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Questions tagged [a.m.-g.m.-inequality]

For questions about proving and manipulating the AM-GM inequality. To be used necessarily with [tag:inequality] tag.

2
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4answers
177 views

Proving an equality using given one's, no need for differentiation

Prove that $(\frac ab+\frac bc+\frac ca)(\frac ba+\frac cb+\frac ac)\geqslant9$ The formulas given were $$\frac{a+b} {2}\geqslant\sqrt {ab}$$ $$a^2+b^2\geqslant2ab$$ $$\frac{a+b+c} {3}\geqslant\root3\...
0
votes
1answer
49 views

In any triangle is sinA+sinB+sinC=(3Root3)/2 always

Well I came with an interesting proof. But I just want to verify it From here we will get sinA+sinB+sinC=<(3root3)/2 and from this I get sinA+sinB+sinC>= (3root3)/2 Now the equation to be ...
4
votes
6answers
126 views

prove that : $\dfrac{a^2}{2}+\dfrac{b^3}{3}+\dfrac{c^6}{6} \geq abc$ for $a ,b ,c \in \mathbb{R}^{>0}$

prove that : $\dfrac{a^2}{2}+\dfrac{b^3}{3}+\dfrac{c^6}{6} \geq abc$ for $a ,b ,c \in \mathbb{R}^{>0}$ . I think that must I use from $\dfrac{a^2}{2}+\dfrac{b^2}{2} \geq ab$ but no result please ...
4
votes
2answers
100 views

show this inequality $a_{1}+a_{2}+\cdots+a_{n}\ge\sqrt{3}(a_{1}a_{2}+a_{2}a_{3}+\cdots+a_{n}a_{1})$

let $a_{1},a_{2},\cdots,a_{n}\ge 0,n\ge 3$,and such $$a^2_{1}+a^2_{2}+\cdots+a^2_{n}=1$$ show that $$a_{1}+a_{2}+\cdots+a_{n}\ge\sqrt{3}(a_{1}a_{2}+a_{2}a_{3}+\cdots+a_{n}a_{1})$$ I can prove ...
1
vote
1answer
26 views

Prove $ \sum_{cyc}\frac{x}{\sqrt{x^2+8yz}} \ge 1, \forall x,y,z\gt 0 $

Prove $ \sum_{cyc}\frac{x}{\sqrt{x^2+8yz}} \ge 1, \forall x,y,z\gt 0 $ I feel like the products between different variables (i.e. not x^2, y^2, z^2) give this inequality the $ \gt $ sign and I don't ...
1
vote
1answer
28 views

Finding the upper bound of a 3-variable function.(inequality)

The question is: $\forall x,y,z\in\Bbb R^+ \cup\{0\},\sum_{cyc}x=32$, find the maximum value of $\sum_{cyc}x^3y$. This question was introduced in a "elementary" book talkng about inequalities. The ...
3
votes
1answer
72 views

If positive $a$, $b$, $c$, $d$ satisfy ${1\over a+1}+{1\over b+1}+{1\over c+1}+{1\over d+1}=1$, then $abcd\geq 81$

Let $a,b,c,d>0$ satisfying $${1\over a+1}+{1\over b+1}+{1\over c+1}+{1\over d+1}=1$$ Prove that $abcd\geq 81$ I've tried to apply arithmetic geometric mean inequality or Cauchy-Schwartz ...
0
votes
2answers
88 views

Let $P$ be a polynomial with positive real coefficients. Prove that if $P(1/x) \geq 1/P(x)$ holds for $x = 1$, then it holds for every $x > 0$.

Let $P$ be a polynomial with positive real coefficients. Prove that if $$ P\left( \frac{1}{x} \right) \geq \frac{1}{P(x)} $$ holds for $x = 1$, then it holds for every $x > 0$. What I did: I was ...
2
votes
3answers
96 views

Prove that $1/a(1+b)+1/b(1+c)+1/c(1+a) ≥ 3/(1 + abc)$ if $a, b,$ and $c$ are positive real numbers.

