If you have searched for the term , you are likely no longer a beginner. You are an intermediate or advanced problem solver looking to conquer symmetric, cyclic, and three-variable inequalities that appear in the IMO, Putnam, and Vietnamese National Olympiads.
Example from the book: Proving $a^2 + b^2 + c^2 + 3abc \ge ab+bc+ca + a+b+c$ for $a,b,c \ge 0$ becomes trivial once you set $p=1$ (by homogeneity) and realize the left minus right is linear in $r$. The mixing variables technique, or "smoothing," is based on a simple but profound idea: If an inequality is symmetric, the extremum often occurs when two variables are equal. secrets in inequalities volume 2 pdf
If you have searched for , you have already taken the first step. Now, do yourself a favor: get a legitimate copy, grab a notebook, and prepare to spend 200 hours learning why $a^2+b^2+c^2 \ge ab+bc+ca$ is just the beginning. Struggling with a specific inequality from Volume 2? The AoPS community has dedicated threads for every major problem in the book. Search the first line of the problem—someone has likely solved it. If you have searched for the term ,
The are not magic tricks—they are systematic, repeatable methods that turn asymmetric chaos into algebraic order. But they require grit. A typical page in Volume 2 takes 30–60 minutes to fully digest. The mixing variables technique, or "smoothing," is based
For decades, the journey from a novice inequality solver to a Master Olympiad competitor has been paved with a few legendary texts. Among them, "Secrets in Inequalities" by Pham Kim Hung stands as a monumental two-volume set. While Volume 1 introduces the foundational theorems (AM-GM, Cauchy-Schwarz, Chebyshev), Volume 2 is where the real magic—and the genuine "secrets"—are revealed.
Volume 2 teaches you how to prove that if you replace two variables $(a, b)$ with their average $\left(\frac{a+b}{2}, \frac{a+b}{2}\right)$, the left-hand side of the inequality changes monotonically. By repeatedly applying this, you "smooth" the variables until they are all equal. If the inequality holds at equality, it holds everywhere.
The "secret" is learning the precise condition for when smoothing works—specifically, when the function is convex in each variable. Most competitors know Schur's inequality of degree 3: $a^3+b^3+c^3 + 3abc \ge a^2(b+c) + b^2(c+a) + c^2(a+b)$. But Volume 2 introduces Schur of degree 4 and the powerful Vornicu-Schur generalization.