalty, so never leave an answer empty.
Where the points are
AMC problems come from four main areas, and it's worth tracking which ones cost you points:
- Algebra: equations, functions, sequences, word problems.
- Geometry: triangles, circles, areas, coordinates. Drawing an accurate diagram solves a surprising number of these.
- Number theory: divisibility, primes, remainders, digits.
- Counting and probability: arrangements, cases, complementary counting.
The first ten or so problems on an AMC 10 are usually quick if you're careful. The middle problems decide most scores, and the last five are hard for nearly everyone. For most students, the fastest way to a better score is getting problems 1 to 15 right every time before spending long on 21 to 25.
A weekly routine
- A little every day. Five problems a day is about 150 a month. That's more than a full year of most school math, and it's how the habit sticks. Boxed's Daily five gives everyone the same five problems each day, climbing from the easy end of the contest to the hard end, with one try each.
- Target your weak topic once or twice a week. In Boxed's Practice, pick a difficulty and one topic, and keep going. After each problem you see the answer and a link to full solutions.
- Read solutions even when you got it right. The official answer often has a shorter method than yours. Those short methods are what you'll reuse.
- Keep a mistake list. Write one line for each problem you missed: the contest, the problem number and the idea you missed. Redo those problems a week later without looking.
- Take a full timed contest every few weeks: 25 problems in 75 minutes. Timing is a skill of its own.
For math team coaches
A short live round makes a good start to a meeting. In Boxed's Live game, the host puts problems on a projector and everyone answers on their own phone. You choose the difficulty and topics before each problem. Right answers score, faster right answers earn up to 50% extra, and the first correct answer earns a bonus. It turns practice into a game without changing what's being practiced. Then spend the rest of the meeting going through the solutions to the problems most people missed.
Where the problems come from
Boxed uses real problems from the AMC 10 (2010 to 2025) and the AIME (2010 to 2026), © Mathematical Association of America. Boxed itself stores only the contest, the problem number, the official answer and a topic tag. The problem text loads from the AoPS Wiki while you play, and each answer links to the full solutions there, which are some of the best free math writing on the web.