Let a, b, c be positive real numbers. prove that: and that equality occurs if and only if a = b = c = 1. What I tried: 1st part: I tried a brute force approach where I make a common denominator for ...
2
votes
1answer
68 views

Finding the greatest value of $a^2b^3c^2$ if $a+b+c=3$ and all numbers are positive

Find the greatest value of $a^2b^3c^2$ if $a+b+c=3$ and all numbers are positive. Here is my attempt using $\text{AM-GM inequality}$: $$AM=\frac{a+b+c+a+b+c+b}{7}$$ $$GM=\sqrt[7]{a^2b^3c^2}$$ We ...
-1
votes
2answers
54 views

Find min of $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}+4\sqrt{2}\sqrt{\frac{ab+bc+ac}{a^2+b^2+c^2}}$ [closed]

Given the three real numbers a, b, c are not negative, in which at most some are equal to zero. Find min of $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}+4\sqrt{2}\sqrt{\frac{ab+bc+ac}{a^2+b^2+c^2}}$ ...
4
votes
3answers
91 views

Show that $a^{2014}+b^{2014}\geq a^{2013}+b^{2013} $.

Let $ a, b\in \mathbb {R}_{+} $ s.t. $a^{22}+b^{22}=a^{3}+b^{3} $. Show that $a^{2014}+b^{2014}\geq a^{2013}+b^{2013} $. By Chebyshev's inequality we obtain $a^{19}+b^{19}\leq 2\Leftrightarrow b^{19}...
0
votes
1answer
29 views

Extremes of the function in a certain set

I need to find the extreme values of a function $$f(x,y,z)=x^2+y^2+z^2$$ on the set $S=\{(x,y,z) \in \mathbb R^3: z=xy+2\}$. Set $S$ is not compact, so we cannot be sure the local extrema exist. ...
5
votes
1answer
123 views

to prove this inequality $\sum a^3b+3\ge 2(ab+bc+ca)$

let $a,b,c>0$ and such $a+b+c=3$,show that $$a^3b+b^3c+c^3a+3\ge 2(ab+bc+ac)$$ This problem is from my question when $n=3$ case,I found not to prove it. show this inequality with $\sum_{i=1}^{n}...
2
votes
0answers
74 views

How to prove using AM-GM $ \frac{4}{abcd} \ge \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} $ [duplicate]

Let $a, b, c$ and $d$ be positive numbers such that $a+b+c+d =4$, prove that $$ \frac{4}{abcd} \ge \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} $$ I am supposed to prove my using AM-GM, ...
0
votes
3answers
51 views

Is this inequality true? $u^2+v^2+s^2+t^2\geq (u+v)(s+t)$

Is this inequality true in $\mathbb{R}$? $$u^2+v^2+s^2+t^2\geq (u+v)(s+t)$$ I don't know if this is a well-known result. If you have a counterexample or a relevant reference I would appreciate it.
2
votes
3answers
45 views

If $x$ and $y$ are acute, and $\sin y = 3 \cos (x+y) \sin x$⁡, then find the maximum value of $\tan y$

Given $x,y$ are acute angles such that $$\sin y = 3 \cos(x+y)\sin x$$ Find the maximum value of $\tan ⁡y$. Attempt: We have $$\begin{aligned} 3(\cos x \cos y - \sin x \sin y) \sin x & = \sin ...
1
vote
1answer
67 views

Maximum value of function $f(x)=\frac{x^4-x^2}{x^6+2x^3-1}$ when $x >1$

What is the maximum value of the $$f(x)=\frac{x^4-x^2}{x^6+2x^3-1}$$ where $x > 1$ . My try Unable to solve further.
2
votes
4answers
80 views

How to find the range of $y=\frac{x^2+1}{x+1}$ without using derivative?

