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Questions tagged [tangent-line-method]

For proofs inequalities by Tangent Line method.

7
votes
1answer
145 views

Proof of an interesting inequality

I think this question was asked here before, but I am unable to find it at the moment. Apologies if this is due to my ineptitude. Anyway, the question is as follows: let $n>1$ be an integer number ...
3
votes
5answers
128 views

Prove that $(a - 1)^3 + (b - 1)^3 + (c - 1)^3 \ge -\dfrac{3}{4}$ where $a+b+c=3$

If $a$, $b$, $c$ are non-negative numbers such that $a + b + c = 3$ then prove $$(a - 1)^3 + (b - 1)^3 + (c - 1)^3 \ge -\frac{3}{4}$$ Here's what I did. Let $c \ge a \ge b$. We have that \begin{...
1
vote
1answer
120 views

Prove $\sum \sqrt{\frac{a^2}{6a^2+5ab+b^2}}\le \frac{\sqrt{3}}{2}$

Let $a,b,c\in R^+$ prove that the inequality $$\sqrt{\frac{a^2}{6a^2+5ab+b^2}}+\sqrt{\frac{b^2}{6b^2+5bc+c^2}}+\sqrt{\frac{c^2}{6c^2+5ca+a^2}}\le \frac{\sqrt{3}}{2}$$ My try:$$\sum\limits_{cyc} \sqrt{...
1
vote
2answers
123 views

$\frac{a}{a^a+1}+\frac{b}{b^b+1}+\frac{c}{c^c+1}\leq \frac{3}{2}$ with $abc=1$

Let $a,b,c>0$ such that $abc=1$ then we have : $$\frac{a}{a^a+1}+\frac{b}{b^b+1}+\frac{c}{c^c+1}\leq \frac{3}{2}$$ My try : The original inequality is equivalent to : $$a(b^b+1)(c^c+1)+b(a^a+1)(...
1
vote
2answers
54 views

3-variable symmetric inequality

Given $a,b,c>0$ satisfying $a^2+b^2+c^2=3$. Prove that $$2\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)+3(a+b+c)\geq 15.$$ I've tried to use the inequality $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}...
2
votes
2answers
66 views

Find maximum value by using AM-GM inequality

I have a problem: Find the maximum value of $P=\frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}$ with $x,y,z>0$. Is there anyway to ...
2
votes
5answers
80 views

Does $\sum\limits_{i=1}^n x_i = 1$ imply $\sum\limits_{i=1}^n x_i^2 \geq \frac{1}{n}$?

Suppose we have real numbers $x_1, ..., x_n$ which satisfy $x_1 + ... + x_n = 1$. Do we have the lower bound $x_1^2 + ... + x_n^2 \geq \frac{1}{n}$? It seems intuitive that we can minimize this by ...
4
votes
3answers
107 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}...
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 $\...
7
votes
4answers
1k views

Proving a three variables inequality

Given that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$ show that:$$(a+1)(b+1)(c+1)\ge 64$$ My attempt: First I tried expanding the LHS getting that$$abc+ab+bc+ca+a+b+c \ge 63$$ I applied Cauchy-Schwarz on ...
0
votes
1answer
27 views

Transform parabola to be tangent to line in point and through other point

Sorry for stupid question, but I give up. I've spent whole weekend to solve that and no results. Please help. I think I know the solution, but it doesn't work form me. So now I am not sure. I am not ...
4
votes
1answer
39 views

Understanding the proof of an inequality

Basically the method applied is the following, we fix $a=\frac{a_1+a_2+...+a_n}{n}$, if: $$f(x)\ge f(a)+f'(a)(x-a) $$This inequality holds for all x, then summing up the inequality will give us the ...
0
votes
1answer
118 views

Inequalities - Tangent line trick

I will first state the "trick": we fix $a=\frac{a_1+a_2+...+a_n}{n}, \ $If $f$ is not convex we can sometimes prove:$$f(x)\ge f(a)+f'(a)(x-a) $$ If this manages to hold for all x, then summing up ...
2
votes
2answers
60 views

