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The following question was asked at MSE without any solution: Show that the equation $x^3+y^3+z^3-2xyz=1$ have infinitely many integer solutions $(x,y,z)$. A more general question was also propo …

The following interesting problem was asked at Aops and I wonder if it was based on some research paper: Let $K$ be a convex body in $\mathbb R^2$, such that the diameter of $K$ is less than $\sqrt2$ …

A plane is colored with two colors. It's an easy exercise to prove that it's always possible to find an equilateral triangle whose vertices have all the same color. Does anyone know any proof or re …

Let $f(G)$ denote the number of $K_4$ in a graph $G$ and $e(G)$ denote the number of edges of $G$. Consider two simple graphs $G_1$ and $G_2$ having the same set $V$ of $n$ vertices and let $H_1(U)$ …

Are there infinitely many positive integers which are neither a sum nor a difference of two perfect powers? This question was proposed some years ago at KoMaL. It's easy to see that the odd nu …

It was asked at the Bulletin of the American Mathematical Society Volume 64, Number 2, 1958, as a Research Problem, if a Hurwitz polynomial with real coefficients (i.e. all of its zeros have negative …

Does anyone know any reference or proof for the following problem? Let $m$ and $n$ be positive integers, $m,n \geq 2$. Each positive integer is coloured in one of $m$ different colours. Is it possibl …

Does anyone know if the following problem has ever been studied? Let $a$ and $b$ be two real numbers and consider the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$$ where $n$ is a posit …

The sequence $x_1, x_2, ..., x_n$ of positive integers contains at least $\frac {2n}{3}+1$ distinct numbers and each of them appears at most three times. How to prove that there is a permutation $y …

Let $p(x) \in \mathbb{Z}[x]$, such that $\deg (p) \ge 3$. Can we always find $q(x) \in \mathbb{Z}[x]$, such that $\deg (q) < \deg(p)$ and $p(q(x))$ is reducible over $\mathbb{Q}[x]$? Is there a …

Is it possible to find the smallest positive real number $c$ (or at least the smallest positive integer $c$) such that the following result holds for all functions $f$ satisfying some conditions? Let …

I asked this question at MSE, but I think it's more appropriated to MO. Let $x_{n}$ be a sequence, such that $x_{n+1}= \dfrac{nx_{n}^2+1}{n+1}$ and $x_n>0$ for all $n$. There is a positive inte …

Let $x_{1},x_{2},\cdots,x_{n}>0$, show that $$\left(\sum_{k=1}^{n}x_{k}\cos{k}\right)^2+\left(\sum_{k=1}^{n}x_{k}\sin{k}\right)^2\le \left(2+\dfrac{n}{4}\right)\sum_{k=1}^{n}x^2_{k}$$ This ques …

I saw this problem some years ago, don't remember the source: Let $P$ be a Jordan polygon (i.e. the only points of the plane belonging to two edges are the polygon vertices) that can be tiled with pa …

Thomassen proved that the vertex set of every planar graph can be decomposed into two sets inducing a 1-degenerate graph and a 2-degenerate graph, respectively (C. Thomassen, Decomposing a planar grap …