Why is this true: $\|x\|_1 \le \sqrt n \cdot \|x\|_2$?
Stuck, help please, I am really new to this.I opened the 2-norm and multiplied by $n$, then I am thinking to square both sides.The problem is that I do not know how to open $(x_1 + x_2 + ... +...
View Article(Dis)Prove $\sum_{i=1}^n\sum_{j=1}^n{(|x_{i}-x_{j}|-|y_{i}-y_{j}|)^2}\geq 4$
Let $n\ge 4$ and two vectors $x$ and $y$ in $\mathbb{R}^n$ that satisfy$\sum_{i=1}^{n}{x_{i}^2}=\sum_{i=1}^{n}{y_i}^2=1$$\sum_{i=1}^{n}{x_{i} y_i}=0$$\sum_{i=1}^{n}{x_{i}}=\sum_{i=1}^{n}{y_i}=0$With...
View ArticleProve $\int_{\Bbb R_+} \frac{1}{1+x} a(x) \mathrm dx > 0$ given $\int_{\Bbb...
For a function $a(x)$ that is defined for $x \in \Bbb R_+$ and satisfies $\int_{\Bbb R_+} a(x) dx = 0$, is it possible to prove that $\int_{\Bbb R_+} \frac{1}{1+x} a(x) \mathrm dx > 0$ if we know...
View ArticleHow to lower bound $\tau$ based on the expression of $H$?
Let $A=\{a_{ij}\}_{1\le i,j\le n}$ be an $n$ by $n$ normalized symmetric Gaussian random matrix with $E[a_{ij}]=0$ and $E[a_{ij}^2]=1/n$. Ordering its eigenvalues by $\lambda_1\le \lambda_2\le \cdots...
View ArticleAn inequality for $J_n(b,c,d,e)$ under $be=r^2cd$, where $1
Let $n\in\mathbb Z_{\ge12}$, $h_n(x)=x^n(1-n\ln x)$ for $0<x\le1$, and $h_n(0)=0$. For $0\le s\le1$, let $H_n(s)=\max_{0\le a\le1-s}\{h_n(a+s)-h_n(a)\}$.For $b,c,d,e\ge0$ with $b+c\le1$ and...
View ArticleIs $A_{n+2}^2>A_nA_{n+4}$?
Let $g(t)=\cosh t\ln(2\cosh t)-t\sinh t$. For $n\ge2$, consider$$A_n=\frac{\sqrt n}{2^n}\sum_{k=1}^{n-1}\frac{n!}{k!(n-k)!}g\left(\left(k-\frac n2\right)\ln2\right).$$My conjecture is...
View ArticleAn asymptotic formula for $c_n^*/b_n^*$
Let $g(t)=\cosh t\ln(2\cosh t)-t\sinh t$ and $J_n(b,c)=(bc)^{n/2}\sum_{k=1}^{n-1}\frac{n!}{k!(n-k)!}g\left((k-n/2)\ln(b/c)\right)$. For $n\ge7$,...
View ArticleConjecture $ \sum _{i=1}^{n-1}\lfloor \frac{i^2}{n}\rfloor \ge...
Let $n \in \mathbb {N^*}$ and $$S_n = \sum_{k=1}^{n-1} (k^2 \bmod n)$$with the first values 0, 1, 2, 2, 10, 13, 14, 12, 24, 45, 44, 38, 78, 77, 70, 56, 136, 129, 152, 130, 182, 209, 184, 148, 250, 325,...
View ArticleBehavior of sub-solution of differential inequality.
Let $f:[0,\infty) \to \mathbb{R}$ obey the differential inequality$$f'' - 2\alpha f'+ 2\alpha f \leq 0$$where $0 < \alpha < 2$ is some constant. If $f(0) = 0$ and $f'(0) = 1$, can I say that...
View ArticleGiven $abc = 1$ for $a,b,c>0$, prove that $\sum\frac{a}{b(c+a)} \ge \frac{3}{2}$
Given $abc = 1$ for $a,b,c>0$, prove that$$\sum_{cyc}\frac{a}{b(c+a)} \ge \frac{3}{2}$$I have been trying this for ages now! Beginning with Titu's lemma,$$\sum_{cyc}\frac{a}{b(c+a)} =...
View ArticleInequality between incomplete beta and gamma functions
Let the regularized incomplete beta and gamma functions be defined as usual:\begin{equation}I_p(z,w) = \frac a {B(z,w)} \int_0^p t^{z-1} (1-t)^{w-1} \,\mathrm...
View ArticleProve $\sqrt{a^2 + (b-c)^2} + \sqrt{b^2 + (a-c)^2} + \sqrt{c^2 + (a-b)^2} \le...
Problem Statement by Nguyen Van Hoa, dated 8 Oct 2026Let $a, b, c$ be the side lengths of a triangle. Prove that the following inequality holds:$$\sqrt{a^2 + (b-c)^2} + \sqrt{b^2 + (a-c)^2} + \sqrt{c^2...
View ArticleConjecture on iterating $200 + x / 200$ [closed]
Conjecture 201: For every integer $x$ from $1$ to $100$, if we repeatedly apply$$f(x) = 200 + \frac{x}{200}$$exactly $100$ times, is it guaranteed that$$\operatorname{round}(f^{100}(x)) = 201?$$If so,...
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 ArticleProof of Triangle Inequality for $d(g; x, y) = \big(|x-y|^4 + g\,| x \times y...
I am seeking assistance in proving that a function, denoted as $d(g; x, y)$, defined on $\mathbb{R}^2 \times \mathbb{R}^2$ and parameterized by the non-negative real number $g$, may satisfy the...
View ArticleHow to find...
Disclaimer: This is repost of deleted one due to deletion of OP's account. I answered it, and I want different approach. My answer is included in this question.Question) Let $a,b,c$ be three...
View ArticleClosed form for the limit of $a_{j+1}=(a_j+b_j)/2$,...
Following these questions on iterated means$^{[1]}$$\!^{[2]}$$\!^{[3]}$, let $a_0=a$, $b_0=b$, where $a>b>0$, and$$a_{j+1}=\frac{a_j+b_j}{2},\...
View Articleminimize the trace of a matrix function
G,H are $n \times n$ Positive definite symmetric matrix and $LGL^{T}=H$, L is $n \times n$ as well, find the L to minimize the $$g(L)=tr((I-L)G(I-L)^{T})$$I learned how to calculate the max of trace by...
View ArticleOn Proving $ a^6 + b^6 + c^ 6 − 3a ^2 b^ 2 c^ 2 + 2(a^ 2 + bc)(b ^2 + ca)(c^...
Show that $ a^6 + b^6 + c^ 6 − 3a ^2 b^ 2 c^ 2 + 2(a^ 2 + bc)(b ^2 + ca)(c^ 2 + ab) ≥ 0 $ where $a,b,c$ are real numbers.On this problem, I made some observations and then started to solve.Here one...
View ArticleProving $\sum\limits_{\mathrm{cyc}}\sqrt{12ab+12ac+1}\ge 15$ for...
Problem. Let $a,b,c$ be non-nagative real numbers with $ab+bc+ca+abc=4.$ Prove that$$\color{black}{\sqrt{12ab+12ac+1}+\sqrt{12bc+12ba+1}+\sqrt{12ca+12cb+1}\ge 15.}$$Source: Tran Ngoc Khuong...
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