Math Contest Strategies: 20 Tips to Maximize Your Score
Two students walk into the same math contest with the same mathematical knowledge. One scores significantly higher than the other. What separates them is rarely the ability to solve harder problems. It is how they manage the time they have, which problems they attempt and in what order, how they check their work, and how they handle the moment when a problem resists their first approach.
Contest performance is two things simultaneously: mathematical skill and contest skill. Both skills can be learned. Most students spend nearly all of their preparation time on the first and almost none on the second. This guide is about the second.
These 20 strategies apply across AMC 8, AMC 10/12, MATHCOUNTS, AIME, and Olympiad-level competitions. Some are format-specific — the scoring model for AMC is different from AIME, and those differences change optimal strategy. Most are universal. All of them are actionable. Use the competition calendar to identify which contests you are targeting this year.
AIME answers are integers from 000 to 999. Guessing is essentially never productive, but there is no penalty for a wrong answer, so always submit something — even if it is just your best estimate based on partial work.
Quick Reference: All 20 Strategies at a Glance
Use this table as a pre-contest checklist. Full explanations follow in each section below.| # | Category | Strategy at a Glance |
|---|---|---|
| 1 | Time Management | Survey the full problem set before writing anything |
| 2 | Time Management | Assign yourself a per-problem time budget |
| 3 | Time Management | Mark and move — never stall on a single problem |
| 4 | Time Management | Reserve the final 5 minutes for review and answer transfer |
| 5 | Problem Selection | Solve the problems you are most confident in first |
| 6 | Problem Selection | Read every problem before skipping any |
| 7 | Problem Selection | On AMC, use the scoring model to decide when to guess |
| 8 | Problem Selection | On olympiad contests, attempt all problems — partial credit exists |
| 9 | Problem Solving | Write everything down — no purely mental computation under pressure |
| 10 | Problem Solving | Draw and label a diagram for every geometry problem |
| 11 | Problem Solving | Try small cases before tackling the general problem |
| 12 | Problem Solving | Look for symmetry, invariants, and extremal cases first |
| 13 | Problem Solving | If stuck, restate the problem in your own words |
| 14 | Checking | Check every numerical answer by substituting back |
| 15 | Checking | Estimate the order of magnitude before finalizing |
| 16 | Checking | Verify special cases: n=0, n=1, equal values, boundary values |
| 17 | Checking | Re-read the problem after solving — answer the question asked |
| 18 | Mental Prep | Warm up with two or three easy problems before the contest begins |
| 19 | Mental Prep | Treat a hard problem as an interesting challenge, not a threat |
| 20 | Mental Prep | Debrief after every contest: what cost points, what worked well |
Part 1: Time Management (Tips 1-4)
Time is the resource that most students mismanage in math contests. The mistakes are predictable: spending eight minutes on a single difficult problem while five easier problems sit unattempted, running out of time before reaching the end of the problem set, or rushing the final few problems and making careless errors. All of these are correctable with deliberate strategy.Tip 1: Survey the Full Problem Set Before Writing Anything
The first 90 seconds of a math contest are more valuable than most students realize. Before attempting any problem, read through the entire set — not to solve, but to categorize. Which problems look immediately accessible? Which will require more work? Which should you come back to? This survey accomplishes two things. First, it prevents the single most costly time management error: getting absorbed in a hard early problem while missing several easier later ones. Second, it activates pattern recognition — your brain begins processing problems in the background even while you are working on others. In practice: On a 25-problem AMC 10, spend 60 to 90 seconds reading the full set. Mark problems by confidence level: circle the ones you expect to solve quickly, put a question mark on the ones you are unsure about, and leave the rest for consideration after the first pass.Tip 2: Assign Yourself a Per-Problem Time Budget
On any timed contest, you should know your per-problem budget before the clock starts. This is simple arithmetic: total time divided by total problems gives your average. AMC 10/12: 75 minutes, 25 problems — 3 minutes per problem, with the understanding that early problems should take 60 to 90 seconds and later problems may warrant 4 to 5 minutes. MATHCOUNTS Sprint: 40 minutes, 30 problems — roughly 80 seconds per problem. AIME: 3 hours, 15 problems — 12 minutes per problem average, with significant variation. The discipline: When your time budget for a problem is up, mark it and move on. The decision to keep working on a problem should be deliberate, not something that happens because you lost track of time.Tip 3: Mark and Move — Never Stall on a Single Problem
The most expensive mistake in any math contest is spending disproportionate time on a single problem while other solvable problems sit untouched. The psychology is understandable: abandoning a problem you have invested several minutes in feels like a loss. The mathematics is clear: it is not. A problem that has resisted three different approaches in five minutes is telling you something. It is either harder than it looks, or it requires a technique you have not yet identified. Neither of those situations improves with an additional five minutes of unfocused effort. Mark it, move on, and return with fresh eyes if time permits. Students who practice this discipline consistently — who can disengage from a stuck problem without frustration — outscore equally skilled students who cannot. This is a trainable habit, and it should be practiced in every mock contest.Tip 4: Reserve the Final 5 Minutes for Review and Answer Transfer
The final five minutes of a math contest are dedicated to two things: reviewing answers you are uncertain about, and ensuring your answer sheet is filled in correctly. Neither of these should happen under genuine time pressure. Answer transfer errors — selecting the wrong bubble, writing the wrong digit in an AIME box, skipping a row on the answer sheet — are among the most frustrating sources of lost points because they do not reflect mathematical errors. They reflect poor time management. Build a hard stop into your pacing: when five minutes remain, close the problem you are working on and shift to review and transfer.Part 2: Problem Selection (Tips 5-8)
