Problem 1-8 Meticulous Manipulations

Problem Posts

[Difficulty 2] [PDF]

QUESTION
(2012 PMO Area Stage) If $x+y+xy=1$, where $x$, $y$ are nonzero real numbers, find the value of $$xy+\frac{1}{xy}-\frac{y}{x}-\frac{x}{y}$$

Laconic Solution Sketch
Manipulate.

Problem 1-5 Trivial Inequality

Problem Posts

Problem 1-5 (Algebra) [Difficulty 2] [PDF]

QUESTION
(Sipnayan 2011 High School Elims, Average Question 2) Suppose $x$ and $y$ are real numbers such that $$2x^2+y^2-2xy+12y+72\le0$$. What is the value of $x^2y$?

Overview/Laconic Solution Sketch
Group into squares and apply the Trivial Inequality.