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Olympiad Preparer

Native lessons, formulas, and contest strategies for math olympiads, Science Olympiad, and computing competitions — built into IntelliPlan so you can study, schedule, and ask Plani for help in one place.

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Lesson 1 · Number theory

Modular arithmetic essentials

For olympiad problems, reduce early and track remainders.

  • Fermat's little theorem: if p is prime and gcd(a,p)=1, then a^(p−1) ≡ 1 (mod p).
  • Chinese remainder: combine congruences when moduli are coprime.
  • Euclid's lemma: if p|ab and p is prime, then p|a or p|b.
Example: Find 7^100 mod 11 → 7^10 ≡ 1, so 7^100 ≡ 1 (mod 11).
Lesson 2 · Combinatorics

Counting without double-counting

  • PIE: |A ∪ B| = |A| + |B| − |A ∩ B| — extend to 3+ sets for derangements.
  • Stars and bars: non-negative integer solutions to x₁+…+x_k = n is C(n+k−1, k−1).
  • Bijection trick: map hard counts to a simpler set (paths, binary strings).
Lesson 3 · Geometry

Power of a point & similar triangles

On circles: if chords AB and CD meet at P inside, PA·PB = PC·PD. Look for cyclic quads and angle chasing before coordinates.

  • Mass points for cevians and ratios.
  • Homothety for tangent circles.
  • Complex numbers / coordinates when synthetic stalls.
Lesson 4 · Algebra

Inequalities & polynomials

  • AM-GM for positive reals: (a+b)/2 ≥ √(ab).
  • Vieta's: sum/product of roots without solving.
  • Telescoping sums and functional equations (injectivity tests).
Resources

AMC → AIME → USAMO path

  • AMC 10/12: speed + pattern recognition (25 problems, 75 min).
  • AIME: integer answers 000–999; no guessing penalty — write clean solutions.
  • USAMO/USA(J)MO: 6 proof problems; clarity and lemmas beat cleverness.

Use IntelliPlan to block weekly mock time and log which topics you miss.

Lesson 1 · Framework

SciOly event map

Split events into study (Anatomy, Astronomy), build (Bridges, Wright Stuff), and lab (Chem Lab, Circuit Lab). Assign 1–2 lead partners per event.

Lesson 2 · Study events

Active recall binders

  • One binder per event with official rules highlighted.
  • Turn each section into Q→A flashcards in Learn.
  • Weekly 25-min practice tests under time pressure.
Lesson 3 · Build events

Iterate with a log

Record mass, score, and failure mode every build. Small changes → measurable deltas. Photo each version for tournament appeal docs.

Lesson 4 · Lab events

Procedure fluency

  • Memorize sig fig rules and unit conversions.
  • Practice titration / multimeter workflows until muscle memory.
  • Pair a "reader" and "doer" during invitational runs.
Lesson 1 · USACO / IOI

Complexity first

Before coding, name target complexity (O(n log n) sorts, O(n) two-pointer, O(n²) only when n ≤ 5000).

Lesson 2 · Patterns

Core algorithm kit

  • Prefix sums & difference arrays
  • Binary search on answer
  • DFS/BFS, DSU, shortest paths
  • DP: knapsack, LIS, interval DP
Lesson 3 · Practice

Contest simulation

3-hour blocks, no IDE autocomplete, read all problems first, attack the median-difficulty problem second.

Universal

12-week olympiad block

  1. Weeks 1–4: topic sprints (one domain per week).
  2. Weeks 5–8: mixed mocks + error log.
  3. Weeks 9–11: timed sets only.
  4. Week 12: light review, sleep, logistics.
During the contest

Time boxing

  • Skim all problems in 5–10 minutes.
  • Mark easy / medium / hard.
  • Never spend >½ the time on one problem until others are attempted.
  • Write partial credit — a lemma or special case often scores.
With IntelliPlan

Integrate prep into your week

  • Import school deadlines so olympiad blocks don't collide with exams.
  • Use Scheduler for 45-min daily drills.
  • Ask Plani to explain a solution you missed on a mock.
  • Save chat notes to your AP Library.