Questions tagged [tangent-line-method]

For proofs inequalities by Tangent Line method.

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Inequality $\sum_{k=1}^{n} \frac{\log(a_k)}{1+a_{k}^{2}} \leqslant 0$

Let $a_1,a_2,...,a_n$ be positive real numbers such that $a_1 \cdot a_2 \cdot ... \cdot a_n=1$. Prove that $\sum_{k=1}^{n} \frac{\log(a_k)}{1+a_{k}^{2}} \leqslant 0$. I tried using Jensen inequality, ...
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2 votes
2 answers
77 views

Max $P= \frac{2ab+3b^2}{(a+3b)^2}+\frac{2bc+3c^2}{(b+3c)^2}+\frac{2ca+a^2}{(c+3a)^2}$

Let: $a,b,c>0$. Find the maximum value of: $$P= \frac{2ab+3b^2}{(a+3b)^2}+\frac{2bc+3c^2}{(b+3c)^2}+\frac{2ca+a^2}{(c+3a)^2}$$ Here are my try: I tried to use tangent line trick, then I got: $$\...
Lục Trường Phát's user avatar
0 votes
3 answers
74 views

How to prove $\frac{a}{a^2+3}+\frac{b}{b^2+3}+\frac{c}{c^2+3}\le \frac{ab+bc+ca+3}{8}$ when $a+b+c=3$

Let $a,b,c\ge 0: a+b+c=3.$ Prove that $$\frac{a}{a^2+3}+\frac{b}{b^2+3}+\frac{c}{c^2+3}\le \frac{ab+bc+ca+3}{8}$$ I'm looking for a smooth proof by using classical inequalities as AM-GM, Cauchy-...
Dragon boy's user avatar
0 votes
1 answer
90 views

Prove $\sum_{\mathrm{cyc}}\sqrt{\frac{a^3+a^2+1}{a^2+a+1}}\ge3$ for $abc=1$ [closed]

Let $a$, $b$, $c\ge0$, $abc=1$, prove that \[\sqrt{\frac{a^3+a^2+1}{a^2+a+1}}+\sqrt{\frac{b^3+b^2+1}{b^2+b+1}}+\sqrt{\frac{c^3+c^2+1}{c^2+c+1}}\ge3.\] The following inequality fails: \[\sqrt{\frac{a^...
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4 votes
4 answers
255 views

Inequality $\frac{a^3}{3ab^2+2c^3} +\frac{b^3}{3bc^2+2a^3} +\frac{c^3}{3ca^2+2b^3} \geq \frac{3}{5} $

I have trouble with solving this inequality: Prove $\frac{a^3}{3ab^2+2c^3} +\frac{b^3}{3bc^2+2a^3} +\frac{c^3}{3ca^2+2b^3} \geq \frac{3}{5}$ for a,b,c>0. Using Cauchy-Schwartz I got this: $\frac{a^...
yslpaul's user avatar
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8 votes
3 answers
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Prove or disprove $\sum\limits_{1\le i < j \le n} \frac{x_ix_j}{1-x_i-x_j} \le \frac18$ for $\sum\limits_{i=1}^n x_i = \frac12$($x_i\ge 0, \forall i$)

Problem 1: Let $x_i \ge 0, \, i=1, 2, \cdots, n$ with $\sum_{i=1}^n x_i = \frac12$. Prove or disprove that $$\sum_{1\le i < j \le n} \frac{x_ix_j}{1-x_i-x_j} \le \frac18.$$ This is related to the ...
River Li's user avatar
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5 votes
4 answers
252 views

Using Rearrangement Inequality .

