# Tagged Questions

53 views

### A tough inequality problem with condition $a+b+c+abc=4$

If, $a+b+c+abc=4$, with $a,b,c$ being positive reals, then prove or disprove the following inequality: $$\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{a+c}}+\frac{c}{\sqrt{a+b}}\geq\frac{a+b+c}{\sqrt2}$$ I ...
34 views

### Solve a system of inequalities

$$\begin{cases} \log_{2}^{2}(-\log_{2}x) + \log_{2}\log_{2}^{2}x \leq 3 & \\-4 |x^2-1|-3\geq \frac{1}{x^2-1}& \end{cases}$$ What I've tried: Make substitution $t=x^2-1$ and solve second ...
113 views

### Find value range of $2^x+2^y$

Assume $x,y \in \Bbb{R}$ satisfy $$4^x+4^y = 2^{x+1} + 2^{y+1}$$, Find the value range of $$2^x+2^y$$ I know $x=y=1$ is a solution of $4^x+4^y = 2^{x+1} + 2^{y+1}$ , but I can't go further more. I ...
160 views

### An Inequality Problem with not nice conditions

How to show that $\dfrac{a^3}{a^2+b^2} + \dfrac{b^3}{b^2+c^2} + \dfrac{c^3}{c^2+a^2} \ge \dfrac32$, where $a^2+b^2+c^2=3$, and $a,b,c > 0$ ?
19 views

### Proving elementary inequalities with equations

Assume $b > 0,\ d > 0$. Assume: $$\frac{a}{b} < \frac{c}{d}$$. Prove that: $$\frac{a}{b} < \frac{a + c}{b + d} < \frac{c}{d}$$. I would like to find an intuitive way to solve ...
56 views

### How to show $a+b+ad\geq c+d+bc$ given $a\geq c$ and $a+b\geq c+d$?

Let $0\leq a,b,c,d\leq 1$ and $a\geq c$ and $a+b\geq c+d$. Show that $a+b+ad\geq c+d+bc.$ Of course we have $a+b\geq c+d$, but how to relate $ad$ and $bc$?
119 views

### Does $xy\geq x+y$?

I just see the GM-AM inequality. But I would like to compare $xy$ with $x+y$ for any $(x, y)\in\mathbb{R}^2$. It looks like $xy>x+y$ since the first one is multiplication and the second one is ...
54 views

### How to prove this inequation?

$$1+\frac{2}{3n-2}\leqslant \sqrt[n]{3}\leqslant 1+\frac{2}{n}, n\in \mathbb{Z}^{+}$$ How to prove this inequation?
45 views

100 views

### My proof of the inequality of arithmetic and geometric means

Doing the exercises from the Apostol's Calculus I give my proof that the geometric mean is less than or equal the arithmetic mean ($G \le M_1$). I followed the hints from the book and I think I've ...
46 views

### Where is a flaw in these logical implications?

We have a theorem: If $a \le x < a + {\frac yn}$ for $y > 0$ and all natural $n \ge 1$ then $x = a$. Suppose I derive that $a < x$ and $x < a + {\frac yn}$ for all $n \ge 1$. In other ...
34 views

### Implications using inequality signs <= and <

Suppose we have a theorem that says: If $A \le X \le B$ and $A$, $B$ both have property $p$ then $X$ has property $p$. I'm working on some problem and I derive that $A < X < B$ and $A$, $B$ ...
80 views