Move parabola to make it tangent to lines

could anyone help me with something very easy (I hope). I have two lines. For example: $y = \frac { x + 4.5 } { 6 }$ (which is green line on the graph) and $y = 1.2 x$ (which is red line on the ...
0
votes
4answers
48 views

Prove using squared number property

$$ If \sum_{i=1}^{10} x_i=10 $$ Prove that $$ \sum_{i=1}^{10} x_i^2\ge 10 $$
0
votes
0answers
48 views

How to manipulate the following inequality [duplicate]

there was this solved question i was doing $a,b,c$ are positive real numbers with sum $3$ . prove that $\sqrt{a}+\sqrt{b}+\sqrt{c} \geq ab +bc+ca $ in the solution the author ,in first step ,said ...
1
vote
1answer
34 views

Geometry proof of an ratio related problem by a random tangent to a pair of fixed parallel tangents on an ellipse

A pair of fixed parallel tangents on 2 points (R, R' respectively) of an ellipse are intercepted by a tangent generated from a random point P on the same ellipse. This random tangent meet parallel ...
7
votes
3answers
230 views

Monotonicity of the function $(1+x)^{\frac{1}{x}}\left(1+\frac{1}{x}\right)^x$.

Let $f(x)=(1+x)^{\frac{1}{x}}\left(1+\frac{1}{x}\right)^x, 0<x\leq 1.$ Prove that $f$ is strictly increasing and $e<f(x)\leq 4.$ In order to study the Monotonicity of $f$, let $$g(x)=\log f(x)=...
10
votes
4answers
265 views

Prove that if $a+b+c+d=4$, then $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq256$

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 = ...
0
votes
2answers
175 views

Prove that: $\sum\limits_{cyc}\frac{1}{(b+c)^2+a^2}\leq \frac{3}{5}$

Given three positive numbers a,b,c satisfying $a+b+c=3$. Show that $\sum\limits_{cyc}\frac{1}{(b+c)^2+a^2}\leq \frac{3}{5}$ Things I have done so far: $$a+b+c=3\Rightarrow b+c=3-a;0<a<3$$ $$\...
6
votes
5answers
86 views

$\frac{a^2} {1+a^2} + \frac{b^2} {1+b^2} + \frac{c^2} {1+c^2} = 2.$ Prove $\frac{a} {1+a^2} + \frac{b} {1+b^2} + \frac{c} {1+c^2} \leq \sqrt{2}.$

$a, b, c ∈ \mathbb{R}+.$ WLOG assume $a \leq b \leq c.$ I tried substitution: $x=\frac{1} {1+a^2}, y=\frac{1} {1+b^2}, z=\frac{1} {1+c^2},$ so $x \geq y \geq z$ and $(1-x)+(1-y)+(1-z)=2 \to x+y+z=1.$ ...
1
vote
2answers
49 views

Proving the following inequality without using AM-GM inequality. [duplicate]

Let $x,y,z \in \Bbb R^+$ such that $x+y+z=3$. Prove the inequality $\sqrt x+\sqrt y+\sqrt z \ge xy+yz+zx$ I tried to prove that $\sqrt x+\sqrt y+\sqrt z-(xy+yz+zx)\ge 0$ I squared the equality, ...
0
votes
1answer
91 views

Prove: $2^{a}+ 2^{b}+ 2^{c}\leqq 3$ [closed]

Prove: $2^{a}+ 2^{b}+ 2^{c}\leqq 3$ with $\sqrt{2}\left ( a+ b+ c \right )= \sqrt{a^{2}+ 4}+ \sqrt{b^{2}+ 4}+ \sqrt{c^{2}+ 4}$ By AM_GM, we have: $$\sum_{cyc}\log_2a=\log_2abc\leq\log_2\left(\frac{a+...
0
votes
3answers
60 views

If $a_i>0$ and $\sum_{i=1}^n a_i = 1$, then $\sum_{i=1}^n \frac{1}{a_i} \geq n^2$?

