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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
votes
2answers
77 views

A very strange algebra problem with four numbers with an average 1.

Here is a Russian algebra problem from a 1999 olympiad. I can’t solve it, please help! Find all possible values of $a,b,c,d$, if they are positive reals, their average is 1, and $$\dfrac{3-a+b(-a-ac)}...
7
votes
5answers
86 views

How to compare logarithms $\log_4 5$ and $\log_5 6$?

I need to compare $\log_4 5$ and $\log_5 6$. I can estimate both numbers like $1.16$ and $1.11$. Then I took smallest fraction $\frac{8}{7}$ which is greater than $1.11$ and smaller than $1.16$ and ...
2
votes
1answer
42 views

How to prove inequality $(a^2+b^2+c^2)^3\ge6(a^3+b^3+c^3)^2$ when $a+b+c=0$?

I know the identity $a^3+b^3+c^3-3abc = 1/2 (a+b+c) [(a-b)^2+(b-c)^2+(c-a)^2]$. So when it comes to this problem where $a+b+c=0$, you get $(a^2+b^2+c^2)^3\ge54(abc)^2$ when $a+b+c=0$ (because $ a^3+b^...
7
votes
1answer
84 views

For positve $a$, $b$, $c$, $d$ with $a+b+c+d\leq 1$, prove that $\sqrt[4]{(1-a^4)(1-b^4)(1-c^4)(1-d^4)}\geq255\cdot a b c d .$

Let $a,b,c,d\in\mathbb R_+$ such that $a+b+c+d\leqslant1$. Prove that$$ \sqrt[4]{\smash[b]{(1-a^4)(1-b^4)(1-c^4)(1-d^4)}}\geqslant255·abcd. $$ My observations: I can see that all of $a,b,c,d$ are ...
-1
votes
2answers
46 views

Prove the following inequalities for any positive real numbers x,y

Given the following inequality, I'm finding it difficult to demonstrate that it holds for all real numbers: $xy^3$ $\leq \frac{1}{4}x^4 + \frac{3}{4}y^4$ I have normalized the equation to avoid ...
0
votes
1answer
86 views

Prove the inequality $\sqrt{\frac{a^2+1}{b+c}}+\sqrt{\frac{b^2+1}{a+c}}+\sqrt{\frac{c^2+1}{a+b}}\ge 3$

Let $a;b;c\in R^+$ such that $ab+bc+ca>0$. Prove that $$\sqrt{\frac{a^2+1}{b+c}}+\sqrt{\frac{b^2+1}{a+c}}+\sqrt{\frac{c^2+1}{a+b}}\ge 3$$ I have seen the similar question is $$\frac{a^2+1}{b+c}+\...
2
votes
3answers
65 views

Minimize value of $P=\frac{a^2+b^2+c^2}{ab+2bc}$

Let $a;b;c\in R^+$. Find the minimize value of $$P=\frac{a^2+b^2+c^2}{ab+2bc}$$ By Wolframalpha i known $P_{\text{min}}=\frac{2}{\sqrt{5}}$ at $b=\sqrt{5}a$ or $c=2a$ so i used AM-GM's inequality but ...
8
votes
1answer
52 views

Inequality, how to know intuition behind it

I was solving the following inequality For $a$, $b$, $c$ and $d$ being positive real numbers which goes as $$ \frac{a}{a+b} + \frac{b}{b+c} + \frac{c}{c+d} + \frac{d}{d+a} \leq \frac{a}{b+c} + \frac{...
4
votes
2answers
72 views

P.T. $\frac{1}{a^3(b+c)} +\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)} \ge \frac 32$

If $abc=1$ where $a,b,c$ are positive real. Prove that ,$\frac{1}{a^3(b+c)} +\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)} \ge \frac 32$. I tried to multiply the LHS by $abc$ to make the relation homogeneous ...
0
votes
0answers
35 views

How to prove the following inequality for a function?

I have function $$D(M\gamma)=\frac{M\gamma K}{K+\gamma(M\gamma-1)}\tag{1}$$ where $K$ is some positive constant. In this case, how to show that $$D(M(\gamma-1))<D(M\gamma),~~\text{for } \gamma<\...
1
vote
2answers
94 views

How to prove this inequality using AM-GM?

