HMIC 2014

Consider a regular $n$-gon with $n>3$, and call a line acceptable if it passes through the interior of this $n$-gon. Draw $m$ different acceptable lines, so that the $n$-gon is divided into several smaller polygons. (a) Prove that there exists an $m$, depending only on $n$, such that any collection of $m$ acceptable lines results in one of the smaller polygons having 3 or 4 sides. (b) Find the smallest possible $m$ which guarantees that at least one of the smaller polygons will have 3 or 4 sides.

2014 triangles have non-overlapping interiors contained in a circle of radius 1 . What is the largest possible value of the sum of their areas?

Fix positive integers $m$ and $n$. Suppose that $a_{1}, a_{2}, \ldots, a_{m}$ are reals, and that pairwise distinct vectors $v_{1}, \ldots, v_{m} \in \mathbb{R}^{n}$ satisfy

$$\sum_{j \neq i} a_{j} \frac{v_{j}-v_{i}}{\left\|v_{j}-v_{i}\right\|^{3}}=0$$

for $i=1,2, \ldots, m$. Prove that

$$\sum_{1 \leq i<j \leq m} \frac{a_{i} a_{j}}{\left\|v_{j}-v_{i}\right\|}=0$$

Let $\omega$ be a root of unity and $f$ be a polynomial with integer coefficients. Show that if $|f(\omega)|=1$, then $f(\omega)$ is also a root of unity.

Let $n$ be a positive integer, and let $A$ and $B$ be $n \times n$ matrices with complex entries such that $A^{2}=B^{2}$. Show that there exists an $n \times n$ invertible matrix $S$ with complex entries that satisfies $S(A B-B A)=(B A-A B) S$.