Questions tagged [tangent-line-method]
For proofs inequalities by Tangent Line method.
132
questions
0
votes
1
answer
128
views
Show that $\frac{1-xy-x}{x+y+3} + \frac{1-zy-y}{z+y+3}+ \frac{1-xz-z}{x+z+3} \geq \frac{5}{11}$
The problem
a) Show that $\frac{ab}{a+b}+ \frac{cd}{c+d} \leq \frac{(a+c)(b+d)}{a+b+c+d}$, with equality for $ad=bc$ (solved already)
b) Let the real numbers $x,y,z \in (0, \infty)$ with $x+y+z=1$. ...
- 752
3
votes
2
answers
172
views
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, ...
user1009942
2
votes
2
answers
82
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:
$$\...
- 783
0
votes
3
answers
81
views
How to prove $\frac{a}{a^2+3}+\frac{b}{b^2+3}+\frac{c}{c^2+3}\le \frac{ab+bc+ca+3}{8}$ when $a+b+c=3$
Let $a,b,c\ge 0: a+b+c=3.$ Prove that $$\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-...
0
votes
1
answer
91
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^...
user1034536
4
votes
4
answers
258
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^...
- 319
8
votes
3
answers
284
views
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 ...
- 40.3k
5
votes
4
answers
258
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 ...
- 709
5
votes
3
answers
122
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) $$(a^2+2)(b^2+2)(c^2+2)\ge 9(...
- 81
0
votes
1
answer
164
views
Inequality with $\sum a^5+8\sum ab$
For every positive real numbers $a,b,c$ for which $a+b+c=3$ we have:
$$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 ...
- 501
4
votes
3
answers
160
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
$$\frac{1}{a^2+3}+\frac{1}{b^2+3}+\frac{1}{c^2+3}\leq\frac{27}{28}...
- 2,402
0
votes
1
answer
71
views
Is this Factorization?
I'm doubtful about the some parts of the solution to this question:
Suppose that the real numbers $a, b, c > 1$ satisfy the condition $$ {1\over a^2-1}+{1\over b^2-1}+{1\over c^2-1}=1 $$ Prove ...
- 1,516
2
votes
7
answers
273
views
Proving $(1+a^2)(1+b^2)(1+c^2)\geq8 $
I tried this question in two ways-
Suppose a, b, c are three positive real numbers verifying $ab+bc+ca = 3$. Prove that $$ (1+a^2)(1+b^2)(1+c^2)\geq8 $$
Approach 1:
$$\prod_{cyc} {(1+a^2)}= \left({...
- 1,516
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
$$ {a^{k+1}\...
- 1,516
4
votes
3
answers
295
views
Inequality with a High Degree Constraint
This question-
Suppose that $x, y, z$ are positive real numbers and $x^5 + y^5 + z^5 = 3$. Prove that $$ {x^4\over y^3}+{y^4\over z^3}+{z^4\over x^3} \ge 3 $$
The inequality has a high degree ...
- 1,516