The only thing I know with this equation is $y=\frac{x^2+1}{x+1}=x+1-\frac{2x}{x+1}$. Maybe it can be solved by using inequality.
15
votes
4answers
199 views

Finding maxima of a function $f(x) = \sqrt{x} - 2x^2$ without calculus

My question is how to prove that $f(x) = \sqrt x - 2x^2$ has its maximum at point $x_0 = \frac{1}{4}$ It is easy to do that by finding its derivative and setting it to be zero (this is how I got $x_0 ...
0
votes
2answers
67 views

Inequality problem: with $x,y,z>0$, show that $\frac{x^5}{y^3}+\frac{y^5}{z^3}+\frac{z^5}{x^3}\geq x^2+y^2+z^2$

I am studying AM-GM inequalities in school and have this problem: With $x,y,z>0$ show that $\frac{x^5}{y^3}+\frac{y^5}{z^3}+\frac{z^5}{x^3}\geq x^2+y^2+z^2$.
1
vote
1answer
101 views

Inequality. $\sum_{cyc}(\frac{1}{a+b+\sqrt{2a+2c}})^3 \le \frac{8}{9}$

Problem. When $a, b, c>0, a, b, c \in \Bbb R, 16(a+b+c)\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$, Prove that $$\sum_{cyc}(\frac{1}{a+b+\sqrt{2a+2c}})^3 \le \frac{8}{9}$$ My approach: If we let $x=a+...
2
votes
3answers
80 views

Proving that the arithmetic and geometric means of a collection of non-negative numbers lies between their minimum and maximum values

Consider non-negative real numbers $a_1, a_2, a_3, ... , a_n$. How can I prove that both the arithmetic mean (AM) and the geometric mean (GM) of $a_1, a_2, a_3, ... , a_n$ are contained in the ...
10
votes
2answers
412 views

show this inequality with $\sum_{i=1}^{n}a_{i}=n$

Let $n\ge 3$ be postive number,$a_{i}>0,i=1,2,\cdots,n$,and $\displaystyle\sum_{i=1}^{n}a_{i}=n$,show that $$a^3_{1}a_{2}+a^3_{2}a_{3}+\cdots+a^3_{n}a_{1}+n\ge 2(a_{1}a_{2}\cdots a_{n-1}+a_{2}a_{...
0
votes
2answers
61 views

Algebraic inequality for positive reals $a,b,c$

The problem is from a previous maths olympiad and the last step is to prove the inequality $$4a^4bc + a^4c^2 + 9a^3bc^2 + 4a^3b^3 + 9a^2b^3c + a^2b^4 + 9ab^2c^3 + 4ab^4c + 4b^3c^3 + b^2c^4 +4a^3c^3 ...
6
votes
5answers
111 views

If $a+b=1$ find the greatest value for $a^2b^3$

I have been trying for some time on this question but i am new to inequalities so I am unable to solve it. I tried am gm but failed. Any help would be apriciated
2
votes
1answer
138 views

Prove $ \frac{a^2}{b + c} + \frac{b^2}{c + a} + \frac{c^2}{a + b} + \frac{36}{a + b + c} \geq 20 $

Show that if $a,b,c > 0$, such that $ab + bc + ca = 1$, then the following inequality holds: $$ \frac{a^2}{b + c} + \frac{b^2}{c + a} + \frac{c^2}{a + b} + \frac{36}{a + b + c} \geq 20 $$ What I ...
0
votes
1answer
92 views

Prove $x_1^2 + x_2^3 + … + x_{n - 1}^n + \frac{1}{x_1^2 x_2^3 … x_{n - 1}^n} \geq n + (x_1 - 1)^2 + 2(x_2 - 1)^2 + … + (n - 1)(x_{n - 1} - 1)^2$

Prove that if $x_1, ..., x_{n-1}$ are positive numbers and $n \geq 2$, than the following inequality holds: $x_1^2 + x_2^3 + ... + x_{n - 1}^n + \frac{1}{x_1^2 x_2^3 ... x_{n - 1}^n} \geq n + (x_1 - ...
2
votes
4answers
147 views

if $x-y =\sqrt{x}-\sqrt{y}$ with $x\neq y$ then $(1+\frac{1}{x})(1+\frac{1}{y})\geq 25$?

let $x\neq y$ be positive real numbers such that :$x-y= \sqrt{x}-\sqrt{y}$ , I have tried to prove this inequality $(1+\frac{1}{x})(1+\frac{1}{y})\geq 25$ that i have created but i didn't got it. ...
3
votes
2answers
123 views

Inequality with $(x+y)(y+z)(z+w)(w+x)=1$

Let $x,y,z,w>0$ and such that $$(x+y)(y+z)(z+w)(w+x)=1.$$ Show that $$\sqrt[3]{xyz}+\sqrt[3]{yzw}+\sqrt[3]{zwx}+\sqrt[3]{wxy}\le 2.$$ I'm trying to use Holder's inequality $$(\sqrt[3]{xyz}+\sqrt[3]...
0
votes
1answer
60 views

Further terms of Taylor expansion for improved inequalities?

