P·01
AMC 8
Foundations of contest thinking
Number sense, clever counting, and geometric intuition. Students learn that contest problems are puzzles with structure — and that structure can be found.
- Number Theory
- Counting
- Geometry
- Logic
Lucid · Institute of Competition Mathematics
MasterCompetitionMathematics.
A complete learning ecosystem for competition mathematics — interactive lessons, adaptive practice, mock contests, and AI coaching, from AMC 8 through the IMO.
AMC 8 → AMC 10/12 → AIME→ USAMO → IMO
§ 01 · Programs
Six programs.
One trajectory.
Every program is a stage of the same ascent — from first principles to the frontier of pre-college mathematics.
P·02
AMC 10/12
Speed, precision, depth
Algebraic technique, combinatorial identities, and trigonometry executed cleanly under time pressure.
P·03
AIME
Where problems become puzzles
Multi-step problems demanding synthesis across domains. We train the decomposition instinct: reduce, transform, conquer.
P·04
USAMO
The art of proof
Olympiad geometry, inequalities, functional equations, and number theory — argued with complete rigor, written to be read.
P·05
IMO Preparation
The summit
Modeled on national team preparation: daily problem sets, mock olympiads, and AI-guided review built on frameworks from former IMO medalists.
P·06
Advanced Problem Solving
Beyond the syllabus
Research-style seminars for students who have outgrown the contest calendar — Putnam preparation, mathematical writing, and open problems.
§ 02 · The Lucid Method
Reasoning is
a discipline.
Five stages, in strict logical order. Each one is a prerequisite for the next — like lemmas building toward a theorem.
Lemma 1
Foundations
Every technique rests on first principles. We rebuild algebra, geometry, combinatorics, and number theory from those principles up — so nothing is memorized that can instead be derived.
Lemma 2
Pattern Recognition
Students learn to see structure: invariants, symmetry, extremal cases, parity. The trained instinct that turns a blank page into a plan.
Lemma 3
Proof Writing
From intuition to rigor. Students write, critique, and rewrite arguments until precision becomes second nature — the skill that separates AIME qualifiers from olympiad medalists.
Lemma 4
Creative Problem Solving
Non-routine problems with no labeled method. The core olympiad skill: constructing an approach that did not exist before you sat down.
Theorem
Timed Competition Practice
Full simulations under authentic constraints, followed by forensic review of every decision — the ones that worked, and the ones that almost did.
§ 03 · Guided Reasoning
Reasoning,
not recall.
Watch a competition problem dissolve under structured thought. This is how every Lucid lesson works — questions, not answers, until the answer is inevitable.
AIME-style · Number Theory
Find the number of ordered pairs (a, b) of positive integers such that lcm(a, b) = 23·57.
Step 1 · Observe
Structure first
Both a and b must divide 23·57, so write a = 2x₁5y₁ and b = 2x₂5y₂. The problem is secretly about exponents.
Step 2 · Reduce
Split by independence
The lcm condition becomes max(x₁, x₂) = 3 and max(y₁, y₂) = 7 — two independent conditions. Count each, then multiply.
Step 3 · Count
A one-line lemma
Pairs with max(m, n) = k number exactly 2k + 1: either m = k (k + 1 choices for n), or n = k with m < k (k more).
Step 4 · Conclude
Multiply and finish
(2·3 + 1)(2·7 + 1) = 7 × 15.
Answer = 105 ∎
§ 04 · Why Lucid
Built differently.
No cohorts to keep up with and no coach to schedule around — just you, the curriculum, and an AI that won’t let you fake understanding.
100% virtual, always on
Every lesson, problem, and mock contest lives on the platform. Log in whenever you have twenty minutes — there’s no seat to book.
Coached, not lectured
The AI coach asks before it answers — Socratic hints, never the final answer, so understanding is earned, not copied.
One continuous curriculum
Six programs, one trajectory: AMC 8 through IMO Prep, each stage a strict prerequisite for the next.
Priced like software
Plans start at $5/month — a virtual platform shouldn’t cost what an hour of private tutoring does.
§ 05 · Curriculum
From first principles
to the IMO.
A single continuous path. Each milestone unlocks the next — no stage skipped, no gap left unproved.
Stage 0 · Months 0–6
Foundations
Rigorous re-derivation of school mathematics. Fluency drills, first proofs, and the habit of asking why.
Stage 1 · AMC 8
Contest Fluency
First exposure to competition structure. Speed with accuracy, pattern libraries, honest error analysis.
Stage 2 · AMC 10/12
Technique Under Pressure
The full toolbox — Vieta, telescoping, mass points, generating intuitions — executed in 75 minutes.
Stage 3 · AIME
Synthesis
Problems that cross domain boundaries. Decomposition strategy, answer-extraction discipline, three-hour endurance.
Stage 4 · USAMO
Rigor
Complete written proofs, graded to olympiad standard. Inequalities, olympiad geometry, functional equations.
Stage 5 · IMO
Mastery
Elite-level training: daily problem sets, mock olympiads, and an AI coach built on frameworks from those who have medaled.
§ 06 · Pricing
Choose your intensity.
Every tier includes the full curriculum platform, problem bank, mock contests, and the AI coach.
Starter
$5/month
- Full curriculum platform: lessons, bank, contests
- AI coach, Socratic hints on every problem
- Monthly mock competition, scored
- Progress tracking & mastery roadmap
Most chosen
Plus
$10/month
- Everything in Starter
- Unlimited AI coach access
- Weekly mock competitions, scored
- Written solutions to every problem
- Full progress analytics
Pro
$15/month
- Everything in Plus
- Priority AI coach, deeper proof review
- Competition-day strategy guides
- Early access to new content
§ 07 · FAQ
Open questions.
Students in grades 5–12 who want to compete seriously in mathematics — from first-time AMC 8 entrants to students preparing for national olympiad selection. Ambition matters more than current level; the platform places you by trajectory, not trophy case.
With the free platform. The adaptive engine measures reasoning habits rather than syllabus coverage and recommends the exact entry point on the curriculum roadmap — no placement test anxiety required.
Nothing here is passive. Every lesson is built from interactive blocks — intuition, guided discovery, inline checks that gate progression, worked examples that reveal step by step, and practice with escalating hints. You cannot scrub to the end.
Entirely self-paced and 100% virtual. There's no cohort to keep up with and no seat to book — every lesson, problem set, and mock contest is on the platform whenever you are, and the AI coach is there the moment you get stuck.
Four to eight hours weekly, depending on tier. Competition mathematics is learned by struggling with problems, not by watching solutions — the platform exists to sharpen that struggle, not replace it.
§ 08 · Q.E.D.
Start your journey to
mathematical excellence.
The full curriculum, problem bank, and AI coach are open. Your mastery map begins with the first problem.
∎