Questions tagged [cauchy-schwarz-inequality]

Problems with using C-S (Cauchy-Schwarz inequality)

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Inequality with special case of equality at $\infty$

Prove that if $a,b,c,d$ are positive reals we have: $$\frac{a}{\sqrt{a^2+b^2}}+\frac{b}{\sqrt{b^2+c^2}}+\frac{c}{\sqrt{c^2+d^2}}+\frac{d}{\sqrt{d^2+a^2}}\leq3$$ I think that I have found a equality ...
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Relation between normalized product of integrals of two functions and integral of product of two normalized functions

Could someone let me know if there is any relation between A and B, if $A=\frac{\int^L_0 f(x)g(x)dx}{L}$ and $B=\frac{\int^L_0 f(x)dx\cdot\int^L_0 g(x)dx}{L^2}$
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Maximize $\sum_{i=1}^N \log\left( 1+a_i b_i c_i \right)$ under $\sum_{i=1}^N a_i \to \infty$ and $\sum_{i=1}^N a_i \to 0$

I have a log-sum maximization problem of the form: $$ \max_{\left\{ a_i \right\},\left\{ b_i \right\},\left\{ c_i \right\}} ~ \sum_{i=1}^N \log\left(1+ a_i b_i c_i \right) $$ subject to $$ \sum_{i=1}^...
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Multiplicative energy and Cauchy-Schwartz

Let $A$ be a finite set in a ring, and define $E(A) =\left|\left\{(a, b, c, d) \in {A}^{4}: c a=d b\right\}\right|.$ A number of papers (e.g. here) quote the lower bound $$E({A}) \geq \frac{|{A}|^{4}}{...
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Find all triples $(a, b, c)$ of real numbers such that $a + 4b + 18c =\frac{a^2+b^2+c^2}{6}=2022$

Find all triples $(a, b, c)$ of real numbers such that $$a + 3b + 18c + min(a, b, c) =\frac{a^2+b^2+c^2}{6}=2022$$ There are three cases ( $a$ is min, or $b$ or $c$) Case 1: $a$ is min. We get $$2a + ...
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Prove that $(1+x)^k/k + (1-x)^m/m\geq 1/k +1/m$ without calculus

Note: this has been edited to make the question more general. I want to show that $(1+x)^k/k + (1-x)^m/m$ is minimized at $x=0$ when $k,m\geq 1$ and $-1\leq x \leq1$. Of course, I could take the first ...
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How can I prove that the perimeter is at most 60?

Problem: Let $\Delta$ be a triangle in the plane. Let $P$ be the perimeter of the triangle and $A$ be the area. Let $a,b,c$ be the length of the sides and suppose they are positive integers. Suppose ...
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Inequality involving real numbers with rational powers

If $ a, b, c$ are positive real numbers, not all equal, and n is a negative rational number then show that: $$ a^n (a-b)(a-c)+ b^n(b-c)(b-a)+c^n(c-a)(c-b) > 0 $$ I started by proving this for the ...
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Inequality - norm. [closed]

Why form first inequality we have the second inequality? \begin{equation*} \begin{aligned} &g(x_2)\geq \langle \nabla g(x_2),\frac{1}{L}\lVert \nabla g(x_2)\rVert\rangle-\int_0^1 \langle \...
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Inequality of positive integers [duplicate]

I have to prove the following inequalities for positive integer $n$: $$n^n\geqslant\left(\frac{n+1}2\right)^{n+1}$$ $$ 2^ {n(n+1)}> (n+1)^{n+1} \left(\frac{n}{1}\right)^{n} \left(\frac{n-1}{2}\...
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Clarification in a question regarding vector space and series convergence

The question I am referring to is Hassani's Mathematical Physics Problem 2.18: Using the Schwarz Inequality to show that if $\{\alpha_i\}_{i=1}^{\infty}$ and $\{\beta_i\}_{i=1}^{\infty}$ is in $\...
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2 answers
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Inequality involving moments of a distribution [closed]

Let X be a real random variable. Under what conditions on the distribution do we have that $$\mathbb{E}( X^{2n + 2}) \geq \mathbb{E}( X^{2n}) \mathbb{E}( X^{2})$$ for all integer $n$? I tried using ...
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To prove the inequality of positive rational numbers

