The Integral Cup — Complete Preparation Guide

India’s Premier Mathematics Competition by STEMvibe — 8 tracks, 2 seasons, 25 centers, ₹1 Crore prize pool.

🔗 Official Website · Discord · Registration

Table of Contents

1. Overview & Competition Structure
2. All 8 Tracks at a Glance
3. Which Track Should I Start With? (Beginner Classification)
4. Recommended Track Order for Beginners
5. Track 1: Integration & Analysis
6. Track 2: Linear Algebra & Optimization
7. Track 3: Probability & Statistics
8. Track 4: Game Theory
9. Track 5: Discrete Mathematics
10. Track 6: Number Theory
11. Track 7: AI & Computational Mathematics
12. Track 8: Topology
13. General Preparation Strategy
14. Official Resources

Overview & Competition Structure

All 8 Tracks at a Glance

Season 1 (Feb–Apr 2026)

Season 2 (Jun–Oct 2026)

Which Track Should I Start With?

If you’re a complete beginner, here’s how the tracks stack up from easiest to hardest:

Tier 1 — Start Here (Most Accessible)

Tier 2 — Moderate (Need Some Course Background)

Tier 3 — Challenging (Needs Dedicated Study)

Tier 4 — Advanced (Graduate-Level Depth)

Recommended Track Order for Beginners

Follow this learning sequence to build foundational skills that snowball into harder tracks:

Step 1: Discrete Mathematics (logic, sets, counting — foundation for everything)
   ↓
Step 2: Integration & Analysis (calculus mastery — bridges high school to undergrad)
   ↓
Step 3: Number Theory (builds on discrete math + arithmetic reasoning)
   ↓
Step 4: Linear Algebra & Optimization (vector spaces, matrices — opens applied math)
   ↓
Step 5: Probability & Statistics (requires comfort with integrals + linear algebra)
   ↓
Step 6: Game Theory (applies probability + optimization + strategic reasoning)
   ↓
Step 7: AI & Computational Mathematics (combines linear algebra, probability, and coding)
   ↓
Step 8: Topology (requires real analysis maturity — tackle last)

For Season 1 specifically (choosing 3 of 4): Beginners should pick Integration & Analysis (priority 3), Linear Algebra (priority 2), and Probability & Statistics (priority 1). Skip Game Theory only if you’ve never studied it.

Track 1: Integration & Analysis

What it covers: Limits, continuity, differentiability, definite/indefinite integrals, improper integrals, series convergence, differential equations, Laplace/Fourier transforms.

Learning Path

Resources

Practice Strategy

1. Master all standard integration techniques from JEE Advanced level
2. Build a personal formula sheet for transforms and series tests
3. Practice 10–15 integrals daily under timed conditions
4. Work through improper integrals and convergence problems
5. Attempt proof-based analysis problems (Apostol, Rudin exercises)

Track 2: Linear Algebra & Optimization

What it covers: Vector spaces, linear transformations, eigenvalues/eigenvectors, matrix decompositions, linear programming, convex optimization, high-dimensional geometry.

Learning Path

Resources

Practice Strategy

1. Prove all major theorems yourself (don’t just read them)
2. Do computational exercises: find eigenvalues, decompose matrices by hand
3. Solve linear programming problems using graphical & simplex methods
4. Practice abstract proof problems from Axler’s exercises
5. Study past Integral Cup problems on Discord

Track 3: Probability & Statistics

What it covers: Probability axioms, conditional probability, distributions (discrete & continuous), random variables, expectation, variance, hypothesis testing, Bayesian inference, stochastic processes.

Learning Path

Resources

Practice Strategy

1. Solve combinatorial probability problems daily (Putnam, IMO-style)
2. Master all standard distributions and their relationships
3. Practice deriving results from first principles (MGFs, transformations)
4. Work through Bayesian inference problems step by step
5. Study quant interview probability puzzles (Jane Street, Optiver style)

Track 4: Game Theory

What it covers: Strategic form games, Nash equilibrium, dominant strategies, mixed strategies, extensive form games, cooperative games, mechanism design, auction theory.

