[Some punny title about determinants and fixed points]

Prove the following identity

$\sum_\sigma (-1)^\sigma t^{\mathrm{Fix}\;\sigma} = \det\begin{pmatrix} t & 1 &\cdots& 1& 1\\ 1 & t &\cdots& 1& 1 \\ \vdots & \vdots &\ddots & \vdots & \vdots \\ 1 & 1 &\cdots& t& 1\\ 1 & 1 &\cdots& 1& t \end{pmatrix}$

the sum running through $S_n$ and the matrix being of size $n\times n$.

Its hip to be skew

Show that the determinant of a skew symmetric $2n\times 2n$ matrix is never reducible when viewed as a polynomial in $\binom {2n} 2$ variables.

The order of an automorphism

Let $A\in \mathrm{GL}(n, q)$. What is the maximum order $A$ can have?

Keeping it real

Let $A\in\Bbb{R}^{n\times n}$ be a matrix with a real, simple eigenvalue. Prove that there exists a neighborhood of $A$ in which all matrices have at least one real eigenvalue.

A generic curve

Exhibit a curve in $\Bbb{R}^n$ such that any $(n+1)$-uple of points in its trace is in general position.

Polynomial Decomposition

Decide if there are polynomials $A,B,C,D,E,F$ over $\mathbb{Q}$ such that

$1 + xy + x^2y^2 + x^3y^3 = A(x)B(y) + C(x)D(y) + E(x)F(y)$.

Sum of powers

Provide the most elegant proof to the following claim you can produce. Suppose $z_1,\ldots,z_s$ and $w_1,\ldots,w_r$ are nonzero complex numbers and for every positive integer $n$ it is true that $z_1^n +\cdots+z_s^n = w_1^n+ \cdots+w_r^n$. then $r=s$ and the $w_i$ are a reordering of the $z_j$.