Show that: $$ \left(\frac{a+b}{a+b+c}\right)^{c} \left(\frac{b+c}{a+b+c}\right)^{a} \left(\frac{a+c}{a+b+c}\right)^{b}< \left(\frac{2}{3}\right)^{a+b+c} ,a\ne b\ne c$$ PS: I am supposed to use ...
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1 answer
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Lower bound for $I(a,b)=\int_0^\infty \frac{dx}{\sqrt{a^2+x^2}\sqrt{b^2+x^2}} \label{elliptic integral 2}$

I am studiyng this paper which states that the integral $$I(a,b)=\int_0^\infty \frac{dx}{\sqrt{a^2+x^2}\sqrt{b^2+x^2}} \tag{1}$$ satisfies the inequality \begin{align} \frac{\pi}{2a} \leq I(a,b) \...
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Proving $\frac{x^2+y^2+z^2}{x+y+z}+\frac 32\sqrt[3]\frac{xy+yz+xz}{x+y+z}\geq \frac 52$, for positive values with $xyz=1$, without expansion?

Let, $x,y,z>0$ and $xyz=1$, then prove that $$\frac{x^2+y^2+z^2}{x+y+z}+\frac 32\sqrt[3]\frac{xy+yz+xz}{x+y+z}\geq \frac 52$$ I know that $$x^2+y^2+z^2\geq x+y+z$$ by Cauchy-Schwarz. So, $$\frac{x^...
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To prove the relation between n numbers and their AM - GM [duplicate]

Given $A$ and $G$ to be the arithmetic and geometric mean of n positive real numbers $a_1, a_2,...,a_n$ then for any $k > 0$ show that $$ (k+A)^n \ge\ (k+a_1)...(k+a_n) \ge\ (k+G)^n .$$ I started ...
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Rayleigh inequality sine

Let $A\in\mathbb{C}^{n\times n}$ be diagonalizable, $y\in\mathbb{C}^n\setminus\{0\}$ and $x\in\mathbb{C}^n$ the eigenvector of $\lambda\in\sigma(A)$. Define the Rayleigh quotient $R_A(y)=\frac{\langle ...
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2 answers
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To prove the following inequalities of positive rational numbers

I have to prove the following inequalities: $$ a^ab^bc^c \ge \ (\frac{a+b}{2})^{\frac{a+b}{2}} (\frac{c+b}{2})^{\frac{c+b}{2}} (\frac{a+c}{2})^{\frac{a+c}{2}} $$ $$(a+b)^{c}(c+b)^{a}(a+c)^{b} < \...
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  • 375
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1 answer
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To prove the following inequality of real numbers

If $a_1,...a_n$ are all positive real numbers then prove that $$\left(\frac{a_1 + a_2 + \dots + a_n}{n}\right)^n \ge a_1a_2\left(\frac{a_3 + a_4 + \dots + a_n}{n-2}\right)^{n-2}.$$ I approached the ...
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  • 375
1 vote
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21 views

Positive random variables with harmonic mean equal to mean have variance zero

Suppose $X>0$ is a random variable with $\mathbb{E}X=c$ and $\mathbb{E}X^{-1}=c^{-1}$. By applying Cauchy-Schwarz to $\sqrt{X}$ and $\sqrt{X^{-1}}$ and using the "equality iff linearly ...
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Cauchy-Schwarz inequality for nonsymmetric matrices

I am interested in Cauchy-Schwarz inequalities for inner products of the form $\langle Ax,y\rangle$ where $A$ is some matrix. It is rather easy to see that the Cauchy-Schwarz inequality continues to ...
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Optimal Poincare constant (with constraints)

Given any integrable function $f\, \colon \mathbb{R}_+ \to \mathbb R$, it is relative easy to show that the best constant for which the following Poincar'e-type inequality $$ \int_{\mathbb{R}_+} f^2\,\...
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Proof of Cauchy-Schwarz in a Complex Inner Product space.