Let $a,b,c\in\mathbf R^+$, such that $a+b+c=3$. Prove that $$\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\ge\frac{a^2+b^2+c^2}{2}$$ $Hint$ : Use Rearrangement Inequality My Work :-$\\$ Without ...
arnav_de's user avatar
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5 votes
3 answers
120 views

Prove that: $(a^5-2a+4)(b^5-2b+4)(c^5-2c+4)\ge9(ab+bc+ca)$

Let $a,b,c>0$ satisfy $a^2+b^2+c^2=3$ . Prove that: $$(a^5-2a+4)(b^5-2b+4)(c^5-2c+4)\ge9(ab+bc+ca)$$ My idea is to use a well-known inequality (We can prove by Schur) $$(a^2+2)(b^2+2)(c^2+2)\ge 9(...
SUWG's user avatar
  • 81
0 votes
1 answer
161 views

Inequality with $\sum a^5+8\sum ab$

For every positive real numbers $a,b,c$ for which $a+b+c=3$ we have: $$a^5+b^5+c^5+8(ab+bc+ca)\ge 27.$$ My ideas is: We can apply Chebyshev inequality $$\sum a^5 =\sum a\cdot a^4\ge \frac13 \left(\sum ...
Tashi's user avatar
  • 501
4 votes
3 answers
156 views

cyclic rational inequalities $\frac{1}{a^2+3}+\frac{1}{b^2+3}+\frac{1}{c^2+3}\leq\frac{27}{28}$ when $a+b+c=1$

I've been practicing for high school olympiads and I see a lot of problems set up like this: let $a,b,c>0$ and $a+b+c=1$. Show that $$\frac{1}{a^2+3}+\frac{1}{b^2+3}+\frac{1}{c^2+3}\leq\frac{27}{28}...
Snacc's user avatar
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0 votes
1 answer
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Is this Factorization?

I'm doubtful about the some parts of the solution to this question: Suppose that the real numbers $a, b, c > 1$ satisfy the condition $$ {1\over a^2-1}+{1\over b^2-1}+{1\over c^2-1}=1 $$ Prove ...
Book Of Flames's user avatar
2 votes
7 answers
256 views

Proving $(1+a^2)(1+b^2)(1+c^2)\geq8 $

I tried this question in two ways- Suppose a, b, c are three positive real numbers verifying $ab+bc+ca = 3$. Prove that $$ (1+a^2)(1+b^2)(1+c^2)\geq8 $$ Approach 1: $$\prod_{cyc} {(1+a^2)}= \left({...
Book Of Flames's user avatar
2 votes
1 answer
81 views

Dubious proof of an Inequality

This question was asked to be proved by Hölder's inequality- Let $a, b, c$ be positive real numbers. Prove that for all natural numbers $k$, $(k \ge 1)$, the following inequality holds $$ {a^{k+1}\...
Book Of Flames's user avatar
4 votes
3 answers
288 views

Inequality with a High Degree Constraint

This question- Suppose that $x, y, z$ are positive real numbers and $x^5 + y^5 + z^5 = 3$. Prove that $$ {x^4\over y^3}+{y^4\over z^3}+{z^4\over x^3} \ge 3 $$ The inequality has a high degree ...
Book Of Flames's user avatar
6 votes
3 answers
194 views

A more elementary proof that if $x_i>0$ for $1\leq i\leq n$, and $\sum x_i=1$, then $(x_1+\frac{1}{x_1})\cdots(x_n+\frac{1}{x_n})\geq(n+\frac1n)^n$

For $x_i>0$, $1\leq i\leq n$ and $\sum_i x_i=1$, show that $$\left(x_1+\frac{1}{x_1}\right)\cdots \left(x_n+\frac{1}{x_n}\right)\geq \left(n+\frac{1}{n}\right)^n$$ I think this can be proved easily ...
maomao's user avatar
  • 1,201
2 votes
3 answers
847 views

If $a$, $b$, $c$, $d$ are positive reals so $(a+c)(b+d) = 1$, prove the following inequality would be greater than or equal to $\frac {1}{3}$.

