CN-TST 2023 Problem 12

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