Questions tagged [cauchy-schwarz-inequality]

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

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Cauchy Schwarz equality in complex number

I was reading Rudin Theorem 1.35 and we have Cauchy Schwartz inequality: ${\left| {\sum\limits_{j = 0}^n {{a_j}{b_j}} } \right|^2}\leqslant \sum\limits_{j = 0}^n {{{\left| {{a_j}} \right|}^2}} \sum\...
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56 views

Proving an inequality of three variables

Can someone prove this inequality for the real numbers a,b,c? $$a^2+2b^2+8c^2\geq2a(b+2c)$$ I have tried simple manipulation of the terms to get quadratic expressions, but since one cannot factor the $...
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1answer
77 views

Hard inequality in contest for regional Olympiad 2011

Let a,b,c strictly positive numbers. Prove that $$5(\frac{a}{b}+\frac{b}{c}+\frac{c}{a})+12 \geq 3(\sqrt{\frac{a+8b}{c}}+\sqrt{\frac{b+8c}{a}}+\sqrt{\frac{c+8a}{b}})$$ I have tried to use tangent ...
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1answer
72 views

If $x,y,z>0$ with $x+y+z=\sqrt{3}$, prove that: $\sqrt{x^2+1-yz}+\sqrt{y^2+1-zx}+\sqrt{z^2+1-xy}\ge 3$ (Venezuela 1960)

If $x,y,z>0$ with $x+y+z=\sqrt{3}$, prove that: $$\sqrt{x^2+1-yz}+\sqrt{y^2+1-zx}+\sqrt{z^2+1-xy}\ge 3$$ I tried solving it as follows: The condition we want proved is: $\sqrt{3x^2+3-3yz}+\sqrt{3y^...
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2answers
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Counterexample? Equality holds in the Cauchy-Schwarz Inequality iff $\textbf{x} = \alpha\textbf{y}$ for some $\alpha \in \mathbb{R}$

I found this way of expressing that equality holds in Cauchy Schwarz inequality iff $\textbf{x},\textbf{y}$ and are linearly dependent in a book (as an exercise), but I think it is wrong, so there's a ...
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2answers
67 views

If $a,b,c,d,e$ be real number other than $-2≤a≤b≤c≤d≤e≤2$, prove the following inequation $\frac{1}{b-a}+\frac{1}{c-b}+\frac{1}{d-c}+\frac{1}{e-d}≥4$

Question If $a,b,c,d,e$ be real numbers $-2≤a≤b≤c≤d≤e≤2$, prove the following inequation $$\frac{1}{b-a}+\frac{1}{c-b}+\frac{1}{d-c}+\frac{1}{e-d}≥4$$ I tried to use Cauchy-Schwarz like $$\left(\frac{...
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1answer
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A simple proof of the Cauchy–Schwarz inequality with vectors

I have seen many proofs online, but none of them is as simple as my potentially correct and simple proof. Please let me know if the following is true, and if not where my mistake is. $$\begin{align*} |...
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184 views

$\sqrt{\frac{a^2+4bc}{b^2+c^2}}+\sqrt{\frac{b^2+4ac}{a^2+c^2}}+\sqrt{\frac{c^2+4ba}{b^2+a^2}}\ge 2+\sqrt{2}$

Prove that $\forall a,b,c\ge 0$ then $$\sqrt{\frac{a^2+4bc}{b^2+c^2}}+\sqrt{\frac{b^2+4ac}{a^2+c^2}}+\sqrt{\frac{c^2+4ba}{b^2+a^2}}\ge 2+\sqrt{2}$$ I have some ideas but they didn't lead to simpler ...
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0answers
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$\sum\sqrt{\frac{2a}{b+c}}\le\sqrt[3]{9\sum\frac{a}{b}}$

