Let $P=A_1A_2\cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \cdots A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$. Prove that […]

Minkowski’s Convex Body Theorem for lattices in the plane: Suppose $\mathfrak{L}$ is a lattice in $\mathbf{R}^2$ defined as $\mathfrak{L}=\{m\vec{v_1}+n\vec{v_2}:m,n\in\mathbf{Z}\}$, where $\vec{v_1}$ and $\vec{v_2}$ are linearly independent. Suppose $d$ is the area of a fundamental parallelogram of $\mathfrak{L}$. Then, if $\mathcal{S}$ is a convex and origin-symmetric region with $Area(\mathcal{S})>4d$, then $\mathcal{S}$ contains some point $q\neq 0$ […]

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