😱Je me GALÈRE sur cet oral niveau LYCÉE ! 😰(sup/L1)

Here is a 2025 Mines-Ponts PC math oral exam that deals with an elementary high school level problem. You have to show that if a + b + c = π/2, then sin(a)sin(b)sin(c) is less than 1/8. I give four different solutions, including two at high school level, a very efficient solution at the preparatory school level, and a hint for a generalization using geometric optimization and complex numbers. 00:00 Introduction and Statement 01:40 Summary 02:42 Geometric Interpretation 03:14 Method 1: High School Level (Trigonometry + Parabola) 09:20 Method 2: A Positive Discriminant 11:35 Method 3: Hint (Preparatory Class Level) 12:20 Method 3: Convexity Inequality (GAI) 16:00 Method 3: Variant Without GAI 19:00 Method 4: Geometric Optimization + Complex Numbers 27:18 Method 4: What Does GAI Tell Us? 28:21 Method 4: Differential Calculus and Lagrange Multipliers 32:30 Conclusion and Epilogue