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Lower bound on circumference while upper bound on area

Source: China TST 2023 - Test 2 Problem 6

March 19, 2023
combinatoricscombinatorial geometryChina TSTanalytic geometry

Problem Statement

Prove that there exists some positive real number λ\lambda such that for any D>1∈RD_{>1}\in\mathbb{R}, one can always find an acute triangle △ABC\triangle ABC in the Cartesian plane such that [*] A,B,CA, B, C lie on lattice points; [*] AB,BC,CA>DAB, BC, CA>D; [*] S△ABC<34D2+λ⋅D4/5S_{\triangle ABC}<\frac{\sqrt 3}{4}D^2+\lambda\cdot D^{4/5}.
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