Let $a$, $b$, $c$, $d$ be real positive reals with $(a+c)(b+d) = 1$. Prove that $\frac {a^3}{b + c + d} + \frac {b^3}{a + c + d} + \frac {c^3}{a + b + d} + \frac {d^3}{a + b + c} \geq \frac {1}{3}$. ...
Boris Poris's user avatar
2 votes
2 answers
141 views

Show that $(a^3+a+1)(b^3+b+1)(c^3+c+1)\le 27$

Let $a,b,c\ge 0$ be such that $a^2+b^2+c^2=3$. Show that $$(a^3+a+1)(b^3+b+1)(c^3+c+1)\le 27$$ I want to consider the function $$f(x)=\ln{(x^{3/2}+x^{1/2}+1)}$$ Maybe it isn't the case $f''(x)\le 0$, ...
math110's user avatar
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4 votes
1 answer
98 views

Prove the inequality $\sum_{cyc}\frac{a^3}{b\sqrt{a^3+8}}\ge 1$

Let $a,b,c>0$ and such $a+b+c=3$,show that $$\sum_{cyc}\dfrac{a^3}{b\sqrt{a^3+8}}\ge 1\tag{1}$$ I tried using Holder's inequality to solve it: $$\sum_{cyc}\dfrac{a^3}{b\sqrt{a^3+8}}\sum b\sum \...
math110's user avatar
  • 93.4k
4 votes
3 answers
179 views

If $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\le 1$, prove that $(1+a^2)(1+b^2)(1+c^2)\ge 125$.

QUESTION: Let $a,b,c$ be positive real numbers such that $$\cfrac{1}{1+a}+\cfrac{1}{1+b}+\cfrac{1}{1+c}\le 1$$ Prove that $$(1+a^2)(1+b^2)(1+c^2)\ge 125$$ When does equality hold? MY APPROACH: ...
Stranger Forever's user avatar
3 votes
2 answers
231 views

AM/GM inequalities

I need some help to prove this inequality... I guess one can use Jensen's then AM/GM inequalities. Let $x_1, x_2, x_3, x_4$ be non- negative real numbers such that $x_1 x_2 x_3 x_4 =1$. We want to ...
Clifford's user avatar
  • 179
0 votes
1 answer
223 views

Curve tangent line with unknown gradient

A curve with the equation $x^2-2x+3$ has point b, what is the absis of point b if the tangent line on b passes point (1,1) I solved it by first finding the gradient of the curve which is 2x-2, then I ...
Gary02's user avatar
  • 33
2 votes
1 answer
65 views

Curve tangent line

A curve with the equation $x^2-x+1$ has two tangent lines $a$ and $b$ that intersects at $x=1$, what is $y$? can I determine $y$ when $a$ and $b$ are perpendicular, or the gradient of $a$ and $b$ are $...
Gary02's user avatar
  • 33
4 votes
2 answers
122 views

Given positive real numbers $a$, $b$, $c$, $d$, $e$ with $\sum_{\text{cyc}}\,\frac{1}{4+a}=1$, prove that $\sum_{\text{cyc}}\,\frac{a}{4+a^2}\le1$.

Let $a, b, c, d, e$ be positive real numbers such that $$\dfrac{1}{4+a} + \dfrac{1}{4+b} +\dfrac{1}{4+c} +\dfrac{1}{4+d} +\dfrac{1}{4+e} = 1.$$ Prove that $$\dfrac{a}{4+a^{2}} + \dfrac{b}{4+b^{2}} +\...
Success's user avatar
  • 375
0 votes
3 answers
117 views

Prove that $21(a^2+b^2+c^2)\ge 20 +9(a^3+b^3+c^3)$ [closed]

Let $a,b,c$ be the length of sides of triangle such that $a+b+c=2$. Prove that $$21(a^2+b^2+c^2)\ge 20 +9(a^3+b^3+c^3)$$ It was in my exam. It can be solved easy by BW but it takes alot of time to ...
user774564's user avatar
2 votes
4 answers
85 views

If $abc=1$, then how do you prove $\frac{b-1}{bc+1}+\frac{c-1}{ac+1}+\frac{a-1}{ab+1} \geq 0$?

