Problem - 1855
Place 9 points in a unit square. Prove it is possible to select 3 points from them to create a triangle whose area is no more than $\frac{1}{8}$.
Equally dividing the unit square into 4 horizontal strips. Then each strip is a $\Big(1\times \frac{1}{4}\Big)$ rectangle. By the Pigeonhole principle, at least three points will lie in one stripe. Then, the area of the triangle formed by these points cannot exceed $$\frac{1}{2}\times 1\times\frac{1}{4}=\frac{1}{8}$$