Problem

Source: China TST 2023 - Test 2 Problem 6

Tags: combinatorics, combinatorial geometry, China TST, analytic geometry, lattices



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 $A, B, C$ lie on lattice points; $AB, BC, CA>D$; $S_{\triangle ABC}<\frac{\sqrt 3}{4}D^2+\lambda\cdot D^{4/5}$.