Suppose $a,b,c$ are positive real numbers. Then prove that $$\Big(\frac{a+b}{2}\Big)\Big(\frac{b+c}{2}\Big)\Big(\frac{c+a}{2}\Big)\ge\Big(\frac{a+b+c}{3}\Big)\Big(abc\Big)^\frac{2}{3}\tag{*}$$ My ...
1
vote
1answer
34 views

Show that for positive numbers a,b,c,d, $\sum_{cyc} ab \leq \frac{1}{4}\left(\sum_{cyc} a \right)^2$ and … [duplicate]

Let a,b,c,d be four positive real numbers. Show that $$\sum_{cyc} ab \leq \frac{1}{4}\left(\sum_{cyc} a \right)^2$$ and $$\sum_{cyc} abc \leq \frac{1}{16}\left(\sum_{cyc} a \right)^3$$ My textbook ...
0
votes
1answer
56 views

How does the number $3$ help in this (probably Cauchy-based) inequality?

Given $a,b,c>0$ and $a+b+c=3$. Prove that $$\dfrac{a^2+bc}{b+ca}+\dfrac{b^2+ca}{c+ab}+\dfrac{c^2+ab}{a+bc}\ge3$$ Attempt: Using Cauchy inequality $(a+b\ge2\sqrt{ab})$: $\dfrac{a^2}{b+ca}+b+ca\...
0
votes
2answers
70 views

Maximizing $x^2 y + y^2 z + z^2 x − x^2 z − y^2 x − z^2 y$ for $x$, $y$, $z$ between $0$ and $1$ (inclusive)

Find the maximum value of the expression:- $$x^2 y + y^2 z + z^2 x − x^2 z − y^2 x − z^2 y$$ when $0 \leq x \leq 1$, and $0 \leq y \leq 1$, and $0 \leq z \leq 1$. Please note: This is a initial ...
9
votes
3answers
146 views

Prove that if $a + b + c + d = 4$, then $(a^2 + 3)(b^2 + 3)(c^2 + 3)(d^2 + 3) \geq 256$.

Given $a,b,c,d$ such that $a + b + c + d = 4$ show that $(a^2 + 3)(b^2 + 3)(c^2 + 3)(d^2 + 3) \geq 256$. What I have tried so far is using CBS: $(a^2 + 3)(b^2 + 3) \geq (a\sqrt{3} + b\sqrt{3})^2 = 3(...
2
votes
2answers
63 views

Prove this inequality for given conditions

For all $x,y>0$, $$\frac{1}{(x+1)^2} + \frac{1}{(y+1)^2} \ge \frac{1}{xy+1}$$ I can only think of substituting $x+1$ with $a$ and $y+1$ with $b$. Then the inequality turns into $$(a^2 + b^2) (ab-a-...
0
votes
4answers
95 views

How to prove $(xyz)^{1/3} \le (x+y+z)/3$ using linear algebra

To elaborate on the title, you can prove that $$\sqrt{xy}\le \frac{x+y}{2}$$ in the following way: Is there a way that this can be extended to the inequality in the title and to a general case? For ...
2
votes
2answers
92 views

find the range if $(x_{1}+x_{2}+\cdots+x_{2009})^2=4(x_{1}x_{2}+x_{2}x_{3}+\cdots+x_{2009}x_{1})$

let $x_{i}\ge 0(i=1,2,3,\cdots,2009$,and such $$(x_{1}+x_{2}+\cdots+x_{2009})^2=4(x_{1}x_{2}+x_{2}x_{3}+\cdots+x_{2009}x_{1})=4$$ find the range $\sum_{i=1}^{2009}x^2_{i}$ I guess the range is $[1.5,...
3
votes
4answers
92 views

How to show that: $\left(\frac{mn+1}{m+1}\right)^{m+1} > n^m$

If m and n are positive integers then show that:$$\left(\frac{mn+1}{m+1}\right)^{m+1} > n^m$$I am new in this Course.So i can't able to think how i start a inequalities question by looking it's ...
1
vote
1answer
136 views

If $x+y+z=3$ then $\sum x\sqrt{x^3+3y} \ge 6$

Let $x,y,z>0$ such that $x+y+z=3$. Prove that $$\sum x\sqrt{x^3+3y} \ge 6$$ This trying doesn't help. With Cauchy Schwarz $(\sum x\sqrt{x^3+3y})^2\geq \sum x^2\sum(x^3+3y) = (x^2 + y^2 + z^2)(x^...
0
votes
2answers
73 views