If $abc=1$, then how do you prove $\frac{b-1}{bc+1}+\frac{c-1}{ac+1}+\frac{a-1}{ab+1} \geq 0$? I tried substitution on the bottom (for example $\frac{b-1}{\frac{1}{a}+1}$), but I then a very similar ...
user avatar
2 votes
3 answers
100 views

$\frac{3x+1}{x+1}+\frac{3y+1}{y+1}+\frac{3z+1}{z+1} \le \frac{9}{2}$

I'm having trouble proving that for any $x,y,z>0$ such that $x+y+z=1$ the following inequality is true: $\frac{3x+1}{x+1}+\frac{3y+1}{y+1}+\frac{3z+1}{z+1} \le \frac{9}{2}$ It seems to me that ...
EngineerInProgress's user avatar
1 vote
3 answers
186 views

Maximize $\sum\limits_{k =1}^n x_k (1 - x_k)^2$

Given problem for maximizing \begin{align} &\sum_{k =1}^n x_k (1 - x_k)^2\rightarrow \max\\ &\sum_{k =1}^n x_k = 1,\\ &x_k \ge 0, \; \forall k \in 1:n. \end{align} My attempt: first of ...
taciturno's user avatar
  • 480
1 vote
1 answer
92 views

Inequality $\sum_{cyc}\frac{a}{2a^2+a+1}\leq \frac{3}{4}$

Let $a,b,c\in\mathbb{R^+}$ such that $a+b+c=3$. Then prove that $$\sum_{cyc}\frac{a}{2a^2+a+1}\leq \frac{3}{4}$$ I tried to use tangent line method. Let $$f(x)=\frac{x}{2x^2+x+1}$$ Then $$f'(x)=\frac{...
Mutse's user avatar
  • 671
0 votes
1 answer
178 views

Question regarding Jensen Inequality

Following is the picture of the question regarding the application of Jensen Inequality. Following is the picture my approach to proove the inequality. Can anyone please check if my proof is ...
user 493905's user avatar
1 vote
2 answers
110 views

Prove this inequality with $xyz=1$

let $x,y,z>0$ and such $xyz=1$,show that $$f(x)+f(y)+f(z)\le\dfrac{1}{8}$$ where $f(x)=\dfrac{x}{2x^{x+1}+11x^2+10x+1}$ I try use this $2x^x\ge x^2+1$,so we have $$2x^{x+1}+11x^2+10x+1\ge x^3+11x^...
math110's user avatar
  • 93.4k
6 votes
3 answers
133 views

Prove that $ a^2+b^2+c^2 \le a^3 +b^3 +c^3 $

If $ a,b,c $ are three positive real numbers and $ abc=1 $ then prove that $a^2+b^2+c^2 \le a^3 +b^3 +c^3 $ I got $a^2+b^2+c^2\ge 3$ which can be proved $ a^2 +b^2+c^2\ge a+b+c $. From here how can I ...
Chris's user avatar
  • 748
4 votes
3 answers
129 views

Given $a, b, c>0$, prove $\frac{a^4}{a+b}+\frac{b^4}{b+c}+\frac{c^4}{c+a}\geq \frac{1}{2}(a^{2}c+b^{2}a+c^{2}b)$

Given $a,b,c>0$, prove that $$\frac{a^4}{a+b}+\frac{b^4}{b+c}+\frac{c^4}{c+a}\geq \frac{1}{2}(a^{2}c+b^{2}a+c^{2}b).$$ My attempt: I have that $$\frac{a^4}{a+b}+\frac{c^2(a+b)}{4}\geq a^{2}c$$ $$\...
Dave Robin's user avatar
3 votes
6 answers
137 views

Prove $\frac{1+a^2}{1-a^2}+\frac{1+b^2}{1-b^2}+\frac{1+c^2}{1-c^2}\ge \frac{15}{4}$

Let $1>a>0$, $1>b>0$, $1>c>0$ and $a+b+c=1$. Prove that $$ \frac{1+a^2}{1-a^2}+\frac{1+b^2}{1-b^2}+\frac{1+c^2}{1-c^2}\ge \frac{15}{4}. $$ I saw the following solution. Let $x=\frac{...
baranka's user avatar
  • 73
1 vote
3 answers
152 views

For $a,b,c\in\left[\frac{1}{\sqrt{6}}, 6\right]$: $\sum_{cyc}\frac{4}{a+3b}\geq \sum_{cyc}\frac{3}{a+2b}$

