Guides

Knowing the technique is the easy half. Recognising it is the paper.

Each guide takes one question a student actually asks and answers it against real problems from the 89-problem archive. Every claim in them is re-derived by the build before the page ships.

All guides

• How to check an answer when the clock is against you 9 September 2026 · Re-reading your working reproduces the error that made it. Five checks that are independent of the derivation, and the order to run them under time pressure.
• Complex numbers for olympiad geometry: when to reach for them 8 September 2026 · Complex coordinates turn a geometry problem into algebra you can finish. The configurations where that trade pays, the six-line dictionary, and where it stalls.
• Jensen's inequality: how to set one up on a competition problem 7 September 2026 · Jensen is one line of theory and three lines of setup, and the setup is where it goes wrong. The shape that calls for it, and the two checks that save you.
• Ceva or Menelaus: when to use which 6 September 2026 · Both theorems say a product of three ratios equals 1. Deciding which one you are looking at takes one question, and a counting rule when that question is hard.
• How to prove something is impossible in a competition 4 September 2026 · Failing to find an example is not a proof that none exists. The five obstructions that rule out every attempt at once, ordered by what they cost you.
• Generating functions: turning a counting problem into a product 3 September 2026 · Generating functions explained as a method: one factor per constraint, multiply, read the coefficient. The dictionary, and how to extract by hand.
• How to read a hard problem without giving up 2 September 2026 · Most hard problems are abandoned before they are ever attempted. Five things to do to a statement with a pen, and the rule that decides when to stop.
• Induction proofs where plugging in n+1 is not the step 2 September 2026 · Four inductive steps that plugging in n plus one will never close: a second predecessor, a stronger claim, a chosen deletion, and a base case that does not reach.
• Power of a point: when to use it, and when not to 31 August 2026 · Power of a point is a recognition problem, not a formula problem. Five tells that call for it, one that does not, and how to find the circle nobody drew.
• Telescoping sums: how to spot them before you compute 30 August 2026 · A telescoping sum announces itself before you do any algebra. Four signals that the middle will cancel, and the habit that kills the sign errors.
• How hard is the AIME compared to the AMC 10? 29 August 2026 · The AIME is not a harder AMC 10. It is a different paper, and four numbers say by how much: time per question, answer space, penalty, and reward for a guess.
• Vieta jumping, explained simply: the jump and when to reach for it 29 August 2026 · Vieta jumping is four lines of algebra guarding one idea. The shape that triggers it, the jump itself, and the case that quietly loses marks.
• When you can write WLOG, and when it breaks the proof 29 August 2026 · WLOG is licensed by a symmetry, never by convenience. The one test that says which assumption you may make, and the cyclic trap that fails it.
• Coloring arguments: how to know which coloring to use 27 August 2026 · A coloring argument is only as good as the coloring. The signals that a combinatorics problem wants one, and how the piece decides the palette.
• The extremal principle: how to choose which extreme to take 26 August 2026 · Taking the largest or the smallest is the easy half. The three signals that call for an extreme, and the one question that tells you if you picked the right one.
• Which modulus? The real trick in competition number theory 25 August 2026 · Modular arithmetic tricks are one decision wearing many hats: which modulus. The residue table, how to read the modulus off the problem, and when it fails.
• Why you keep running out of time on the AMC 24 August 2026 · Running out of time is almost never a speed problem. Three different things cause it, the arithmetic tells you which one is yours, and each has its own fix.
• How a double counting proof is actually built 23 August 2026 · Double counting proofs are always the same three steps, and only one of them is hard. The skeleton, five worked examples, and where they go wrong.
• AMC 12 in one month: a plan made of what you cut 22 August 2026 · Four weeks buys around twenty hours. A study plan for the AMC 12 that spends them by naming what to drop: most mocks, the last five questions, every new topic.
• Angle chasing: when to stop and what to reach for instead 21 August 2026 · Angle chasing rarely fails outright. It finishes, and you keep going. Four signals that the chase is over, and what the conclusion says to reach for next.
• AM–GM or Cauchy–Schwarz: how to know which one to use 20 August 2026 · The two inequalities solve different shapes, not different difficulties. A shape test that picks one in seconds, and the equality case that settles the ties.
• What a complete olympiad solution actually looks like 19 August 2026 · A complete solution is not a longer one. The four beats every finished proof moves through, the five gaps that quietly lose marks, and a self-check.
• How to get better at proofs when you can only compute 18 August 2026 · Computation is not the wrong skill for proofs, it is an unfinished one. Five moves that turn arithmetic you can already do into an argument that holds.
• AMC 10: how many questions do you need for AIME? 17 August 2026 · The AIME cutoff is a percentile, not a number you can look up. The scoring rule is fixed, though, and it says exactly what a target score costs.
• How to spot a pigeonhole problem in under a minute 16 August 2026 · Pigeonhole is easy to understand and hard to recognise. Four signals that tell you to reach for it, and the three questions that turn a signal into boxes.
• When to use an invariant in a competition problem 16 August 2026 · An invariant proves impossibility, not possibility. The four signals that a problem wants one, the ladder of candidates to test, and the one-move test.