MOTS-2024-AdZU-Algebra

Introduction

• Presenter: Louie John D. Vallejo

• Role: Deputy Team Leader, IMO 2015-2017; Assistant Professor, Institute of Mathematics, UPD

• Event: Math Olympiad Training Sessions on November 10, 2024

Algebra Topics

• Key Topics Covered:

  • Solutions to equations/inequalities

  • Functional Equations

  • Proving Inequalities

  • Recursive sequences

  • Various other topics

Warm-Up: Past PMO Algebra Questions

Function Conditions (PMO '13)

• Problem: Let f be a function such that:

  • f(x + y) = f(x)f(y)

  • f(xy) = f(x) + f(y)

• Goal: Find f(π2013).

Solution Approach

• Set x = y = 0 in the second equation to find:

  • f(0) = 0

  • Thus, f(x) must be 0 for all x ∈ ℝ.

Another Function Condition (PMO '13)

Given:

• f(0) = 1;

• f(2xy - 1) = f(x)f(y) - f(x) - 2y - 1.

• Question: What is f(x)?

Solution Steps

• Interchanging variables leads to:

  • f(2xy - 1) = f(x)f(y) - f(y) - 2x - 1.

• Subtracting the equation shows:

  • f(x) = f(y) - 2y + 2x.

• Setting y = 0 gives:

  • f(x) = 2x + 1.

Algebraic Equation Solutions (PMO '13)

Equation:

• Find all x ∈ ℝ such that (p(x))^2 p(x) = 1.

• Equation to Solve: (2 - x^2)x^{2-3} + 2x + 4 = 1.

Solution Result

• 2 - x^2 = 1 ⇒ x = ±1.

• Remaining equation: x^2 - 3x + 4 = 0 gives:

  • x = 2√2 or x = ?2. (Indeterminate for x = ?2)

Domain of Functions (PMO ’25)

Given:

• Function f(x) has the domain (-1, 1).

• Question: What is the domain of f(3 - x^3 + x)?

Solution Steps

• Identify values of x such that:

  • -1 < 3 - x^3 + x < 1.

Inequalities (PMO '14)

Given Condition:

• 9a + a ≥ x for any positive a.

• Question: What is the largest possible value of x?

Solution Approach

• Using AM-GM Inequality:

  • 9a + a ≥ 2sqrt(9a^2);

  • Result implies: x ≤ 3.

Solutions to Equalities (PMO '25)

Problem on Minima:

• Find the minimum of (18a + 1/3b)(3b + 1/8a) expressed as m/n.

Solution Strategy

• Expand expression:

  • 54ab + 9/4 + 1 + (1/24)ab; complete the square.

• Using AM-GM on last terms gives:

  • Minimum = 25/4.

Exercise Examples

• Exercise (PMO ’26):

  • Minimize 1/x + 2 + 3/y + ... under given constraints.

• Exercise (PMO ’25):

  • Find max value of f(px) = cos(2πx/3).

Vieta’s Formulas - Theory

• Polynomials of degree n have n complex roots;

• Formulas relate coefficients and roots directly.

Quadratic Polynomials

• For ax^2 + bx + c = 0:

  • Roots r and s yield r + s = -b/a, rs = c/a.

Applications in Competition Problems

• Theorems, inequalities, and functional equations are common in competitions.

Conclusion

• Various algebraic techniques for problem-solving are critical for Math Olympiad success.