For $a,b,c\in\left[\frac{1}{\sqrt{6}}, 6\right]$ prove that $$\frac{4}{a+3b}+\frac{4}{b+3c}+\frac{4}{c+3a}\geq\frac{3}{a+2b}+\frac{3}{b+2c}+\frac{3}{c+2a}.$$ I can't really find a way to exploit the ...
John WK's user avatar
  • 989
4 votes
4 answers
123 views

Cauchy-Schwarz inequality for $a_1^4 + a_2^4 + \cdots + a_n^4 \geqslant n$

Let $a_1+a_2,...,a_n \in \mathbb{R}.$ Show that if $a_1+a_2+...+a_n=n$, then $$a_1^4+a_2^4+...+a_n^4 \geqslant n.$$ The proposed solution for this was the following: Using the Cauchy-Schwarz ...
user avatar
0 votes
1 answer
29 views

How to prove $\sum_{i=1}^{k}x_i^2\ge k$ if $\sum_{i=1}^{k}\frac{1}{x_i}=k$ and $\min (x_i-1)^2$ is sufficiently small?

How to prove $\sum_{i=1}^{k}x_i^2\ge k$ if $\sum_{i=1}^{k}\frac{1}{x_i}=k$ and $\min (x_i-1)^2$ is sufficiently small? In $k=2$ case it is true. $\frac{1}{x}+\frac{1}{y}=2$ implies $y = \frac{x}{2x-...
C.X.Neo's user avatar
  • 21
1 vote
3 answers
174 views

Minimize $\frac{2}{1-a}+\frac{75}{10-b}$

Let $a,b>0$ and satisfy $a^2+\dfrac{b^2}{45}=1$. Find the minimum value of $\dfrac{2}{1-a}+\dfrac{75}{10-b}.$ WA gives the result that $\min\left(\dfrac{2}{1-a}+\dfrac{75}{10-b}\right)=21$ with ...
mengdie1982's user avatar
  • 13.8k
1 vote
3 answers
150 views

How to prove the harmonic-geometric mean inequality by solving an optimization?

The harmonic-geometric mean inequality is defined as follows $$ \frac{n}{\sum_{i=1}^n \frac{1}{x_i}} \leq (\Pi_{i=1}^{n}x_i)^{\frac{1}{n}}\tag{1} $$ Given the following linear programming problem $$ ...
user avatar
3 votes
8 answers
185 views

Given positives $a, b, c$, prove that $\frac{a}{(b + c)^2} + \frac{b}{(c + a)^2} + \frac{c}{(a + b)^2} \ge \frac{9}{4(a + b + c)}$.

Given positives $a, b, c$, prove that $$\large \frac{a}{(b + c)^2} + \frac{b}{(c + a)^2} + \frac{c}{(a + b)^2} \ge \frac{9}{4(a + b + c)}$$ Let $x = \dfrac{b + c}{2}, y = \dfrac{c + a}{2}, z = \dfrac{...
Lê Thành Đạt's user avatar
1 vote
1 answer
102 views

Prove that $\sum_{i=1}^n\frac{\sum_{i'=1}^na_{i'}^p - a_i^p}{\sum_{i'=1}^na_{i'}^q - a_i^q}\le n\cdot\frac{\sum_{i=1}^na_i^p}{\sum_{i=1}^na_i^q}$.

Given positives $$\large a_1, a_2, \cdots, a_{n - 1}, a_n$$ $(n \in \mathbb Z^+, n \ge 3)$. Prove that for all naturals $p$ and $q$ such that $p \ge q$, $$\large \sum_{i = 1}^n\frac{\displaystyle \...
Lê Thành Đạt's user avatar
4 votes
3 answers
169 views

Inequality $\frac{x^3}{x^2+y^2}+\frac{y^3}{y^2+z^2}+\frac{z^3}{z^2+x^2} \geqslant \frac{x+y+z}{2}$

Help to prove this Inequality: If x,y,z are postive real numbers then: $\dfrac{x^3}{x^2+y^2}+\dfrac{y^3}{y^2+z^2}+\dfrac{z^3}{z^2+x^2} \geqslant \dfrac{x+y+z}{2}$ I tied to use analytic method ...
samad's user avatar
  • 49
0 votes
1 answer
73 views

