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vitrual triangles, a^{2024}+b^{2024}> c^{2024}, ineq. system

Source: 2024 Mathematics Regional Olympiad of Mexico West P6

October 21, 2024
geometryinequalitiesgeometric inequality

Problem Statement

We say that a triangle of sides a,b,ca,b,c is virtual if such measures satisfy {a2024+b2024>c2024,b2024+c2024>a2024,c2024+a2024>b2024\begin{cases} a^{2024}+b^{2024}> c^{2024},\\ b^{2024}+c^{2024}> a^{2024},\\ c^{2024}+a^{2024}> b^{2024} \end{cases} Find the number of ordered triples (a,b,c)(a,b,c) such that a,b,ca,b,c are integers between 11 and 20242024 (inclusive) and a,b,ca,b,c are the sides of a virtual triangle.
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