Background: We know the classical exponential inequality \begin{align*} 1 + x \le e^x \qquad \text{or} \qquad x \le e^{x - 1} \end{align*} Taking $x_i = y_i/\overline{y}$, where $y_i \ge 0$ and $\...
1
vote
1answer
56 views

Find $a^3 + b^3 +c^3, $ given $a+b+c=12$ and $a^3 \cdot b^4 \cdot c^5 = 0.1 \cdot (600)^3$

$a+b+c=12$ and $a^3 \cdot b^4 \cdot c^5 = 0.1 \cdot (600)^3$. Find $a^3 + b^3 +c^3 = ?$ My approach is to use AM-GM inequality. Is it correct?
4
votes
1answer
83 views

How to show $7^{th}$ degree polynomial is non-positive in $[0,1]$

Let $0\le x\le 1$, show that inequality $$99x^7-381x^6+225x^5-415x^4+157x^3-3x^2-x-1\le 0$$ This problem comes from the fact that I solved a different inequality.I tried to solve it by factorizing ...
3
votes
1answer
120 views

If $ab+bc+ca=3$ for non-negative $a$, $b$, $c$, show that $\sum_{cyc}a^2b^2+\sum_{cyc}\frac{12a^2b^2c^2}{(a+b)^2}\ge 12abc$

Problem Let $a,b,c\ge 0$,and such $ab+bc+ca=3$, show that $$\sum_{cyc}a^2b^2+\sum_{cyc}\dfrac{12a^2b^2c^2}{(a+b)^2}\ge 12abc\tag{1}$$ A few hours ago, I asked for an error inequality. Wrong ...
2
votes
3answers
111 views

Simple two variable am-gm inequality [duplicate]

Given $x,y \in \Bbb{R}$, show that:$$x^2+y^2+1\ge xy+y+x $$ I tried using the fact that $x^2+y^2 \ge 2xy$ But then I'm not sure how to go on, Also tried factoring but didn't help much, also tried ...
2
votes
0answers
104 views

Prove $x_1x_2…x_n \ge (n-1)^n$ [closed]

Let $x_1x_2…x_n$ be positive real numbers such that $ $$\tfrac{1}{1+x_1}$$ + $$\tfrac{1}{1+x_2}$$ +… + \tfrac{1}{1+x_n}=1 $ Prove that $x_1x_2…x_n \ge (n-1)^n$ Please can someone help with this ...
0
votes
2answers
48 views

A simple proof for $\prod_{i=1}^d a_i+\prod_{i=1}^d b_i \le \prod_{i=1}^d (a_i^d+b_i^d)^{1/d}$? [duplicate]

Let $a_1,\dots,a_d,b_1,\dots,b_d$ be positive real numbers. Then $$ \prod_{i=1}^d a_i+\prod_{i=1}^d b_i \le \prod_{i=1}^d (a_i^d+b_i^d)^{1/d}$$ and equality holds if and only if $\frac{a_i}{a_j}=\frac{...
0
votes
1answer
45 views

Does $\alpha_j-(\Pi \alpha_i)^{1/d}=\beta_j-(\Pi \beta_i)^{1/d} $ force $\alpha_i=\beta_i$?

Let $d>1$, and let $\alpha_1,\dots,\alpha_d,\beta_1,\dots,\beta_d$ be positive real numbers. Is there an easy proof of the following claim: If $\alpha_j-(\Pi_{i=1}^d \alpha_i)^{1/d}=\beta_j-(\...
0
votes
2answers
41 views

Choosing suitable weights for AM-GM

Consider the following example: Prove that for all positive reals $a,b,c,d.$ $$a^4b+b^4c+c^4d+d^4a\ge abcd(a+b+c+d) $$ Solution: We apply AM-GM$$\frac{23a^4b+7b^4c+11c^4d+10d^4a}{51}\ge \sqrt[51]{a^{...
1
vote
1answer
40 views

Application of weighted AM-GM

Weighted AM-GM is usually stated as follows: given the non-negative reals $a_1,a_2,\dots,a_n$ and $\omega_1,\omega_2,\dots,\omega_n\ge 0$ with $\omega_1+\omega_2+\dots+\omega_n=1$ we have:$$\...
2
votes
1answer
52 views

Proof of an inequality: is it correct?