Let $a$, $b$ and $c$ be positive numbers. Prove that: $$\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}}\le\sqrt[3]{9\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$$ It is from ...
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4answers
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Prove the limit equation: $\lim_{x\rightarrow\infty}e^{-x}\sum_{n=0}^{\infty}f_{n}\left(x\right)=-\gamma.$ FYI$,\quad\gamma$ is Euler's constant

Define a function sequence $\left \{ f_{n} \right \}_{n= 0}^{\infty}$ on $\left ( 0, \infty \right )$ as $$f_{0}\left ( x \right )= \ln x,\quad f_{n+ 1}\left ( x \right )= \int_{0}^{x}f_{n}\left ( t \...
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When can we deduce $(mA+nB+pC)>(mX+nY+pZ)$ from $(A+B+C)>(X+Y+Z)$?

I am trying to prove an inequality in the form $(mA+nB+pC)>(mX+nY+pZ)$. I can prove the inequality $(A+B+C)>(X+Y+Z)$ and I wonder if there is any condition, under which we can deduce $(mA+nB+...
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Given three non-negative numbers $a,b,c$ so that $ab+bc+ca=3\geq a\geq b\geq c.$ Prove that $\frac{a+b}{2}+b+c+\frac{c+a}{\sqrt{3}}\leq\sqrt{3}+3$

Given the non-negative numbers $a, b, c$ so that $ab+ bc+ ca= 3\geq a\geq b\geq c.$ Show that $$\frac{a+ b}{2}+ b+ c+ \frac{c+ a}{\sqrt{3}}\leq\sqrt{3}+ 3$$ Source: StackMath/@RiverLi_ https://math....
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1answer
88 views

Inequality over real numbers [closed]

Let $$ x_1,\ x_2,\ldots,\ x_9\ $$ be non-negative real numbers for which holds: $${x_1}^2+{x_2}^2+\ldots+\ {x_9}^2 \geq25 $$ Prove that there are three of these numbers whose sum is at least 5.
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Find the value of $\int_0^1 f(x)\, dx$

Let $f(x)$ be continous on $[0; 1]$, satisfy the following condition: $$f(1) = 0$$ $$\int_0^1 [f'(x)]^2 \, \mathrm{d}x = 7$$ $$\int_0^{1} x^2f(x) \, \mathrm{d}x = \dfrac{1}{3}$$ Find the value of $\...
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1answer
43 views

Is the Cauchy–Schwarz inequality valid for the Lebesgue space $L^2(0,T,L^2(\Omega))$?

Let $n \in \mathbb{N}$ and $\Omega \subset \mathbb{R}^n$ be open and bounded set. It is known that if $f, g \in L^2(\Omega)$ then by Cauchy–Schwarz inequality we have $$ \Bigg| \int_{\Omega} f(x)\...
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Algebraic inequality $\sum \frac{x^3}{(x+y)(x+z)(x+t)}\geq \frac{1}{2}$

The inequality is $$\frac{x^3}{(x+y)(x+z)(x+t)}+\frac{y^3}{(y+x)(y+z)(y+t)}+\frac{z^3}{(z+x)(z+y)(z+t)}+\frac{t^3}{(t+x)(t+y)(t+z)}\geq \frac{1}{2},$$ for $x,y,z,t>0$. It originates from a 3-D ...
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1answer
28 views

Prove an inequality with ODE's

I need to prove that $\int_0^1 (u(x))^2 dx \leq \frac{1}{3} \int_0^1 (f(x))^2dx$ knowing that u solves the ODE $-u''(x) + u'(x) = f(x)$ for $x \in (0;1)$, and $u\in C_0^2((0;1))$ (u is $C^2$ and $u(0)=...
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2answers
73 views

Prove inequality using Cauchy-Schwarz

I have some difficulties to show that $\int_0^1 u^2(x) dx \leq \frac{1}{2} \int_0^1 (u'(x))^2 dx$ for some $u \in \{u \in C([0;1]) \cap C^2((0;1)) | u(0)=u(1) = 0 \}$ . I tried to write $u(x) = \int_0^...
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22 views

Reference for Lemma 2 on Mean Value Theorem Wiki

I'm looking for a reference for Lemma 2 on the Mean Value Theorem Wikipedia page. I've also stated it below: Lemma 2 Let $v:[a,b]\rightarrow\mathbb{R}^m$ be a continuous function on $[a,b]$, then we ...
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79 views

Maximum and minimum value of $\frac{8x - 3y}{\sqrt{4x^2+y^2}}$?

