IOQM study plan

Build a one-page divisibility and congruence toolkit (Euclid's lemma, gcd-lcm identity, linear Diophantine criterion) and solve 15 problems that require each tool at least once.

Drill modular arithmetic until you can reduce large powers and factorials modulo n without hesitation, completing 12 such computations under a 4-minute timer each.

Prove and apply the factor-counting and divisor-sum formulas for n = product of prime powers, then use them on 10 constructed numbers with three or more prime factors.

Write out the standard algebraic identities (difference of squares, sum/difference of cubes, Sophie Germain, x+1/x substitutions) and match each to two competition problems it unlocks.

Master the Chinese remainder theorem and the lifting-the-exponent lemma: state each precisely, then solve 8 problems where they are the intended route.

Handle p-adic valuations fluently by computing v_p of factorials, binomial coefficients, and products, and by solving 10 problems on trailing zeros and divisibility of factorials.

Classify and solve functional equations by substituting x = 0, x = y, and x = -y, building a checklist of standard substitutions and applying it to 8 FE problems.

Practice polynomial remainder and factor theorems together with Vieta's formulas on 10 problems, including symmetric-function manipulations of roots.

Solve 12 problems on perfect squares, quadratic residues, and parity arguments, writing a one-line justification for why each non-obvious residue is excluded.

Derive and apply AM-GM, Cauchy-Schwarz, and the rearrangement inequality to 10 optimization problems, stating equality conditions every time.

Work through 8 quadratic-equation problems with integer-root constraints, using the discriminant and Vieta conditions to bound parameters.

Attempt two full-length mixed algebra and number theory mock sections under strict time, then re-solve every missed problem without notes.

Reconstruct classical triangle geometry from scratch: angle bisector theorem, Apollonius, Stewart, and the sine and cosine rules, proving each and then applying each to 3 problems.

Solve 12 circle problems using cyclic quadrilaterals, the power of a point, and the tangent-secant theorem, drawing a clean diagram for each.

Practice directed-angle chasing on 10 configurations, writing each angle equality with its justification (same arc, tangent-chord, spiral similarity).

Introduce the standard auxiliary constructions (reflect, rotate, extend, draw the circumcircle) and record which configuration each construction serves.

Apply triangle centers (centroid, orthocenter, incenter, circumcenter) and their standard properties to 10 problems, including the Euler line and nine-point circle facts you can justify.

Handle homothety, spiral similarity, and the radical axis on 8 problems, identifying the center or axis before computing anything.

Master counting fundamentals: permutations with repetition, circular arrangements, stars and bars, and inclusion-exclusion, applying each to 5 problems.

Set up bijections to convert awkward counting problems into clean ones, completing 8 such conversions with a written description of the map.

Solve 10 recurrence problems by deriving characteristic equations and closed forms, and verify each formula against the first four terms.

Use invariants, monovariants, and coloring arguments on 10 problems, stating the invariant explicitly before the argument.

Practice extremal and pigeonhole reasoning on 10 problems, identifying the pigeons and holes in writing each time.

Revisit geometry through coordinates and complex numbers on 6 problems, choosing the method that minimizes algebra.

Take two full-length IOQM-style mocks under exam conditions, then spend at least 90 minutes on each doing a full error analysis.

For every missed problem, write the intended insight in one sentence and re-solve it cold two days later.

Audit your time allocation across the four domains and adjust the per-section budget based on where points were actually lost.

Patch the two weakest subtopics identified by the mocks with 10 targeted problems each.

Take one final full-length mock and one half-length speed set, aiming to beat your Week 7 score on both.

Review the tool-to-trigger sheet, construction log, and error dossier in three spaced sessions rather than one long cram.

Rehearse exam-day mechanics: answer-sheet discipline, order of attempting sections, and a rule for when to abandon a problem.

Re-solve the 15 hardest problems you have encountered, with no notes, to confirm retention.