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Here is one I've thought of a while ago for my combinatorics classes, but never ended up using. Let $C_{0},C_{1},C_{2},\ldots$ be the Catalan numbers. Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. …

Here is an olympiad-level problem on elementary number theory: Let $a$ be an integer and $n$ a positive integer. Prove that \begin{align} \left(a-1\right)^n \cdot n! \mid a^{n-1} \prod_{i=1}^n \left( …

to Question 1: Yes. To prove this, let me fix a positive integer $n$ and denote your matrix (whose determinant $f_{n}$ is) by $A$. The notation $\left[ k\right] $ shall be used for the set $\left\{ …

Yes, it does. Here is a stronger claim: Theorem 1. Let $F$ be a field of characteristic $\neq2$. Let $c\in F$. Let $r\in F$ be a nonzero square, and let $n_{1},n_{2}\in F$ be such that the pol …

Here is the answer I hinted at in a comment, in real detail. Took me a while, but I had no idea how tiresome such arguments are to expose... Yes, it is true: see Corollary 6 (b) below. The proof reli …

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Let $\left(U, d\right)$ be a finite ultrametric space -- that is, $U$ is a finite set, and $d : U \times U \to \mathbb{R}_{\geq 0}$ is a metric on $U$ such that every $x, y, z \in U$ satisfy $d\left(x …

To your first question: No, it is false. For $n = 256 - 13$, the number $a_n$ has $\nu_2\left(a_n\right) = 7 < 8 = \nu_2 \left(n+13\right)$. HOWEVER, it is almost correct: namely, it is correct whene …

We have $1\equiv u\left( p-r\right) \equiv u\left( -r\right) =-ur\operatorname{mod}p$, so that $p\mid1+ur=ur+1$. Hence, $\left\lceil \dfrac{ur}{p}\right\rceil =\dfrac{ur+1}{p}$. We need to prove t …

Let $q$ be a prime power. I will use the notations of Keith Conrad's Carlitz extensions paper (but I'll work over $\mathbb{F}_q$ rather than $\mathbb{F}_p$). The most general question I'm asking here …

I am looking for references (both of the readable and of the historical kind!) for the following result (which I formulate in one of its least general forms, so as not to complicate the discussion). I …

Persi Diaconis gave a talk about probabilities of carries at MIT about half a year ago. Unfortunately I was so busy trying to spot symmetric functions in the talk that most of it drifted past my mind, …

First of all, let's formalize the definition: Let $\lambda$ be a partition. For every integer $i \geq 1$, let $m_i\left(\lambda\right)$ be the number of appearances of $i$ in $\lambda$. Then your fanc …

The answer is Yes. The only reason for the appearance of formally real fields and $\mathbb R$ in the question is to rule out roots of unity; in fact, we have the following (cleaner and more general) s …

This question comes from the explicit construction of a smooth projective model of a hyperelliptic curve. Nevertheless it is fully elementary and, to me, more interesting than hyperelliptic curves. N …