Problem selection is about maximizing expected points within the time you have. It is not about solving problems in order, and it is not about attempting every problem. It is about making deliberate choices about where to allocate effort.Tip 5: Solve the Problems You Are Most Confident In First
Confidence-first ordering means doing your most accessible problems before your hardest ones. This is not the same as easiest-first, because problem difficulty is personal — a number theory problem that stumps one student may be straightforward for another who has spent a month on divisibility. Working confidently in your first pass serves two purposes. It banks points efficiently, ensuring you do not run out of time before completing the problems you were fully capable of solving. It also builds momentum: the mental state that comes from solving several problems correctly is genuinely useful when you encounter harder material later in the contest.Tip 6: Read Every Problem Before Skipping Any
A problem that looks hard from the first sentence is sometimes completely accessible once you have read the full statement. Problem writers occasionally front-load complexity in the setup and make the actual question quite direct. Skipping a problem based on its first two lines is a decision made with incomplete information. Read the full problem, assess what it is actually asking, then decide whether to attempt it now or return to it later. This takes 15 to 20 seconds and frequently changes the prioritization.Tip 7: On AMC, Use the Scoring Model to Decide When to Guess
The AMC 10 and AMC 12 award 6 points for correct answers and 0 points for incorrect or blank answers. This means there is no penalty for guessing, but the calculus is not as simple as ‘always guess on everything you skip.’ The threshold for guessing: can you eliminate at least one option? If yes, guess. Even if the problem is completely opaque and you cannot eliminate anything, a guess still has positive expected value.| Contest | Correct | Wrong / Blank | Guessing Threshold |
|---|---|---|---|
| AMC 10/12 | +6 pts | 0 pts (no penalty) | Guess if you can eliminate 1 or more choices |
| MATHCOUNTS Sprint | +1 pt | 0 pts | Always attempt — no penalty |
| AIME | +1 pt | 0 pts | Always attempt — answer is 000 to 999 |
| USAMO / Olympiad | 0-7 pts per problem | 0 pts | Attempt all — partial credit available |
Tip 8: On Olympiad Contests, Attempt All Problems — Partial Credit Exists
Olympiad-style contests — USAMO and international Olympiads — award partial credit based on proof quality and progress toward a solution. A complete but inelegant proof of one subcase is worth more than no written work. The selection strategy for Olympiad contests is different from multiple-choice formats. Read all problems in the first few minutes. Identify the one or two problems where you have the clearest path: where you can see at least the opening moves of a solution. Work those first, completely and carefully. Then return to the harder problems with whatever time remains, writing up as much rigorous reasoning as you can construct. Olympiad principle: A half-page of careful, correct reasoning toward a partial result earns points. A half-page of confident but incorrect assertions earns nothing. Rigor matters more than quantity of written work.Part 3: In-Problem Solving Strategies (Tips 9-13)
These tips govern how you work through individual problems — the habits and approaches that separate clean, efficient solving from circular or error-prone effort.Tip 9: Write Everything Down — No Purely Mental Computation Under Pressure
Under time pressure, students frequently attempt to hold intermediate steps in working memory rather than writing them down. This is almost always a mistake. Working memory degrades under pressure, and the mental effort required to maintain those steps crowds out the problem-solving attention you need. Write every step. This is not a sign of weak mathematics — it is the practice of every serious competition mathematician. Written work also makes checking efficient: instead of reconstructing a solution from scratch, you can scan what you wrote and verify each step independently. Practical habit: Use your scratch paper liberally. Start each problem in a new area of the page. Label which problem each block of work belongs to. This makes returning to a problem after moving on significantly faster.Tip 10: Draw and Label a Diagram for Every Geometry Problem
Geometry problems without a diagram are solved more slowly and with more errors than geometry problems with one. Draw the diagram large enough to label clearly. Mark all given lengths, angles, and relationships on the diagram as you read the problem statement. Add auxiliary lines — angle bisectors, altitudes, circumscribed circles — as they become relevant. A diagram that accurately reflects the given information often reveals the solution path without requiring any algebraic manipulation. If a geometry problem seems to resist every approach, redraw the diagram from scratch, taking extra care with proportions. An inaccurate initial diagram is a more common source of wrong approaches than most students realize.Tip 11: Try Small Cases Before Tackling the General Problem
When a problem involves a general formula, a pattern, or a condition that holds for some number n, computing the first few cases by hand is almost always the fastest route to an insight. Small cases reveal structure that the general problem obscures. This technique applies in combinatorics (count the arrangements for n=2, n=3, n=4 before looking for a general formula), number theory (test small values of n in a divisibility condition), and algebra (evaluate a functional equation at x=0, x=1, x=-1 before trying to find the general solution). Students who skip directly to the general case often spend time developing machinery that a two-minute small-cases check would have rendered unnecessary.Tip 12: Look for Symmetry, Invariants, and Extremal Cases First
Before choosing a solution technique, spend 30 seconds asking three structural questions about any problem you find difficult:- Is there symmetry? Can variables be swapped without changing the problem? If so, the solution likely exploits that symmetry.
- Is there an invariant? Is there a quantity that stays constant as the configuration changes? Invariants often unlock problems that look dynamic.
- What happens at the extremes? What does the solution look like when variables are at their maximum or minimum? Extremal cases often reveal the general structure.
