Competition problem — Geometry — Maths Olympiad Prep

Library / /2 of 17

Geometry Difficulty 4.9 AIME Prove it Ukraine

For which real numbers $x > 1$ there exists a triangle with sides of lengths $x^4 + x^3 + 2x^2 + x + 1$ , $2x^3 + x^2 + 2x + 1$ and $x^4 - 1$ ?

Solution

Дійсно, при всіх $x > 1$ , як нескладно переконатися, мають місце нерівності
$x^4 + x^3 + 2x^2 + x + 1 > 2x^3 + x^2 + 2x + 1,$
$x^4 + x^3 + 2x^2 + x + 1 > x^4 - 1,$
$x^4 + x^3 + 2x^2 + x + 1 < (x^4 - 1) + (2x^3 + x^2 + 2x + 1).$

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

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