Inequality $\sum_{cyc}a^3\frac{(16a^2-10ab+12b^2)}{(13a^2+5b^2)^2}\geq \frac{a+b+c}{18}$

It's a variant of Inequality $\sum\limits_{cyc}\frac{a^3}{13a^2+5b^2}\geq\frac{a+b+c}{18}$ : Let $a,b,c>0$ then we have : $$\sum_{cyc}a^3\frac{(16a^2-10ab+12b^2)}{(13a^2+5b^2)^2}\geq \frac{a+b+...
Miss and Mister cassoulet char's user avatar
0 votes
1 answer
93 views

Find all positive real solutions of the system

Find all positive real solutions of the system of equations $$\begin{cases} x_1+x_2+...+x_{1994}=1994 \\ x_1^4+x_2^4+...+x_{1994}^4=x_1^3+x_2^3+....+x_{1994}^3 \end{cases}$$ ''By Hölder, we have in ...
Lambert macuse's user avatar
7 votes
8 answers
855 views

Prove that the minimum values of $x^2+y^2+z^2$ is $27$ with given condition $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1$.

Question: Prove that the minimum values of $x^2+y^2+z^2$ is $27$, where $x,y,z$ are positive real variables satisfying the condition $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1$. From AM$\ge$ GM, we ...
Primo Raj's user avatar
  • 111
4 votes
4 answers
175 views

Proving $\sum_{cyc} \sqrt{a^2+ab+b^2}\geq\sqrt 3$ when $a+b+c=3$

Good evening everyone, I want to prove the following: Let $a,b,c>0$ be real numbers such that $a+b+c=3$. Then $$\sqrt{a^2+ab+b^2}+\sqrt{b^2+bc+c^2}+\sqrt{c^2+ca+a^2}\geq 3\sqrt 3.$$ My attempt: I ...
ArtOfProblemSolving's user avatar
2 votes
1 answer
104 views

Inequality with 5 cyclic variables

For postive real numbers $a$,$b$,$c$,$d$ and $e$, prove that $$\frac{4a^3}{a^2+2b^2+\frac{2b^3}{a}} + \frac{4b^3}{b^2+2c^2+\frac{2c^3}{b}} + \frac{4c^3}{c^2+2d^2+\frac{2d^3}{c}}+ \frac{4d^3}{d^2+2e^2+...
trombho's user avatar
  • 1,591
2 votes
1 answer
80 views

inequality under condition $x+y+z=3$

$x$, $y$ and $z$ being three positive real numbers such that $x+y+z=3$ It is asked to prove that $$ \dfrac{\sqrt x}{y + z}+ \dfrac{\sqrt y}{x + z} + \dfrac{\sqrt z}{y + x} \ge \dfrac 3 2$$ I tried ...
ahmed's user avatar
  • 1,273
4 votes
2 answers
202 views

Prove that $\sum \frac{x}{x^2+7}\le \frac{3}{8}$

Let $x,y,z>0$ such that $xy+yz+xz=3$. Show that $$\frac{x}{x^2+7}+\frac{y}{y^2+7}+\frac{z}{z^2+7}\le \frac{3}{8}$$ We have: $$x+y+z\ge \sqrt{3\left(xy+yz+xz\right)}=3\rightarrow \frac{3}{8\left(x+...
DVdivi's user avatar
  • 413
0 votes
1 answer
115 views

Chebyshev's sum inequality

Given $a,b,c,d>0$ satisfying $a+b+c+d=4$. Prove that $$\dfrac{1}{8+a^2}+\dfrac{1}{8+b^2}+\dfrac{1}{8+c^2}+\dfrac{1}{8+d^2}\leq \dfrac{4}{9}.$$ I've tried solving by assuming that $0<a\leq b\...
Steven Tran's user avatar
1 vote
2 answers
73 views

Stuck on this cyclic 3-variables-inequality with constraint

Problem Statement: For $a,b,c>0$ and $a+b+c=3$, I'd want to prove that $$\frac{1}{a^2+b+c}+\frac{1}{b^2+a+c} +\frac {1}{c^2+a+b} \leq1 .$$ I am a beginner when it comes to inequalities. This ...
Quadro's user avatar
  • 335