Learning Path

Resources

Practice Strategy

1. Start with simple 2×2 games — find all equilibria by hand
2. Draw extensive-form game trees for sequential games
3. Solve mixed-strategy equilibrium problems
4. Study auction formats (first-price, second-price, Vickrey)
5. Practice creating novel game scenarios and analyzing them formally

Track 5: Discrete Mathematics

What it covers: Logic, set theory, relations, functions, combinatorics, graph theory, recurrence relations, generating functions, Boolean algebra.

Learning Path

Resources

Practice Strategy

1. Master proof techniques: induction, contradiction, pigeonhole
2. Solve counting problems daily — permutations, combinations, stars & bars
3. Practice recurrence relations → derive closed forms
4. Study graph theory through algorithmic problems (e.g., CP graph problems)
5. Attempt Putnam/Olympiad combinatorics problems

Track 6: Number Theory

What it covers: Divisibility, primes, modular arithmetic, Euler’s theorem, quadratic residues, continued fractions, Diophantine equations, algebraic number theory basics.

Learning Path

Resources

Practice Strategy

1. Build strong modular arithmetic intuition — solve 5 problems daily
2. Prove major theorems (Euler’s, Fermat’s, CRT) yourself
3. Study Diophantine equations systematically (linear → quadratic → higher)
4. Work through Olympiad NT shortlists (ISI, CMI, IMO, Putnam)
5. Use Project Euler to blend computation with theory

Track 7: AI & Computational Mathematics

What it covers: Numerical methods (root-finding, interpolation, numerical integration), optimization algorithms, machine learning mathematics, computational complexity, algorithmic problem-solving.

Learning Path

Resources

Practice Strategy

1. Implement numerical methods from scratch in Python (Newton’s method, Gaussian elimination, etc.)
2. Understand error analysis — truncation vs rounding errors
3. Study the math behind ML: gradient descent convergence proofs, regularization theory
4. Solve computational problems with pen + paper first, then verify with code
5. Read research papers on numerical methods for competition-level depth

Track 8: Topology

What it covers: Topological spaces, open/closed sets, continuity, connectedness, compactness, quotient spaces, fundamental group, homology basics.

Learning Path

Resources

Practice Strategy

1. Do NOT skip prerequisites — you need real analysis (metric spaces, continuity, compactness in ℝⁿ) first
2. Read Munkres chapters 1–4 methodically, doing every exercise
3. Build a counterexample notebook — for every theorem, know where it fails if conditions are relaxed
4. Practice proof-writing: “Show that X is compact,” “Prove f is continuous,” etc.
5. Discuss problems on math forums (StackExchange, AoPS) for feedback on your proofs

General Preparation Strategy

8-Week Preparation Plan (Per Season)

Tips for All Tracks

1. Join the Discord — The Integral Cup Discord has past problems in #archive, syllabus in #announcements, and sample papers in #resources
2. Priority strategy matters — Assign your strongest track as priority 3 (weight = 3×), second-strongest as priority 2, and weakest as priority 1 for maximum score
3. Round 2 is open-book — Focus on deep understanding over memorization; know where to find things and how to apply them
4. Round 3 includes presentations — Practice explaining proofs verbally; communication matters at the finale
5. Form study groups — Connect with participants from your center via Discord
6. Track the sponsors — Optiver, QRT, and Jane Street sponsor this; their quant interview prep resources overlap heavily with Tracks 1–4

Recommended Books Across All Tracks

Official Resources

• 🌐 Official Website
• 📋 Examination Details
• 🎯 Tracks
• 📝 Registration
• 💬 Discord Community
• ▶️ YouTube — STEMvibe Foundation
• 📸 Instagram
• 💼 LinkedIn
• 📧 Contact: integralcup@stemvibe.org