If $a_i>0$ and $\sum_{i=1}^n a_i = 1$, is $\sum_{i=1}^n \frac{1}{a_i} \geq n^2$? I'm doing an inequality exercise. If I can confirm that's true, then my proof is done. I wrote down some examples ...
1
vote
6answers
586 views

A better way to prove this inequality [duplicate]

Exercise 1.1.6. (b) For positive real numbers $a_1, a_2, ... , a_n$ prove that $$(a_1+a_2+ \ldots +a_n)\Big(\frac{1}{a_1}+\frac{1}{a_2}+ \ldots +\frac{1}{a_n}\Big) \geq n^2.$$ From $AM \geq GM$: $$...
0
votes
2answers
73 views

Let $\sum\frac{1}{a^3+1}=2$. Prove that $\sum\frac{1-a}{a^2-a+1}\ge 0$

Let $a,b,c$ are nonnegative real numbers such that $\frac1{a^3+1}+\frac1{b^3+1}+\frac1{c^3+1}+\frac1{d^3+1}=2$. Prove the inequality $$\frac{1-a}{a^2-a+1}+\frac{1-b}{b^2-b+1}+\frac{1-c}{c^2-c+...
1
vote
2answers
81 views

Prove the inequality $\left(1+\frac{1}{a_1(1+a_1)}\right)…\left(1+\frac{1}{a_k(1+a_k)}\right)\ge\left(1+\frac{1}{p(1+p)}\right)^k$

Let $a_1, a_2,...,a_k$ are any positive real numbers. Prove the inequality $$\left(1+\frac{1}{a_1(1+a_1)}\right)\left(1+\frac{1}{a_2(1+a_2)}\right)...\left(1+\frac{1}{a_k(1+a_k)}\right)\ge$$ $$\ge\...
2
votes
3answers
115 views

Prove the inequality $\frac{a}{1+a^2}+\frac{b}{1+b^2}+\frac{c}{1+c^2}\le\frac{3\sqrt3}{4}$ [duplicate]

Let $a,b,c$ are nonnegative real numbers such that $a^2+b^2+c^2=1$. Prove the inequality $$\frac{a}{1+a^2}+\frac{b}{1+b^2}+\frac{c}{1+c^2}\le\frac{3\sqrt3}{4}$$ I tried the method of Lagrange ...
2
votes
4answers
372 views

Proving that $\frac {1}{3x^2+1}+\frac {1}{3y^2+1}+\frac {1}{3z^2+1}\geq \frac {3}{16 } $

Let $x,y,z\geq 1$ and $x+y+z=6$. Then $$\frac {1}{3x^2+1}+\frac {1}{3y^2+1}+\frac {1}{3z^2+1}\geq \frac {3}{16 }. $$ I tried to use Cauchy- Schwartz inequality but it doesn't work.
2
votes
1answer
95 views

Show the following inequality.

Let $a,b,c \in \mathbb R^+$ and $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} =1$$ Show that $$(a^2 -3a +3)(b^2-3b+3)(c^2-3c+3) \ge 27$$ I tried using using the AM-GM inequality and some algebraic ...
0
votes
5answers
110 views

$\frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{x+y} \ge \frac{1}{2}(x+y+z)$ [duplicate]

let $x,y,z$ be positive real numbers $$\frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{x+y} \ge \frac{1}{2}(x+y+z)$$ then how to prove above by Cauchy Schwarz -Inquality
0
votes
4answers
50 views

Prove an inequality $\frac{x^2}{1+y}+\frac{y^2}{1+x}\ge 1$ where $x,y \ge 0$ and $x+y=2$.

I was trying to solve the following: $$ \frac{x^2}{1+y}+\frac{y^2}{1+x}\ge 1$$ $$x^2(1+x)+y^2(1+y)\ge (1+y)(1+x)$$ $$x^3+x^2+y^3+y^2 \ge 1+y+x+xy$$ $$(x+y)(x^2-xy+y^2)+x^2+y^2 \ge 1+y+x+xy$$ ...
2
votes
1answer
72 views

Find max value of $a$

Let $a, x, y, z \in \mathbb{R}_{>0}$ such that $xyz=1$. Find maximum value of $a$ such that is satisfied the inequality $$ \sum_{cyc} \frac{x}{(x+1)(y+a)} \ge \frac{3}{2(1+a)} $$ After doing some ...
2
votes
3answers
122 views

