2023 Singapore MO Open Q14: Two methods Exploring Real Number Solutions and Trigonometry

Description: Join us as we dive into a fascinating problem from the 2023 Singapore MO Open section. Consider the set of all possible (𝒙,𝒚) of real numbers that satisfy the equation (𝒙−𝟒)^𝟐+(𝒚−𝟑)^𝟐=𝟗. Our goal is to find the largest possible value of 𝒚/𝒙, denoted as S, and calculate the value of ⌊𝟕𝑺⌋.

In this video, we'll present two distinct methods for solving this problem. The first method explores the use of trigonometric identities, revealing a clever approach to finding the value of S. We'll explain the steps involved and highlight the insights gained from applying trigonometry to this equation.

Next, we'll delve into the second method, which revolves around the concept of discriminants in quadratic equations. By analyzing the discriminant, we'll unveil an alternative path to determine S. We'll provide a step-by-step explanation of this method, allowing you to grasp the underlying principles and gain a deeper understanding of quadratic equations.

Throughout the video, we encourage you to embrace resilience and maintain a "keep trying" spirit. Mathematical problem-solving often requires multiple attempts and different strategies. So, give it a try, have fun, and witness the excitement of discovering the solution to this intriguing problem. Join us for an educational and compelling journey into the realm of real number solutions and trigonometry!

[realanalysis]

(x-4)^2 = x^2-8x+16. You wrote x^2-16x+16