Library / Belarus — Selection and Training Session /18 of 78

Belarus — Selection and Training Session

Algebra Difficulty 5.1 AIME, harder Prove it Belarus

Consider all polynomials $P(x)$ with real coefficients that have the following property: for all real $x$ and $y$ one has
$|y^2 - P(x)| \le 2x \iff |x^2 - P(y)| \le 2|y|.$

Determine all possible values of $P(0)$ .

Solution

3. See IMO-2014 Shortlist, Problem A5.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.