Find the minimum of expression: $\frac{2-x}{3+x}+\frac{2-y}{3+y}+\frac{2-z}{3+z}$

If $x+y+z=1$ and $x,y,z$ are positive numbers, Find the minimum of expression: $$\frac{2-x}{3+x}+\frac{2-y}{3+y}+\frac{2-z}{3+z}$$ My solution: $$\left[\frac{2-x}{3+x}+\frac{2-y}{3+y}+\frac{2-z}{3+...
0
votes
2answers
93 views

How to prove $|\sum_{i=1}^n a_i|\le \sqrt{n} \sqrt{\sum_{i=1}^n a_i^2}$

Let $n$ be a natural number and $a_1,a_2,\ldots,a_n$ are real numbers. Then prove that $|\sum_{i=1}^n a_i|\le \sqrt{n\cdot \sum_{i=1}^n a_i^2}$. At first I tried to prove for $n=2$ i.e, $|a_1+a_2|\...
0
votes
2answers
104 views

Let $a_1, a_2,\ldots, a_{100}$ be non-zero real numbers such that$a_1+ a_2+\cdots+ a_{100}=0$. Then $\sum_{i=1}^{100} a_i 2^{a_i}$

Let $a_1, a_2,\ldots, a_{100}$ be non-zero real numbers such that $$a_1 + a_2 + \cdots + a_{100} = 0$$ Then A - $\sum_{i=1}^{100} a_i 2^{a_i}\ge0$ and $\sum_{i=1}^{100} a_i 2^{-a_i}\ge0$ B- $\...
3
votes
6answers
156 views

How to prove $(a_1+a_2+\dots a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_n}\right)\ge n^2$? [duplicate]

let $a_1,a_2,\dots ,a_n$ be a positive real numbers . prove that $$(a_1+a_2+\dots a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_n}\right)\ge n^2$$ how to prove this inequality ..is this ...
1
vote
2answers
102 views

Inequality on tangent and secant function

Let $\{\alpha, \beta, \gamma, \delta\} \subset \left (0,\frac {\pi}{2}\right)$ and $ \alpha + \beta+\gamma+\delta = {\pi}$. Prove that $\sqrt {2} \left(\tan \alpha +\tan \beta +\tan\gamma+\tan \delta\...
0
votes
1answer
85 views

Inequality : $\sum_{cyc}\left(\frac{a^3+1}{a^2+1}\right)^4 \geq \frac{1}{27}\left(\sum_{cyc}\sqrt{ab}\right)^4$

Let $\{a, b, c\} \subset \mathbb{R}^+$, where $a+b+c=3$. Prove that: $$\left(\frac{a^3+1}{a^2+1}\right)^4 + \left(\frac{b^3+1}{b^2+1}\right)^4 + \left(\frac{c^3+1}{c^2+1}\right)^4 \geq \frac{1}{27}\...
2
votes
2answers
99 views

Chebyshev Inequality toughnut [closed]

Let $a^2 + b^2 + c^2 + d^2 = 1$, where $(a,b,c,d \geq 0)$. Prove that: $$ \frac{a^{2}}{b+c+d}+\frac{b^{2}}{a+c+d}+\frac{c^{2}}{b+a+d}+\frac{d^2}{b+c+a} \geq \frac{2}{3}$$
5
votes
2answers
166 views

$ \sin^2x_1+\dots \sin^2x_{10}=1$ implies $ 3(\sin x_1+\dots \sin x_{10})\leq \cos x_1 +\dots +\cos x_{10}. $ [duplicate]

Suppose that $x_1,.\dots x_{10}\in[0,\frac{\pi}{2}]$ and that $$ \sin^2x_1+\dots \sin^2x_{10}=1. $$ Prove that $$ 3(\sin x_1+\dots \sin x_{10})\leq \cos x_1 +\dots +\cos x_{10}. $$
0
votes
1answer
60 views

Prove the inequality

Thesis: $$\frac{a}{a^2+1}+\frac{b}{b^2+1}+\frac{c}{c^2+1}\leq 9/10$$ Assumptions: $$a+b+c=1$$ $$ a,b,c \geq -\frac{3}{4}$$ Can someone give me a hint? I suppose there is a tricky way, not using ...
2
votes
2answers
94 views

