Problem
ALG-B2-M01-P007 Separate estimates are not enough
Nonzero \(x,y\) satisfy \(x^2-x>y^2\) and \(y^2-y>x^2\). What is the sign of \(xy\)?
Hint 1. First add the inequalities.
Hint 2. Then carefully multiply the transformed inequalities.
Adding gives \(-x-y>0\), so \(1-x-y>1>0\). Now multiply the inequalities \(x^2-x>y^2\) and \(y^2-y>x^2\). This is legitimate because \(x^2\) and \(y^2\) are positive, and the left sides are larger than them. We obtain \((x^2-x)(y^2-y)>x^2y^2\). Expanding and cancelling gives \(xy(1-x-y)>0\). Since \(1-x-y>0\), we get \(xy>0\).
A. Source analysis. Main objects: inequalities, order, an extremal element, or an invariant. The obvious first move usually gives only a local estimate. The hidden observation is to choose the right nondecreasing quantity, or to add/multiply inequalities only after signs are controlled. The needed step is an ordering, an invariant, a product transformation, or a boundary case.
F. Difficulty justification. Regional level 6: one must add the conditions and then safely multiply them after checking positivity.
G. Why this is not a one-step exercise. Neither of the two inequalities alone determines the sign of \(xy\).