Setup: Given a complex inner product space $V$ the Cauchy-Schwarz inequality is $$ |\langle x,y \rangle|\leq \sqrt{\langle x,x \rangle} \sqrt{\langle y,y \rangle}. $$ I know that the RHS is just the ...
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3 votes
2 answers
127 views

Showing $\frac{xy+yz+xz}{x+y+z}>5-\sqrt{4(x^2+y^2+z^2)+6}$ for $x,y,z>0$ and $xyz=1$

Let $x,y,z>0$ with $ xyz=1$ then prove that, $$\frac{xy+yz+xz}{x+y+z}>5-\sqrt{4(x^2+y^2+z^2)+6}$$ Let $$5≤\sqrt{4(x^2+y^2+z^2)+6}\implies x^2+y^2+z^2≥\frac {25-6}{4}=\frac {19}{4}$$ then the ...
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Prove that $\int ^1_0x^2f(x)dx\cdot\int ^1_0f(x)dx\leq \int ^1_0xf(x)dx\cdot\int^1_0f(x)^2dx$

Let $f:[0,1]\to (0,+\infty)$ be a decreasing function. Prove that: $$\int ^1_0x^2f(x)dx\cdot\int ^1_0f(x)dx\leq \int ^1_0xf(x)dx\cdot\int^1_0f(x)^2dx$$ My approach: I think it can be solved by C-S ...
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2 votes
2 answers
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Proving $\int_0^1\vert(f(x)^2)'\vert dx\leq \int_0^1( f''(x))^2dx$, where $f(0)=f'(0)=0$ [duplicate]

Suppose that $f\in C^{2}\left[ 0,1\right] $ and $f\left( 0\right) =f^{\prime }\left( 0\right) =0$. Prove that $$ \int_{0}^{1}\left\vert \left( f\left( x\right) ^{2}\right) ^{\prime }\right\vert dx\leq ...
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3 answers
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Prove that $(xy+yz+xz)(x^2+y^2+z^2+x+y+z)≥2(x+y+z)^2$

Let, $x,y,z>0$ such that $xyz=1$, then prove that $$(xy+yz+xz)(x^2+y^2+z^2+x+y+z)≥2(x+y+z)^2$$ My progress: Using the Cauchy-Schwars inequality I got $$(xy+yz+xz)(x^2+y^2+z^2+x+y+z)≥2(xy+yz+xz)(x+...
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1 vote
2 answers
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Is this nice-looking inequality actually trivial?

Let, $x,y,z>0$ such that $ xyz=1$, then prove that $$(xy+yz+xz)(x^2+y^2+z^2+xy+yz+xz)≥2(x+y+z)^2 $$ I tried to use the inequality $$x^2+y^2+z^2≥xy+yz+xz$$ Then I got, $$(xy+yz+xz)(x^2+y^2+z^2+xy+...
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3 answers
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If $x_1+x_2+x_3+\dots+x_n=1$ then show that $x_1^2+x_2^2+x_3^2+\dots+x_n^2\ge\frac{1}{n}$. [closed]

Basically, if the sum of $n$ numbers is $1$ then prove that the sum of their squares is greater than or equal to $1/n$.
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Proving summation given vectors and orthonormal basis using triangle inequality and cauchy-schwartz inequality

Let $\{y_1,y_2,...,y_n\}$ be a set of vectors in $\Bbb R^n$ and let $\{u_1,u_2,...,u_n\}$ be an orthonormal basis of $\Bbb R^n$ prove that for every $\lambda_1,...,\lambda_n$ $\in$ $\Bbb R$ $||\sum_{i=...
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Cauchy-Schwarz Inequality definition meaning

In Linear algebra, I have an inequality theorem that states, "If $x$ and $y$ are vectors in an inner product space $V$, then $$\langle x,y\rangle^{2} \leqslant \langle x,x\rangle\langle y,y\...
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1 answer
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Discriminant involving variance-covariance matrices

I'm trying to show a quadratic equation of the form $$\frac{1}{2} a' \Sigma^{-1} a x^2 - a' \Sigma^{-1} \iota x + \frac{1}{2} \iota' \Sigma^{-1} \iota$$ has a real solution. Here $\Sigma$ is an $N \...
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2 votes
2 answers
56 views

