Mastering Functional Equations in Math Olympiads

Mastering Functional Equations in Math Olympiads

🏆 Dive deep into the realm of Math Olympiads and conquer functional equations like a true champion! 💡 Join us as we unveil expert strategies, solve intricate problems, and empower you to excel in the realm of competition math. 🧠 From understanding the fundamentals to mastering advanced techniques, this guide is your roadmap to success in navigating functional equations in Olympiad contests. 🌟 Equip yourself with the skills and knowledge needed to tackle any challenge that comes your way! #MathOlympiad #FunctionalEquations #CompetitionMath #ProblemSolving #Mathematics #OlympiadPreparation #Mathletes #Logic #CriticalThinking #Education #MathIsFun #Puzzle #MathematicsEducation

📘 Topics Covered:

Understanding the basics of functional equations
Step-by-step solutions for common types: substitutions
Advanced techniques of algebraic manipulations and change of variable
Solving system of functional equations
Practical calculation of function f(x)

Timestamps:
0:00 Introduction
1:30 Simplifying expressions
3:06 Replacement of variable
5:40 Solving functional equation
8:40 Finding f(x)

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Thanks for Watching!!
@infyGyan

[RashmiRay-cy]

Let us label the given equation as (1). In (1), let us replace x by 1/(1-x). This gives the equation f(1/(1-x)) + f((x-1)/x) = 2(1-x^2)/x. Let us label this as (2). Then, let us replace x in (1) by (x-1)/x. This gives us f((x-1)/x) + f(x) = [2x(2-x)]/(x-1). Let us call this (3). Then, (1) + (3) - (2) gives 2 f(x) = 2 [(x+1)/(x-1)]. Thus, f(x) = (x+1)/(x-1).

[ramunasstulga]

Equation f(20-x)=f(22+x) works for all real numbers, we know there is only two values where f(x)=0, then the sum of those values: A) -1; B) 20; C) 21; D) 22; E) 42 ?