$X>0$, $Y>0$ and $X^2+Y ^3\geq X^3+Y^4$. Prove that $X^3+Y ^3\leq 2$ [duplicate]

$X>0$, $Y>0$ and $X^2+Y ^3\geq X^3+Y^4$. Prove that $X^3+Y ^3\leq 2$ First I tried: $0<X\leq1$ and $0<Y\leq1$, but this in not the only case for $X^2+Y ^3\geq X^3+Y^4$ and get nowhere. ...
-3
votes
2answers
245 views

Inequality $\sqrt{xy+yz+zx} \ge \frac {8}{15} (x+y+z)$ [closed]

By Titu's inequality: $\sum_{cyc} \frac {1}{x+y} \ge \frac {(1+1+1)^2}{2(x+y+z)} = \frac {9}{2(x+y+z)}$ Then, to prove: $ \frac {3}{x+y+z} + \sum_{cyc} \frac {1}{x+y} \ge \frac {4}{\sqrt{xy+yz+zx}}$...
0
votes
4answers
76 views

Prove that $a_n \gt b_n$ $\forall $ $n \ge 6$

Given that $$a_n =\left(1^2+2^2+3^2+\cdots+n^2\right)^n$$ and $$b_n=n^n \times (n!)^2$$ Then prove that $a_n \gt b_n$ $\forall $ $n \ge 6$ My attempt: I tried using induction, but I could not ...
1
vote
1answer
62 views

Generalized Sum over Product Inequality : Related to Cauchy Schwarz's and AM-GM Inequality.

Consider set $S = \left \{ a_i \mid 1 \leq i \leq n \right \}$ where $a_i \in \mathbb{R}$ and fixed integer $t$ I do not know whether the following inequality is true or not: $$\sum_{P \subset S, \...
2
votes
4answers
97 views

How to prove AM-GM by induction 3

Let $a_1;a_2;...;a_n\ge 0$. Prove that $$\frac{\sum ^n_{k=1}a_k}{n}\ge \sqrt[n]{\prod ^n_{k=1}a_k}$$ We will prove it's true with $n=k$. Indeed we need to prove it's true with $n=k+1$ WLOG $a_1\le ...
2
votes
5answers
65 views

Prove that $\log_n(n+1)\geq\log_{n+1}(n+2)$ for $n>1$

Prove that $\log_n(n+1)\geq\log_{n+1}(n+2)$ for $n>1$. So far I only know that $\log_n(n+1)>\frac{\log_n(n+2)}{\log_n(n+1)}$ Since $n>1$, LHS must be greater than RHS. Is there any other ...
7
votes
2answers
912 views

Largest possible value of trigonometric functions

Find the largest possible value of $$\sin(a_1)\cos(a_2) + \sin(a_2)\cos(a_3) + \cdots + \sin(a_{2014})\cos(a_1)$$ Since the range of the $\sin$ and $\cos$ function is between $1$ and $-1$, shouldn'...
2
votes
1answer
134 views

Prove that: $\frac{bc}{a^2+1}+\frac{ac}{b^2+1}+\frac{ab}{c^2+1}\leq \frac{3}{4}$ [duplicate]

Given three positive numbers a,b,c satisfying $$a^2+b^2+c^2=1$$ Prove that: $$\frac{bc}{a^2+1}+\frac{ac}{b^2+1}+\frac{ab}{c^2+1}\leq \frac{3}{4}$$ The things I have done so far: $$\sum \limits_{cyc}\...
1
vote
2answers
255 views

Find the maximum of the value$(n^{n-1}-1)\sqrt[n]{a_{1}a_{2}\cdots a_{n}}+\sqrt[n]{\frac{a^n_{1}+a^n_{2}+\cdots+a^n_{n}}{n}}$

Let $n$ be give a positive integer, $a_{i} \ge 0$,such that $a_{1}+a_{2}+\cdots+a_{n}=n$. Find the maximum value of $$(n^{n-1}-1)\sqrt[n]{a_{1}a_{2}\cdots a_{n}}+\sqrt[n]{\dfrac{a^n_{1}+a^n_{2}+\...
2
votes
2answers
154 views

Prove that: $xy\sqrt{z}+yz\sqrt{x}+zx\sqrt{y}\geq x+y+z$

Let $x$,$y$ and $z$ are positive and $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\leq 3$$ Prove that: $$xy\sqrt{z}+yz\sqrt{x}+zx\sqrt{y}\geq x+y+z$$ The things I have done so far $$3\geq \sum \limits_{cyc}\...
1
vote
2answers
148 views

