SMO Junior Crash Course Notes

Introduction

• Nerves due to approaching SMO Junior exam.
• Tips to score extra points.
• Key point: Olympiad questions are unpredictable.

Key Concepts in Olympiads

Handling Large Numbers

• Don't deal directly with huge numbers.
• Adjust bases or powers to simplify comparisons.
  • Example: Powers of 300, 200, and 100 can be made the same for easier comparison.
• Recognize patterns in problems involving large numbers.

Practice Problems

• First Problem: Compare large powers easily.
• Second Problem: Recognize constants are relevant (e.g., 2023).

Finding Patterns

• Look for patterns in numerical problems.
  • Example: 999 squared + 1999 = 10000 shows a pattern.
• Use algebra to derive solutions when patterns are not obvious.
• For numerical questions, recognize identities like the difference of squares.

Positive Integers vs. Real Numbers

• Positive integers have limited possibilities; use this information to simplify equations.
• Use algebraic manipulation (e.g., shifting terms to recognize squares).

Equations vs. Identities

• Equations change; identities remain constant.
• Understand the difference to solve problems correctly.

Pythagoras Theorem in Geometry

• Useful for geometry questions, focus on relationships not just lengths.
• Example Problem: Use symmetry and algebra to find dimensions without directly calculating square roots.

Divisibility Tests and Remainders

• Understand divisibility rules for larger numbers (e.g., 3, 6, 11).
• Upgrade tests to use segments of digits rather than only single digits.

Highest Common Factor (HCF) and Lowest Common Multiple (LCM)

• Use prime factorization to find HCF and LCM logically.
• Understand the relationship between HCF and LCM in problem-solving.

Counting Factors Using Prime Factorization

• Use the formula for counting factors based on prime powers.
• Identifying the number of factors can streamline solutions.

Common Sense and Problem-Solving Strategies

• For strange-looking problems, sometimes the simplest approach is best.
• Analyze unusual situations carefully to avoid logical errors.
• Remember to enjoy the process of problem-solving and reasoning.

Conclusion

• Ten ideas presented for SMO Junior preparation.
• Emphasize reasoning over rote formula memorization.
• Encourage discussion and questions for deeper understanding.