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

• Lifting the exponent lemma explained for competitions 18 October 2026 · The lifting the exponent lemma in one line, the p = 2 version that catches almost everyone, and the signals that tell you a contest problem wants it.
• How to start a functional equation problem 17 October 2026 · A functional equation gives you no obvious first move. The substitutions to try first, how to guess the answer early, and the check people skip at the end.
• Cyclic quadrilateral tricks for olympiad geometry 16 October 2026 · The cyclic quadrilateral is rarely drawn for you. Five ways to prove four points concyclic, where hidden ones sit, and what each one hands you back.
• How to prove an olympiad inequality, step by step 15 October 2026 · Most failed inequality proofs pick a tool in the first minute. The order of attack that picks it last: equality case, degree, substitution, shape, audit.
• Schur's inequality: when to use it, and how to tell AM–GM will not do 14 October 2026 · Schur's inequality is the tool for symmetric cubics that AM–GM cannot finish. How to recognise one, the second equality case that gives it away, and IMO 1964/2.
• The ratio lemma in olympiad geometry: when to reach for it 13 October 2026 · The ratio lemma turns an angle split at a vertex into a length split on the opposite side. Four signals that call for it, the proof, and the symmedian in two lines.
• gcd(2^m − 1, 2^n − 1): the proof, and how to spot it 12 October 2026 · Why gcd(2^m − 1, 2^n − 1) = 2^gcd(m,n) − 1: run Euclid's algorithm on the exponents. A full proof, a second proof by orders, and where it turns up.
• Ramsey problems for beginners: one move, four levels 11 October 2026 · Ramsey olympiad problems in order of difficulty, from six people to three colours and number puzzles. One vertex, pigeonhole its edges, then look inside.
• Nesbitt's inequality: three proofs and how to spot it in disguise 10 October 2026 · Nesbitt's inequality takes two lines to prove once you know it. The skill is seeing it inside a harder problem, and choosing which of three proofs to use.
• How to build a problem-solving notebook that actually works 8 October 2026 · Most problem notebooks are copied solutions nobody reopens. Index yours by the clue in the statement, not by topic, and test it from the clue side.
• Olympiad geometry without coordinates: where to start 7 October 2026 · Coordinates are what you reach for when you do not know the synthetic first move. Translate the goal, hunt the cyclic quads, and let power of a point do the lengths.
• What separates a Level 5 problem from a Level 7, side by side 6 October 2026 · Two IMO inequalities, both rated 5 out of 6, sit two rungs apart on the ladder. What actually separates a Level 5 problem from a Level 7, and how to cross.
• Number theory problems that look like algebra, and how to tell 5 October 2026 · An equation, a fraction, a polynomial: it reads like algebra, but the word integer changes the method. Five signals, and the three moves that follow.
• How to revise a topic you understand but keep missing 4 October 2026 · Rereading a topic you already understand fixes nothing. Find the one step that keeps breaking, revise that step, and test it cold. Worked on mods and bounds.
• Similar triangles: the ratio tricks that save time 3 October 2026 · Similar triangles eat time when you write four-term proportions. One scale factor, the k-squared area rule and three shapes to memorise do the work in seconds.
• Extremal graph problems for beginners: the bound and the example 1 October 2026 · An extremal graph problem is two problems: a bound nobody can beat and an example that reaches it. How to spot one, and which vertex to pick first.
• The incircle configuration: five facts worth memorising 30 September 2026 · Five incircle facts cover most contest problems: tangent lengths, r = K/s, the angle BIC, the arc-midpoint lemma, excircle touch points. When each one fires.
• The phantom point technique: how to prove a point you cannot reach 29 September 2026 · When a geometry chase stalls because you cannot get hold of the point, build a second one you can. How phantom points work, and the uniqueness step people drop.
• AMC 8 to AMC 10: what actually changes 28 September 2026 · The AMC 10 is not a harder AMC 8. The clock, the scoring and the length of a question all change, and the scoring flips your most useful AMC 8 habit.
• How many practice problems should you do a day? 27 September 2026 · The daily count is the wrong target. How to size a practice session from your miss rate and review time, and why a mixed set beats a bigger one.
• Greedy arguments in olympiad combinatorics: when they work 26 September 2026 · Greedy is the first idea on every construction problem and the one most often marked zero. How to tell when it works, and the two proofs it always owes.
• When to reach for a rotation or a reflection in a geometry proof 20 September 2026 · Transformations look like magic until you know what triggers them. Five signals in a figure that tell you which rotation, reflection or homothety to use.
• How to write up a proof so it gets full marks 15 September 2026 · Your proof is right and the score is not. What a marker can and cannot credit, the four places marks leak out of a correct argument, and a read-back that finds them.
• When a smoothing argument works, and when it quietly lies 13 September 2026 · Smoothing nudges two variables together and reads the answer off the equal point. Here is the test for when that is valid, and the two ways it fails silently.
• When to reach for the Ravi substitution (and when not to) 12 September 2026 · Ravi turns "these are the sides of a triangle" into "these are positive numbers". The signals that call for it, the mechanics, and what it costs you.
• Constructive vs complementary counting: which one to reach for 11 September 2026 · Two ways to count the same set, and one question that tells you which is cheaper. The signals, the worked pair, and the trap where the obvious tell lies to you.
• What to do when you have no idea how to start 10 September 2026 · The blank page is a procedure failure, not a knowledge failure. Five opening moves ordered by cost, and the rule that says when to abandon one for the next.
• 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.