How prove this inequality $\sum_{cyc}\sqrt{\frac{2b+2c}{a}-1}\ge 3\sqrt{3}$

Let $a,b,c>0 ,2b+2c-a\ge 0,2c+2a-b\ge 0,2a+2b-c\ge 0$ show that $$\sqrt{\dfrac{2b+2c}{a}-1}+\sqrt{\dfrac{2c+2a}{b}-1}+\sqrt{\dfrac{2a+2b}{c}-1}\ge 3\sqrt{3}$$ I try use AM-GM and Cauchy-Schwarz ...
0
votes
2answers
81 views

Prove that $\sum_{cyc}\frac{x}{y^2+z^2}\ge\frac{3\sqrt3}2$

Given $x,y,z$ are positive number satisfy $x^2+y^2+z^2=1$. Prove that $$\frac{x}{y^2+z^2}+\frac{y}{z^2+x^2}+\frac{z}{x^2+y^2}\ge \frac{3\sqrt{3}}{2}$$ I need a way use reduction of many fractions to ...
0
votes
3answers
57 views

Find maximize $A=\sum_{cyc}\sqrt{1+x^2}+2(\sum_{cyc}\sqrt{x})$

Given $x,y,z$ are positive number satisfy $x+y+z\le 3$. Find the value of maximize $$\sqrt{1+x^{2}}+\sqrt{1+y^{2}}+\sqrt{1+z^{2}}+2(\sqrt{x}+\sqrt{y}+\sqrt{z})$$ $\sum (\sqrt{1+x^{2}}+\sqrt{2x})\leq \...
0
votes
3answers
210 views

$a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$

$a, b,c $ are positive real numbers such that $a+b+c = 3$, prove that :$a\sqrt{a+3}+b\sqrt{b+3}+c\sqrt{c+3} \geq 6$ Any ideas ?
1
vote
1answer
106 views

Find the maximum value $\frac{11a+4b}{4a^2-ab+2b^2}+\frac{11b+4c}{4b^2-bc+2c^2}+\frac{11c+4a}{4c^2-ca+2a^2}$

For the positive real numbers $a,b,c$ satisfy $ab+bc+ca=3abc$. Find the maximum value $$P=\frac{11a+4b}{4a^2-ab+2b^2}+\frac{11b+4c}{4b^2-bc+2c^2}+\frac{11c+4a}{4c^2-ca+2a^2}$$ i tried all methods ...
3
votes
1answer
77 views

Prove that $\frac{(2a+b+c)^2}{2a^2+(b+c)^2}+\frac{(2b+c+a)^2}{2b^2+(c+a)^2}+\frac{(2c+a+b)^2}{2c^2+(a+b)^2} \le 8$

Prove that $$\frac{(2a+b+c)^2}{2a^2+(b+c)^2}+\frac{(2b+c+a)^2}{2b^2+(c+a)^2}+\frac{(2c+a+b)}{2c^2+(a+b)^2} \le 8$$. MY ATTEMPT:I want to make a relation between $a,b,c$. By trial I found that if we ...
2
votes
3answers
163 views

Prove that $\sum\limits_{cyc} a^7 \geq \sum\limits_{cyc}a^4b^3$ [duplicate]

Prove that $a^7+b^7+c^7\ge a^4b^3+b^4c^3+c^4a^3$ SOURCE : "A Brief Introduction to Olympiad Inequalities" by Evan Chen It was one of the practice problems. Equality case is easy. I tried AM-GM ...
5
votes
7answers
252 views

Proof that $\frac{a}{a+3b+3c}+\frac{b}{b+3a+3c}+\frac{c}{c+3a+3b} \ge \frac{3}{7}$ for all $a,b,c > 0$

So I am trtying to proof that $\frac{a}{a+3b+3c}+\frac{b}{b+3a+3c}+\frac{c}{c+3a+3b} \ge \frac{3}{7}$ for all $a,b,c > 0$. First I tried with Cauchy–Schwarz inequality but got nowhere. Now I am ...