Prove that $\frac{6(a^2 + b^2 + c^2)}{a + b + c} \geq \frac{(a + b)^2}{b + c} + \frac{(b + c)^2}{c + a} + \frac{(a + c)^2}{a + b}$

Prove that if $a,b,c$ are the lengths of the edges of a given triangle, then the following inequality holds: $\frac{6(a^2 + b^2 + c^2)}{a + b + c} \geq \frac{(a + b)^2}{b + c} + \frac{(b + c)^2}{c + ...
0
votes
6answers
112 views

Prove that inequality $\frac{2ab}{a+b}+\sqrt{\frac{a^2+b^2}{2}}\ge \sqrt{ab}+\frac{a+b}{2}$

Let $a;b\ge 0$. Prove that inequality $$\frac{2ab}{a+b}+\sqrt{\frac{a^2+b^2}{2}}\ge \sqrt{ab}+\frac{a+b}{2}$$ My try: $LHS-RHS=\frac{2ab}{a+b}-\frac{a+b}{2}+\sqrt{\frac{a^2+b^2}{2}}-\sqrt{ab}\ge 0$ ...
0
votes
1answer
40 views

Find the maximum constant such that the inequality

Let $a;b>0$. Find the maximum constant such that the inequality $$\frac{1}{a^2+b^2}+\frac{1}{a^2}+\frac{1}{b^2}\ge \frac{8+2k}{\left(a+b\right)^2}$$ Let $a=1$ then we have: $-\frac{k-1}{2a^2}\ge 0\...
2
votes
5answers
112 views

Minimum value of $\frac{x^4+5x^2+7}{x^2+3}$

Minimum value of $$f(x)=\frac{x^4+5x^2+7}{x^2+3}$$ we have $f(x)$ as $$f(x)=(x^2+3)+\frac{1}{x^2+3}-1$$ Now by $AM \gt GM$ we have $$(x^2+3)+\frac{1}{x^2+3} \gt 2$$ But equality cannot occur ...
1
vote
1answer
47 views

Show $\prod_{i=1}^{n}x_i^{\alpha_i} \leq \sum_{i=1}^{n}\alpha_ix_i$ using convexity.

Prove that $$\prod_{i=1}^{n}x_i^{\alpha_i} \leq \sum_{i=1}^{n}\alpha_ix_i$$ where $\alpha_i $ positive scalars with $\sum_{i=1}^n\alpha_i=1$ and $x_i$ are positive scalars. I thought to express $\...
1
vote
5answers
70 views

Finding the minimum value of $\log _d a + \log _bd + \log _ac + \log _c b$

The question says to find the minimum value of $$\log _da + \log _bd + \log _ac + \log _cb$$ Given, $a,b,c,d \; \in R^+ -\Bigl(1\Bigl)$ My approach: I used the AM-GM inequality and so we can see that $...
2
votes
2answers
68 views

Prove the inequality $(a^2 b+b^2 c+c^2 a)(a b^2+b c^2+c a^2)\geq9a^2 b^2 c^2$.

For positive real numbers $a, b$ and $c$, how do we prove that $$(a^2 b+b^2 c+c^2 a)(a b^2+b c^2+c a^2)\geq9a^2 b^2 c^2\,?$$
2
votes
4answers
71 views

Algebraic Inequality involving AM-GM-HM

If $$a,b,c \;\epsilon \;R^+$$ Then show, $$\frac{bc}{b+c} + \frac{ab}{a+b} + \frac{ac}{a+c} \leq \frac{1}{2} \Bigl(a+b+c\Bigl)$$ I have solved a couple of problems using AM-GM but I have always ...
0
votes
0answers
28 views

Mixed Means Inequality References?

I've recently stumbled upon a great Mixed Means Inequality (see An "AGM-GAM" inequality). I know it can be used to quickly prove Cauchy-Schwarz. Does anyone know a book that has worked ...
1
vote
3answers
63 views

Prove: $\log_2(x)+\log_3(x)+\log_5(x)>9\log_{30}(x)$

Prove for all $x>1$ $\log_2(x)+\log_3(x)+\log_5(x)>9\log_{30}(x)$ So what I did was: \begin{align} &\frac{\ln(x)}{\ln(2)}+\frac{\ln(x)}{\ln(3)}+\frac{\ln(x)}{\ln(5)}>9\frac{\ln(x)}{\...
5
votes
4answers
123 views

Showing that if $p_1 + \cdots p_n = 1$ then $\displaystyle \sum_{k=1}^n \left(p_k + \dfrac {1}{p_k} \right)^2 \ge n^3+2n+\dfrac 1n$?