Can we prove the inequality without opening the parentheses? $(x+y+z)(xy+yz+xz)(x^2+y^2+z^2)≥6(x^2+y^2+z^2)+3(xy+yz+xz)$

Let, $x,y,z>0$ such that $ xyz=1$, then prove that $$(x+y+z)(xy+yz+xz)(x^2+y^2+z^2)≥6(x^2+y^2+z^2)+3(xy+yz+xz)$$ I tried to use the following inequalities: $$x^2+y^2+z^2≥xy+yz+xz$$ and The Cauchy–...
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2 votes
3 answers
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When the equality in the Cauchy-Schwarz inequality holds? [duplicate]

Q19 deals with Schwartz inequality. $$x_1y_1 + x_2y_2 \le \sqrt{x_1^2 + x_2^2}\cdot \sqrt{y_1^2 + y_2^2}$$ The last part asks for each of the proofs to deduce when equality holds. Specifically, in ...
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1 vote
1 answer
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Inequality of expectation of product [closed]

Suppose non-negative random variables $X_1,X_2,\ldots,X_N$, and they are maybe dependent. Is the following inequality correct? \begin{align} E\left[\prod_{n=1}^{N} X_n^{k_n}\right] \le \max_n E\left[...
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  • 545
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1 answer
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Bound of expectation of product

We know the Cauchy–Schwarz inequality, \begin{align} (E(XY))^2\le E(X^2)E(Y^2). \end{align} I am wodering whether the following holds for positive random variables $X$ and $Y$ \begin{align} (E(X^mY^n))...
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  • 545
0 votes
1 answer
39 views

Prove that $yz+zx+xy\leq (y+z-x)^2+(z+x-y)^2+(x+y-z)^2$.

Suppose $x,y,z$ are real numbers. Prove that $yz+zx+xy\leq (y+z-x)^2+(z+x-y)^2+(x+y-z)^2$. May I ask how to use Cauchy-Schwarz Inequality to obtain $|yz+zx+xy|\leq x^2+y^2+z^2$ ? I only know $x^2+y^2+...
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2 votes
1 answer
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Alternative idea to find minimum of $\sin^4x+\cos^4x$ without derivative

My question is about to find minimum value of $f(x)=\sin^4x+\cos^4x$ without derivation. My trial is listed below $$\sin^4x+\cos^4x=\\(\sin^2x+\cos^2x)^2-2\sin^2x\cos^2x=\\1-2(\frac{\sin^2(2x)}{2})^2\\...
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0 votes
1 answer
99 views

Prove that $\frac{x^2}{x^2+2y}+\frac{y^2}{y^2+2z}+\frac{z^2}{z^2+2x} \ge 1$

Prove that $$\frac{x^2}{x^2+2y}+\frac{y^2}{y^2+2z}+\frac{z^2}{z^2+2x} \ge 1$$ with $xy+yz+zx=3$ and $x,y,z >0$. If I use Cauchy-Schwarz then $\frac{x^2}{x^2+2y}+\frac{y^2}{y^2+2z}+\frac{z^2}{z^2+...
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2 votes
0 answers
63 views

Proving Inequality Over Reals

If $a,b$ and $c$ are real numbers, then $\dfrac{a+b-c}{kc^2+a^2+b^2}+\dfrac{b+c-a}{ka^2+b^2+c^2}+\dfrac{c+a-b}{kb^2+c^2+a^2}\leqslant \dfrac{9}{(k+2)(a+b+c)}$where $(k=\dfrac{3+\sqrt{17}}{4})$ My ...
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3 votes
2 answers
162 views

How can I use the Cauchy-Schwarz inequality in this function of random variables?