Let $x_{1},\cdots, x_{n}>-1$ be real numbers such that $\sum{x_{i}}=n$. Prove that: $$\sum_{i=1}^{n}{\frac{1}{x_{i}+1}}\geq \sum_{i=1}^{n}{\frac{x_{i}}{x_{i}^{2}+1}}$$ My proof: By AM-HM and $\...
1
vote
1answer
112 views

Find the maximize value of $\sqrt{\frac{a}{a+c}}+\sqrt{\frac{b}{b+c}}-\frac{9\sqrt{c^2+1}}{8c}$

Let $a;b;c>0$ such that $ab+bc+ca=1$. Find the maximize value of $$K=\sqrt{\frac{a}{a+c}}+\sqrt{\frac{b}{b+c}}-\frac{9\sqrt{c^2+1}}{8c}$$ I can see: $$a=b=\frac{1}{\sqrt{7}};c=\frac{3}{\sqrt{7}}\...
0
votes
3answers
66 views

Regarding AM-GM inequality

I have to show that if $a_1,a_2,\ldots a_n$ are non-negative real numbers, then $\frac{a_1+a_2+\ldots+ a_n}{n}\geq (a_1a_2\ldots a_n)^{1/n}$. Also equality holds if and only if $a_1=a_2=\ldots=a_n$. ...
0
votes
2answers
37 views

Prove or disprove an inequality problem

Let $n\geq 1$ be an integer and let $a_1,\ldots,a_n$ be positive real numbers, all between $0$ and $1$. Is it possible to prove or disprove: $$ {(\prod_{i=1}^{n}(1-a_i))}{(1+\sum_{i=1}^{n}a_i)}<1 ...
2
votes
2answers
69 views

Proof using AM-GM inequality

The questions has two parts: Prove (i) $ xy^{3} \leq \frac{1}{4}x^{4} + \frac{3}{4}y^{4} $ and (ii) $ xy^{3} + x^{3}y \leq x^{4} + y^{4}$. Now then, I went about putting both sides of $\sqrt{xy}...
0
votes
1answer
29 views

Inequality of small two numbers with power

Let $0<x,y<1$ are two real numbers and $n\in\mathbb N$. Is the following inequality true $$x^n-y^n\leq x^{n+1}-y^{n+1}?$$ I split into two cases: Case1: when $y<x$. Case2:when $x<y.$ ...
0
votes
2answers
27 views

Inequality problem (may be) involving means

If n is a positive integer then how can I prove that $$2^n>1+n \sqrt{2^{n-1}}$$ .Any hint may help.My textbook mentions this problem in category of A.M. ,G.M. , H.M. inequalities.So please give ...
0
votes
1answer
42 views

Inequality involving a kind of Harmonic mean

While revising the Harmonic mean, I came across this inequality which I haven't figured out how to solve, but I think it should be the application of some known inequality. I would be very grateful if ...
6
votes
2answers
94 views

Prove following inequality

Prove that $(\frac{2a}{b+c})^\frac{2}{3}+(\frac{2b}{a+c})^\frac{2}{3}+(\frac{2c}{a+b})^\frac{2}{3} ≥ 3$ What I tried was to use AM-GM for the left side of this inequality, what I got was $3(\frac{8abc}...
1
vote
2answers
61 views

Proving the given inequalities

Q: Prove the given inequalities for positive a,b,c:$(i) \left[\frac{bc+ca+ab}{a+b+c}\right]^{a+b+c}>\sqrt{(bc)^a.(ca)^b.(ab)^c}$$(ii) \left(\frac{a+b+c}{3} \right)^{a+b+c}<a^ab^bc^c<\left(\...