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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 $\sqrt2$ …

I saw this problem some years ago and I would greatly appreciate any reference or solution. Let $X \in \operatorname{M}_n ( \mathbb{R} )$. Prove that there is $Y \in \operatorname{M}_n ( \mathbb{Z} ) …

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 …

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 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 …

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 …

Let $m,n$ be positive integers, $m,n>1$ and $X = \{(x_1,x_2, ..., x_m) \in \mathbb{Z}^m :1 \le x_i \le n, \forall 1 \le i \le m\}$. $A$ and $B$ are two disjoint subsets of $X$, such that if $a \in A$ …

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 …

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 $a$ and $b$ be two real numbers and $p_n(x,y)$ the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i},$$ where $n$ is a positive integer. In a previous post I asked if $p_n(a,b)$ was a rationa …

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 …

Let $k \geq 2$ and $N_1, N_2, ..., N_k$ be positive integers. Let $S=\{(a_1,a_2,...,a_k) \in \mathbb{Z}^k:1 \leq a_i \leq N_i\}$ and $A=\{1,2,...,\prod_{i=1}^{k} N_{i}\}$. Given a bijective map $f: …

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 …

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 …