22 votes
Accepted

Simplest diophantine equation with open solvability

Determining which integers $n$ are a sum of three cubes is a very famous open problem: $$a^3 + b^3 + c^3 = n, \quad a,b,c \in \mathbb{Z}.$$ Conjecturally, $n$ is a sum of three cubes iff $n \not \...
Daniel Loughran's user avatar
17 votes

Is there any published summary of Erdos's published problems in the American Mathematical Monthly journal?

I do not know of such a summary. But JSTOR has spent considerable time indexing the Monthly, including the "Problems and Solutions" sections. So if your library has a JSTOR subscription you can ...
Gerald Edgar's user avatar
  • 40.1k
15 votes

Characterization of a sphere: every "sub-sphere" has two centers

Blaschke's conjecture might be relevant. In particular, if the surface is $S^2$ with a $C^3$ metric, then it follows from Blaschke's conjecture, proved by Green. This states that a metric on the ...
Ian Agol's user avatar
  • 66.6k
14 votes
Accepted

Covering the disk with a family of infinite total measure

No even in dimension 1 (and multiplying the example for $\mathbb{R}$ by the small segment you get a counterexample in $\mathbb{R}^2$). Take the set $A_n\subset \mathbb{R}$ defined as $\bigcup_{k\in \...
Fedor Petrov's user avatar
13 votes

Simplest diophantine equation with open solvability

It's more complicated than the other answers by MattF and DanielLoughran, but the Erdős–Straus conjecture states that for every integer $n \ge 2$, there exist positive integers $x, y, z$ such that $$\...
Glorfindel's user avatar
  • 2,743
9 votes
Accepted

Inequality with symmetric polynomials

This looks like a better fit for Math Stackexchange, because it's the kind of thing one learns from Olympiad problem books . . . One standard approach that has not been mentioned yet: We may assume $...
Noam D. Elkies's user avatar
6 votes

Contest problems with connections to deeper mathematics

$\textbf{Problem IMO 1986}$. Let $n$ be a natural number. If $k^{2}+k+n$ is a prime number for $0\leq k \leq \lfloor\sqrt{n/3}\rfloor$, then show that $k^{2}+k+n$ is a prime for $0\leq k \leq n-2$. ...
5 votes
Accepted

Where can I access American Mathematical Monthly problems given an index?

Unfortunately, I don't think there's any particularly easy way to find a specific problem given its index number, but let me summarize some of the comments (and add some of my own) in a community wiki ...
5 votes

Examples of using physical intuition to solve math problems

My favorite example: it's not too hard to show---see here for example---that if $U$ is a unitary matrix, then $\left|\operatorname{Per}(U)\right|\le1$, where Per denotes the matrix permanent function, ...
4 votes

Inequality with symmetric polynomials

This is easily proven using the Rearrangement Inequality, which says that if we have two sequences of reals, and we are to pair them up in such a way as to maximize the sum of the products of the ...
R.P.'s user avatar
  • 4,745
4 votes

Inequality with symmetric polynomials

Follows from Hölder's inequality (p=6, q = 6/5): $ab^5 + ba^5 \le (a^6+b^6)^{1/6} (b^6+a^6)^{5/6}$
jjcale's user avatar
  • 2,768
4 votes

Is there any published summary of Erdos's published problems in the American Mathematical Monthly journal?

I do not think that anyone ever made an effort to collect ALL problems of Erdos. In the later part of his life he liked to give talks on various conferences under the title "My favorite unsolved ...
Alexandre Eremenko's user avatar
4 votes
Accepted

Covering the disk with a family of infinite total measure - the convex sequel

The planar case is rather simple but I'm still struggling with dimension 3 and higher, so we'd better keep this thread open at least for a while. Lemma 1: Suppose we have finitely many infinite ...
fedja's user avatar
  • 59.5k
4 votes

Method to solve system of exponential sums of the form $a^x+b^x=c$ given more equations than variables

I'm not sure about your method, but such equations can easily be turned into polynomial equations by introducing $u:=a^{x-1}$ and $v:=b^{x-1}$: \begin{cases} u+v=337 \\ au+bv=1267 \\ a^2u+b^2v=4825 \\ ...
Max Alekseyev's user avatar
3 votes
Accepted