For real numbers $\,x, y\neq 0\,$ consider $$\frac{8x - 3y}{\sqrt{4x^2+y^2}}\,.$$ How to find the maximum and minimum value? I've already got the maximum by using the Cauchy–Schwarz inequality $$\big[(...
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In whiche case do we have $\left(\sum_{i=1}^n x_iy_{i}\right)^2 = \sum_{i=1}^n (x_i)^2 \sum_{i=1}^n (y_{i})^2$?

The exercice asked me first to prove that $( \sum_i x_i y_i )^2 \leq (\sum_i x_i^2)(\sum_i y_i^2)$ ( Cauchy inequality) and i managed to prove but then it asked me in which case we have the equality ($...
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A small error in an exercise from Apostol's Calculus I

Recently, I started to teach myself calculus by reading Apostol's Calculus Volum 1. I almost finished the introductory chapter. However, I felt the condition is imperfect for exercise 17 from ...
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1answer
100 views

Inequality with $abc=1$

Let $a;b;c$ be positive real numbers and $abc=1$ Find maximum value of: $P=\dfrac{1}{\sqrt{a^2+2b^2}}+\dfrac{1}{\sqrt{b^2+2c^2}}+\dfrac{1}{\sqrt{c^2+2a^2}}$ I tried to use Cauchy-Schwarz: $a^2+2b^2 \...
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3answers
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How do I prove this trigonometry inequality $\sin^2(y) \leq 2 \sin^2(x) + 2\sin^2(y-x)$?

I'm reading something which claims that for each $x ,y \in \mathbb{R}$, $$ \sin^2(y) \leq 2 \sin^2(x) + 2\sin^2(y-x). $$ Can someone please show me how this is true? Do I just use something like the $\...
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1answer
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When equality holds in C-S integral inequality? [closed]

In general C-S inequality : $(a^2+b^2)(x^2+y^2)\geq (ax+by)^2$ We know that if $\frac{a}{x}=\frac{b}{y}$ then equality holds. Then, when equality holds in C-S integral inequality? : $$ (\int_a^bf(x)g(...
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1answer
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Inequality demonstration using Cauchy-Schwarz inequality [closed]

I've been reading the mml book and had to do some exercices to pratice what I learned. But this exercise doesn't seem to have a solution to me. Can someone help me? The exercise: "Let $n \in \...
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1answer
88 views

Inequality $\dfrac{1}{\sqrt{1+2x}} + \dfrac{1}{\sqrt{1+2y}} + \dfrac{1}{\sqrt{1+2z}} \leq \sqrt{3}$.

Recently I answered this Question which required a proof for the upper and lower bound of a given function. While seemingly a calculus question, I realised that it can be transformed into an ...
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1answer
48 views

Trouble understanding proof of the Cauchy-Schwarz inequality

Hope you're doing well. Well I'm stuggling to understand the proof of Cauchy-Schwarz inequality, especially with the following equation: $$ \frac{1}{\|\mathbf{v}\|^{2}}\|\| \mathbf{v}\left\|^{2} \...
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2answers
108 views

Alternative approaches to prove the following inequality

For $a,b,c \in \mathbb{R^+},$ prove that $$\left(\dfrac{2a}{b+c} \right)^{\frac{2}{3}} + \left(\dfrac{2b}{c+a}\right)^{\frac{2}{3}} + \left(\dfrac{2c}{a+b}\right)^{\frac{2}{3}} \geq 3.$$ I managed to ...
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0answers
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How to prove $\bigl(\int_a^bf(x)dx\bigr)\bigl(\int_a^b\frac{1}{f(x)}dx\bigr)\ge (b-a)^2$?