I have the function $\rho_{\lambda}:RV(\Omega)\rightarrow \mathbb{R}$ defined on the space $RV(\Omega)$ supported over some scenario set $\Omega$: $\rho_\gamma(X)=\frac{1}{\gamma} \log (\mathbb{E}[e^{-...
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1 vote
3 answers
108 views

Prove : $\sqrt{\dfrac{ab}{bc^2+1}}+\sqrt{\dfrac{bc}{ca^2+1}}+\sqrt{\dfrac{ca}{ab^2+1}}\le\dfrac{a+b+c}{\sqrt{2}}$

Let $a,b,c>0$ satisfy $abc=1$, prove that: $$\sqrt{\dfrac{ab}{bc^2+1}}+\sqrt{\dfrac{bc}{ca^2+1}}+\sqrt{\dfrac{ca}{ab^2+1}}\le\dfrac{a+b+c}{\sqrt{2}}$$ My attempt: Let $a=\dfrac{1}{x};b=\dfrac{1}{y};...
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1 vote
1 answer
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Find the error in my workings for the proof of $\left(\frac{x}{x+2y} + \frac{y}{y+2z} + \frac{z}{z+2x}\right)\ge 1$

Here are my workings... By Cauchy-Schwarz, $$\left(\frac{x}{x+2y} + \frac{y}{y+2z} + \frac{z}{z+2x}\right)\left( \frac{x+2y}{x} + \frac{y+2z}{y} + \frac{z+2x}{z}\right) ≥ (1+1+1)^2$$ $$\left(\frac{x}{...
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0 votes
0 answers
43 views

How do I know if I've found the actual maximum value of an expression using the Cauchy inequality?

Example: Let $x,$ $y,$ and $z$ be real numbers such that $x^2 + y^2 + z^2 = 1.$ Find the maximum value of $x + 2y + 2z^2.$ We can try using Cauchy: $$(x^2 + y^2 + z^2)(1 + 4 + 4z^2) = 5 + 4z^2 \ge (...
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0 votes
2 answers
83 views

Solve for $a,b,c,d$ over $a^4+b^4+c^4+d^4=48, abcd=12$

Find the number of ordered quadruples $(a,b,c,d)$ of real numbers such that \begin{align*} a^4 + b^4 + c^4 + d^4 &= 48, \\ abcd &= 12. \end{align*} I think I should apply some inequalities, ...
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9 votes
1 answer
175 views

A lower bound of $\sum_{i=1}^n a_i \sum_{i=1}^n \frac{1}{a_i}$

For fixed $n \ge 2$ , find the maximum of real number $t$ such that $$ \sum_{i=1}^n a_i \sum_{i=1}^n \frac{1}{a_i} \ge n^2 + t \cdot \frac{\displaystyle\sum_{1\le i<j \le n} (a_i - a_j)^2 }{\left( \...
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1 vote
2 answers
61 views

Finding the maximum of $ \sqrt{x + 27} + \sqrt{13 - x} + \sqrt{x}$ for $0\le x \le 13$ using the Cauchy-Schwarz inequality gives two different answers

I am trying to find the maximum of the expression $$\sqrt{x+27} + \sqrt{13-x} + \sqrt{x} \qquad \text{for } 0 \le x \le 13.$$ Clearly using Cauchy is the way to go here, so I tried $$((x + 27) + 2(13-...
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1 vote
1 answer
89 views

Prove that for $a,b,c \in \mathbb{R}^+$ with $abc = 1$, $\frac{1}{a^3(b + c)} + \frac{1}{b^3(a + c)} + \frac{1}{c^3(a + b)} \ge \frac{3}{2}$ [duplicate]

I would like confirmation that I did this proof correctly. If I did, it would be a milestone in my mathematical journey as it would be my first IMO problem. By Cauchy, we have that $$\left( \frac{1}{a^...
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0 votes
1 answer
37 views

Does there exist a "quadratic" reverse triangle inequality?

Let $x,y$ be some vectors. The triangle inequality states $$-\|x - y\| \leq \|x\| - \|y\| \leq \|x - y\|$$ Is it also true that: $$-\|x - y\|^2 \leq \|x\|^2 - \|y\|^2 \leq \|x - y\|^2$$
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1 vote
2 answers
133 views

Find $n\ge 3$ such that $\displaystyle\sum_{k=1}^n x_k = 0$ implies $\displaystyle\sum_{\mathrm{cyc}} x_1x_2\le 0$

Find the number of positive integers $n \ge 3$ that have the following property: If $x_1,$ $x_2,$ $\dots,$ $x_n$ are real numbers such that $x_1 + x_2 + \dots + x_n = 0,$ then $$x_1 x_2 + x_2 x_3 + \...
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