Classifying functions up to suitable pre-composition and/or post-composition

The emphasis on finding solutions to equations $f(x) = y$ is a red herring, and note that in your examples your symmetries aren't acting pointwise on $Y$ as in your general explanation (sometimes they ...
Qiaochu Yuan's user avatar
3 votes

Inequality with symmetric polynomials

This is a very very special case of Muirhead's inequality. It involves two multi-exponents $\newcommand{\bZ}{\mathbb{Z}}$ $\alpha,\beta\in\bZ_{\geq 0}^n$ and the natural action of the symmetric ...
Liviu Nicolaescu's user avatar
3 votes

Inequality with symmetric polynomials

Also follows from Muirhead's inequality since $(6,0)$ majorizes $(5,1)$.
Max Alekseyev's user avatar
3 votes
Accepted

the sum of fractional parts times the ordinary powers

As Gerry already commented, your formulas are equivalent to a reciprocity theorem of Tom Apostol; see, e.g., the bibliography in this paper (Apostol's paper is too old to be on the arXiv): Your sums ...
matthias beck's user avatar
2 votes

List of recently solved mathematical problems

Here https://theorems.home.blog/theorems-list/ is the website you are asking for. It covers all recently solved mathematical problems, which are important (for example, published in a top journal) ...
2 votes
Accepted

Characterize the Monotonicity of a root of a cubic equation

Differentiating $h(x, \eta) = 0$ implicitly, we get $$ \dfrac{dx}{d\eta} = \dfrac{-c\eta^2 + V(2\eta+x)(\eta-x)^2}{\eta (3 V (\mu - x)^2 + c \eta)}$$ The denominator is always positive. On any ...
Robert Israel's user avatar
2 votes

A problem with elementary inequality involving probabilities and Brier scoring rule

Clearly $\sum_{i=1}^{n-1}p_i - (\sum_{i=1}^{n-1}p_i)^2\geq 0$, so we can ignore those terms in the inequality. Since $(\frac{1}{1-p_n})\sum_{i=1}^{n-1}p_i^2 \geq \sum_{i=1}^{n-1}p_i^2$ it suffices to ...
Dap's user avatar
  • 1,338
2 votes

Contest problems with connections to deeper mathematics

IMO 2003-6: Show that for each prime $p$, there exists a prime $q$ such that $n^p − p$ is not divisible by $q$ for any positive integer $n$. This obviously looks like something you want to apply the ...
2 votes
Accepted

Method to solve system of exponential sums of the form $a^x+b^x=c$ given more equations than variables

We can use linear algebra to make this easier. Consider the space of sequences of the form $A a^n + B b^n$. We can use several dual bases for this space. One is the obvious basis of maps that send $A ...
user44191's user avatar
  • 4,961
2 votes
Accepted

Solving an recursive sequence

$$f(x+1) = f(x)^2 - 2,\;\;f(0)=4$$ $$\Rightarrow f(x)=2\cos\left(2^x\arccos 2\right)=2 \cosh \left(2^x \ln \left(\sqrt{3}+2\right)\right)$$
Carlo Beenakker's user avatar
1 vote

Is it possible to find UNSATisfiable solutions to a SAT problem with a SAT problem?

Indeed, it is most likely not possible to find a polynomial-size propositional formula whose models are exactly the non-outputs of $f(x,y)$, the problem for general $f$ is $\Sigma_2^p$-complete. You ...
Tomáš Peitl's user avatar
1 vote

Contest problems with connections to deeper mathematics

Colorado Mathematical Olympiad (1986): Santa Claus and his elves paint the plane with two colors, red and green. Prove that the plane contains two points of the same color, exactly one inch apart. So ...
1 vote

Characterize the Monotonicity of a root of a cubic equation

In reply to the follow-up questions from FTXX, here is an argument. Using Israel's notations, suppose $h(x,\eta)=0$ and $\frac{dx}{d\eta}=0$. That means $$V(\eta-x)^3=c\eta(\eta+x) \tag1$$ and $$-c\...
T. Amdeberhan's user avatar
1 vote

Examples of using physical intuition to solve math problems

Let A, B, C be three points in the plane and assume that P minimizes the sum of the distances to A, B, C. One can proof that then the angles APB, BPC, CPA are all 120 deg. Physically we can imagine a ...

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