This problem is from the book "Cauchy-Schwarz Masterclass": Show that if $p_1 + \cdots p_n = 1$ with each $p_i$ positive, then $\displaystyle \sum_{k=1}^n \left(p_k + \dfrac {1}{p_k} \right)^2 \ge ...
6
votes
1answer
166 views

$ \frac {3 x y z}{(x+y) (y+z) (z+x)} + \sum\limits_{cycl}^{} \left(\frac {x+y}{x+y+ 2 z}\right)^2 \ge \frac {9}{8}.$

For $x,y,z>0,$ I have to prove that $$ \frac {3 x y z}{(x+y) (y+z) (z+x)} + \sum\limits_{cycl}^{} \left(\frac {x+y}{x+y+ 2 z}\right)^2 \ge \frac {9}{8}.$$ I tried to use $$ \sqrt {\frac {1}{3} \...
1
vote
2answers
66 views

How do I prove the following inequality? [duplicate]

How do I proceed to solve the inequality $$\frac{(a^2+b^2)}{(a+b)} + \frac {(b^2+c^2)}{(b+c)} + \frac{(a^2+c^2)}{(a+c)} \geq (a+b+c)$$ where $a , b , c > 0$ I have thought of taking the terms on ...
1
vote
5answers
111 views

Show $\frac{1}{b+c+d} + \frac{1}{a+c+d} + \frac{1}{a+b+d} + \frac{1}{a+b+c} \ge \frac{16}{3(a+b+c+d)}$.

If $a,b,c,d > 0$ and distinct then show that $$ \frac{1}{b+c+d} + \frac{1}{a+c+d} + \frac{1}{a+b+d} + \frac{1}{a+b+c} \ge \frac{16}{3(a+b+c+d)} $$ I tried using HM < AM inequality but am ...
2
votes
4answers
76 views

Prove: $\log_2(x)+\log_x(y)+\log_y(8)\geq \sqrt[3]{81}$

Prove that for every $x$,$y$ greater than $1$: $$\log_2(x)+\log_x(y)+\log_y(8)\geq \sqrt[3]{81}$$ What I've tried has got me to: $$\frac{\log_y(x)}{\log_y(2)}+\log_x(y)+3\log_y(2)\geq \sqrt[3]{81}$...
1
vote
1answer
94 views

An inequality with substitution

If $a,b,c$ positive real numbers, then I have to prove $ \frac {1}{18} \sum\limits_{cycl}^{} \frac{a^2}{b^2} + \sum\limits_{cycl}^{} \frac {a}{2a+b+c} \ge \frac {11}{12}$ We have that $\frac {1}{18} \...
3
votes
0answers
51 views

Intermediate inequalities; is there a way to know if you're getting a “bad deal”?

Let's say I want to prove a contest-style inequality $f(a, b, c) + g(a, b, c) \ge h(a, b, c)$ in some $S \subseteq \mathbb{R}^3$. Suppose I want to make the RHS simpler by applying the AM-GM ...
2
votes
2answers
64 views

Using Method of Undetermined coefficients to find maximum value

Find the maximum value of ($1$+$x$)$^5$($1$$-$$x$)($1$$-$$2$$x$)$^2$ given that $1/2$$<$$x$$<$$1$. The solution states the follows Consider the maximum value of ($A$($1$$+$$x$))$^5$ $*$ (...
2
votes
4answers
69 views

For $x\geq 0$, what is the smallest value of $\frac{4x^2+8x+13}{6(x+1)}$?

I know that I have to use the AM-GM inequality. I tried separating the fraction: $$\frac{4x^2+2x+7}{6(x+1)} + \frac{6(x+1)}{6(x+1)}$$ However, it doesn't seem to make either side of the inequality ...
0
votes
1answer
71 views

Showing that $x+y+z \le 2\left(\dfrac {x^2}{y+z}+ \dfrac {y^2}{x+z}+ \dfrac {z^2}{x+y}\right)$ for positive $x, y, z$?

Please, even more than the solution I would like to understand how get better at solving inequalities. Currently my only method is to just blindly try different manipulations to see if they work. ...