# [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 $n\times n$.

# A two-liner

Is there a ring $A$ such that there are two different ring injections $\Bbb{R}\to A$?

# Finite codimension

Let $E,F$ be Banach (more generally Fréchet) spaces and let $T,S:E\to F$ be continuous linear maps. Suppose that $T$ is surjective and $S$ is compact. Prove that $T+S$ has closed image and $\dim \mathrm{im}(T+S)<\infty$.

# Still expecting my fix

What’s the expected number of fixed points of a uniformly distributed random permutation of $n$ elements?

[redacted]

# What goes up must come down

Let $f \in C^1(\mathbb{R})$ such that $f,f' \in L^1(\mathbb{R})$. Prove that $\int_\mathbb{R}f'=0$.

# 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.