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Ukrainian Mathematical Olympiad, Final Round

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
$x^4 + x^3 + 2x^2 + x + 1 > 2x^3 + x^2 + 2x + 1,$
$x^4 + x^3 + 2x^2 + x + 1 > x^4 - 1,$
$x^4 + x^3 + 2x^2 + x + 1 < (x^4 - 1) + (2x^3 + x^2 + 2x + 1).$

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