Let $f:[a,b]\to \Bbb R$ be a positive and continuous function. Prove that $\bigl(\int_a^bf(x)dx\bigr)\bigl(\int_a^b\frac{1}{f(x)}dx\bigr)\ge (b-a)^2$. By the mean value theorem, there exist $c$ and $\...
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Lipschitz continuity of a certain function

Let $\mathcal{P}=\{P_1,\cdots,P_n\}$ be a collection of point-sets in $\mathbb{R}^d$. Given a point $x\in\mathbb{R}^d$, we define the following (convex) function $F_z(x) = \sum_{i\in[n]}\max_{p\in P_i}...
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2answers
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Prove that $f\left ( x \right )- x^{2021}$ always has at least one root $x_{0}\in\left ( 0, 1 \right )$

Given positive continuous function $f\left ( x \right )$ on the interval $\left [ 0, 1 \right ]$ so that $\int_{0}^{1}f\left ( x \right ){\rm d}x< \frac{1}{2022}.$ Prove that $f\left ( x \right )- ...
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2answers
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Range of $\frac{\cos\theta_1+\cdots+\cos\theta_{10}}{\sin\theta_1+\cdots+\sin\theta_{10}}$ given $\sin^2\theta_1+\cdots+\sin^2\theta_{10}=1$

Given that for $ \theta_i \in \left[0, \dfrac{\pi}{2}\right]$, where $1 \le i \le 10$ , $\sin^2\theta_1+\sin^2\theta_2+\cdots+\sin^2\theta_{10}=1$, find the minimum and maximum value of $$\dfrac{\cos \...
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1answer
68 views

Need help deriving the Schwartz inequality.

I am trying to prove the Schwartz Inequality (through an exercise) in the form of $$ \sum_{i=1}^{n}x_iy_i \le \sqrt{\sum_{i=1}^{n}x_i^2}\sqrt{\sum_{i=1}^{n}y_{i}^2} \tag{1} $$ through a method of ...
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1answer
129 views

Given real polynomials $P(x)$ so that $P(0)=P(1)=0,\int_0^1|P'(x)|{\rm d}x=1.$ For $x\in(0,1)$ prove or disprove that $|P(x)|\leq\frac{1}{2}$

Given real polynomial $P\left ( x \right )$ so that $P\left ( 0 \right )= P\left ( 1 \right )= 0, \int\limits_{0}^{1}\left | {P}'\left ( x \right ) \right |{\rm d}x= 1.$ For $x\in\left ( 0, 1 \right )$...
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0answers
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Using Cauchy-Schwarz and Jensen's inequality

I need a certain estimation from a paper in which it has been described to use the Cauchy-Schwarz inequality and Jensen's inequality. I did some calculations and had a few tries but I don't seem to ...
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4answers
91 views

How to minimize $\Big( \frac{2}{a^2}+\frac{1}{b^2}\Big)(a^2+b^2)$

I've tried completing the square but I can't however i think there's something called a cauchy schwarz inequality that could help me solve this. Help? But how $\frac2{a^2}$ is not a square number for ...
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2answers
118 views

I have not heard of this identity before

I am trying to increase my math abilities by working through the solutions of past Putnam exam problems. I am currently working on Putnam 1985-2a. I have been able to work through to the last part of ...
3
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1answer
57 views

Deriving the equation of the polar axis-aligned ellipsoid

The polar of a convex body $C$ is defined as: For a convex body $C$, its polar $C^*$ is given by $$C^* = \{x\in\mathbb{R}^n: \langle x,c\rangle \le 1, \forall c\in C\}$$ If I start with an ...
7
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1answer
148 views

