Does every triangle satisfy...
Let $a, b, c$ be the sides of a triangle and $A,B,C$ be the corresponding opposite angles and $R$ be its circumradius. Experimental data shows that $\pi^2$ is the largest i.e. optimal constant for...
View ArticleInequality for log-concave distributions
I would really appreciate some help with showing that an inequality holds for a particular class of probability distributions. In what follows, I will describe you the problem and my solution so...
View ArticleAbout $\sum_{i=1}^n \frac{x_i x_{i+1}}{1 - x_i x_{i+1}} < \frac{n}{n^2 - 1}$
Let $n \ge 3$, $a_i \in \mathbb{N}$ and $a_i \ne a_j$. Define $S = \sum_{i=1}^n a_i$, and let $x_i = \frac{a_i}{S}$ (where $x_i > 0$ and $\sum x_i = 1$, and $x_i$ are pairwise distinct). Does the...
View ArticleProving inequality $\sum_{i=1}^{n} \frac{x_i}{\sqrt{(n-1)x_i^2 +...
For $n \ge 2$, let $x_1, x_2, \dots, x_n$ be positive real numbers. I am trying to prove the following inequality:$$\sum_{i=1}^{n} \frac{x_i}{\sqrt{(n-1)x_i^2 + (n+1)(\sum_{j \neq i} x_j)^2}} \ge...
View ArticleGiven three positive numbers $a,b,c$ so that $abc= 1$. Prove...
(A problem proposed by Michael Rozenberg). Given three positive numbers $a,b,c$ so that $abc= 1$. Prove$$\left ( a- 1+ \frac{1}{b} \right )\left ( b- 1+ \frac{1}{c} \right )\left ( c- 1+ \frac{1}{a}...
View ArticleInequality involving face areas and inradius of a tetrahedron: $\sum 1/S_i^2...
In a tetrahedron $A_1A_2A_3A_4$, let $S_1$ be the area of the face opposite vertex $A_1$, and define $S_2, S_3, S_4$ similarly for the other faces. Let $r$ be the inradius of the tetrahedron.I am...
View Article4$">A "musical comma" inequality: prove that $\sqrt[8]{5}\sqrt[3]{35}>4$
In musical tuning, intervals are represented by frequency ratios. An octave corresponds to a ratio of $2/1$. Other "pure" intervals are derived from small primes: the prime $3$ gives the perfect fifth...
View ArticleProve $a^6+b^6+c^6 + \frac14(1 - a^2 - b^2 - c^2)^3 \ge \frac{11}{180}$...
Problem. Let $a, b, c \ge 0$ with $a^2 + b^2 + c^2 + \frac12(a + b + c)^2 \le 1$. Prove that$$a^6+b^6+c^6 + \frac{(1 - a^2 - b^2 - c^2)^3}{4} \ge \frac{11}{180}.$$Equality case: $a = b = c =...
View ArticleGiven three positive numbers $a,b,c$ so that $a\ge b\ge c$. Prove that...
Given three positive numbers $a, b, c$ so that $a\ge b\geq c$. Prove that$$\sum_{cyc}\frac{a+ b\sqrt{\frac bc}}{a\sqrt{\frac bc}+ b}\ge 3.$$I made this inequality.Firstly, we need to have one general...
View ArticleGiven two positive numbers $b$ and $c$, prove $\left ( \frac{3}{b}- 1 \right...
Given two positive numbers $b,\,c$. Prove $\left ( \dfrac{3}{b}- 1 \right )(3- b)^{2}+ \left ( \dfrac{b}{c}- 1 \right )(b- c)^{2}+ (c- 1)^{3}\geqq 0$ .My problem is given a solution by user...
View ArticleA question about the columns in the triangle of Stirling numbers of the first...
It is known that the rows of both Stirling number triangles (first kind ${n \brack k}$ and second kind ${n \brace k}$) are log-concave sequences. This is also true for the main diagonals in the...
View ArticleFor any two partitions $\mathcal{A},\mathcal{B}$ of $\{0,1\}^k$, prove...
For any two partitions $\mathcal{A},\mathcal{B}$ of $\{0,1\}^k$, prove $\sum_{A\in \mathcal{A},B\in \mathcal{B}} |A \wedge B|^2\geq 4^k$.
View ArticleIf $n\geq3$ doesn't divide $a_1,a_2,,\ldots,a_n$, or $\sum a_i$, it divides...
$\color{red}{\mathbf{Problem\!:}}$ Let $n\geq3$ be a given positive integer, and $a_1 ,a_2, a_3, \ldots ,a_n$ are all given integers that aren't multiples of $n$ and $a_1 + \cdots + a_n$ is also not a...
View ArticleA lower bound for a sum of minima of products
Let $1/3<r<1/2$, and let $\mathbf p=(p_1,p_2,p_3,p_4)$ and $\mathbf q=(q_1,q_2,q_3,q_4)$ satisfy$$0\le p_i,q_i\le r,\qquad\sum_{i=1}^{4}p_i=\sum_{i=1}^{4}q_i=1.$$For an integer $n\ge1$, what is...
View ArticleHow does one prove the determinant inequality...
Reposted on MathOverflowLet $\,A,B,C\in M_{n}(\mathbb C)\,$ be Hermitian and positive definite matrices such that $A+B+C=I_{n}$, where $I_{n}$ is the identity matrix. Show that...
View ArticleA logarithmic minimum with equal variables for $n= 6$ and unequal variables...
For an integer $n\ge2$, let $a,b,c>0$ satisfy $a+b+c=1$, and let $s_n=1-a^n-b^n-c^n$. Consider$$F_n(a,b,c)=\frac{\displaystyle\sum_{k=1}^{n-1}\binom nk...
View ArticleAsymptotic expansions involving three Gumbel variables
Let $G_1,G_2,G_3\sim\operatorname{Gumbel}(0,1)$ be independent. For $\eta_1,\eta_2,\eta_3\in\mathbb R$ and $\tau>0$,...
View ArticleDoes every triangle with side $a,b,c$ and inradius $r$ satisfy...
Let $a,b,c$ be the sides of a triangle with inradius $r$ and semi-perimeter $s$. Experimental data show that every triangle satisfies$$ 1728r^2|(a-b)(b-c)(c-a)| \le s^5 $$and that $1728$ is the optimal...
View ArticleInequality including the minimum value of the Gamma function and the...
It's a problem I found unconventional :Problem :Let $0<x$ be a real the define :$$f\left(x\right)=\prod_{n=1}^{292}\left(\frac{n+1}{x!+n}\right)^{\frac{1}{n}}$$Shows that :$$f(x)<1+C$$Where $C$...
View ArticleWhy can't we multiply both sides by $x$ when solving $\frac{1}{x} > 1$?...
Given the inequality $\frac{1}{x} > 1$, why can't we simply multiply both sides by $x$ to solve it? Multiplying both sides by $x$ yields $1 > x$, but I'm told this result is incorrect. Someone...
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