Problem
ALG-B2-M01-P019 Sixth powers and sign
Nonzero \(x,y\) satisfy \(x^6-y^6>x\) and \(y^6-x^6>y\). Prove that \(xy>0\).
Hint 1. Add the two inequalities.
Hint 2. If \(xy<0\), choose the positive number and get a contradiction.
Adding gives \(x+y<0\). Suppose \(xy<0\). Without loss of generality, \(x>0\), \(y<0\). Then \(x+y<0\) implies \(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 7: one must choose signs after a global addition of the two conditions.
G. Why this is not a one-step exercise. A single inequality says nothing decisive about the sign of the product.