Prove inequality with $a;b;c \in R$

Let $a;b;c \in \mathbb{R} $ such that $a+b+c=0$ Prove that: $P=\dfrac{a-1}{a^2+8}+\dfrac{b-1}{b^2+8}+\dfrac{c-1}{c^2+8} \geq -\dfrac{3}{8}$ I tried to do this: $\dfrac{8a-8}{a^2+8}+2+\dfrac{8b-8}{b^2+...
5
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1answer
109 views

Prove this inequality (please using AM-GM or Cauchy-Schwarz if possible)

Let $a ; b ; c > 0$ such that $a+b+c=3$ Prove: $P=\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+\dfrac{6abc}{ab+bc+ca} \geq 5$ I tried: $\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}=\dfrac{a^2c+b^2a+c^2b}{abc}=\...
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1answer
57 views

Prove the following inequality for all real numbers a,b,c,d [duplicate]

In this question I think we take the cyclic summation.... I am not familiar with it and could not understand it.Could you please help me and explain it in a easy way. Also how to you go about solving ...
6
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2answers
118 views

Prove a well-known inequality using Cauchy-Schwarz or AM-GM

For $a;b>0$ and $ab \geq 1$ we have a well-known inequality: $\dfrac{1}{1+a^2}+\dfrac{1}{1+b^2} \geq \dfrac{2}{1+ab}$ Which is equivalent to: $\dfrac{(a-b)^2(ab-1)}{(1+a^2)(1+b^2)(1+ab)} \geq 0$ (...
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2answers
107 views

Prove that $ \max(a_1, a_2, \ldots, a_n)\leq 4\min(a_1, a_2, \ldots, a_n)$

For $ n\geq2$ let $ a_1, a_2, \ldots a_n$ be positive real numbers such that $$ (a_1 + a_2 + \cdots + a_n)\left(\frac {1}{a_1} + \frac {1}{a_2} + \cdots + \frac {1}{a_n}\right) \leq \left(n + \frac {1}...
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1answer
41 views

Application of Cauchy Schwarz

I had a question about the inequality in the following picture. I'm working on problem 6.1.4 from Classical Fourier Analysis by Grafakos. I've been told that Cauchy-Schwarz will allow me to conclude ...
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1answer
59 views

Better bound for integral inequality

Let $f:[a,b] \rightarrow \mathbb{R}$ with $f \in C^1$ and $f(a)=0$. We want to show that $$\int_{a}^{b}f^2(x)\mathrm{d}x \leq (b-a)^2\int_{a}^{b}[f'(x)]^2\mathrm{d}x$$ There is a hint namely to ...
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2answers
48 views

Proof that the orthogonal projection onto a subspace satisfies the Cauchy Schwarz inequality

I am trying to prove $\left\|\hat{P}_{s}x\right\|^{2}_{2} \leqslant \left\| x \right\|_{2}\left\|\hat{P}_{s}x \right\|_{2} $ for all $x_1, x_2 \in X$ I am aware that I will need to use the Cauchy ...
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1answer
47 views

Find a lower bound using Cauchy-Schwarz inequality

Let $B$ a square positive definite matrix. Define $$\mu=\frac{(y^TB^{-1}y)(s^TBs)}{(y^Ts)^2}$$ How can I prove that $\mu\ge 1$? So far I get $$\frac{(y^TB^{-1}y)(s^TBs)}{(y^Ts)^2}\ge \frac{(y^TB^{-1}y)...
8
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2answers
230 views

$\frac{a^2+3b^2}{a+3b}+\frac{b^2+3c^2}{b+3c}+\frac{c^2+3a^2}{c+3a}\geqslant 3$

Let $a,b,c>0$ and $a^2+b^2+c^2=3$, prove $$\frac{a^2+3b^2}{a+3b}+\frac{b^2+3c^2}{b+3c}+\frac{c^2+3a^2}{c+3a}\geqslant 3$$ This inequality looks simple but I do not know how to solve it. The ...

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