Newbetuts

Given $~ \mathrm{a,b,c >0 ~ , ~ a+b+c=1 } ~ $ then prove $~ \mathrm{\sum\limits_{cyc} \sqrt{a+b^2 } \geqslant 2 } $

If $x,y,z\in[-1,1]$ and $1+2xyz\geq x^2+y^2+z^2$, then can we infer $1+2(xyz)^n\geq x^{2n}+y^{2n}+z^{2n}$?

Find the maximum of the expression

Find minimum value of $\sum \frac{\sqrt{a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}$

A inequality proposed at Zhautykov Olympiad 2008

Prove $\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$

How can I show that the dot product scales up linearly with the norms of the underlying arguments?

How prove this inequality $\frac{2}{(a+b)(4-ab)}+\frac{2}{(b+c)(4-bc)}+\frac{2}{(a+c)(4-ac)}\ge 1$

Can Cauchy Schwarz inequality be proven using Jensen's inequality?

Let $a,b,c\in \Bbb R^+$ such that $(1+a+b+c)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})=16$. Find $(a+b+c)$

Proof of Cauchy-Schwarz inequality using a particular result from orthonormal sets

Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds.

Proof of one inequality $a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$

Proving inequality $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{3\sqrt[3]{abc}}{a+b+c} \geq 4$

Is $\left(1+\frac1n\right)^{n+1/2}$ decreasing?

How prove this inequality $\sum\limits_{cyc}\frac{x+y}{\sqrt{x^2+xy+y^2+yz}}\ge 2+\sqrt{\frac{xy+yz+xz}{x^2+y^2+z^2}}$

Inequality with five variables

Prove $1<ab+bc+ca-abc<\frac{28}{27}$

New posts in cauchy-schwarz-inequality

Given $~ \mathrm{a,b,c >0 ~ , ~ a+b+c=1 } ~ $ then prove $~ \mathrm{\sum\limits_{cyc} \sqrt{a+b^2 } \geqslant 2 } $

Inequalities help: $(a^3+b^3)^2\leq (a^2+b^2)(a^4+b^4)$

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

If $x,y,z\in[-1,1]$ and $1+2xyz\geq x^2+y^2+z^2$, then can we infer $1+2(xyz)^n\geq x^{2n}+y^{2n}+z^{2n}$?

Find the maximum of the expression

Find minimum value of $\sum \frac{\sqrt{a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}$

A inequality proposed at Zhautykov Olympiad 2008

Prove $\sqrt{3a + b^3} + \sqrt{3b + c^3} + \sqrt{3c + a^3} \ge 6$

How can I show that the dot product scales up linearly with the norms of the underlying arguments?

How prove this inequality $\frac{2}{(a+b)(4-ab)}+\frac{2}{(b+c)(4-bc)}+\frac{2}{(a+c)(4-ac)}\ge 1$

Can Cauchy Schwarz inequality be proven using Jensen's inequality?

Let $a,b,c\in \Bbb R^+$ such that $(1+a+b+c)(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})=16$. Find $(a+b+c)$

Proof of Cauchy-Schwarz inequality using a particular result from orthonormal sets

Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds.

Proof of one inequality $a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$

Proving inequality $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{3\sqrt[3]{abc}}{a+b+c} \geq 4$

Is $\left(1+\frac1n\right)^{n+1/2}$ decreasing?

How prove this inequality $\sum\limits_{cyc}\frac{x+y}{\sqrt{x^2+xy+y^2+yz}}\ge 2+\sqrt{\frac{xy+yz+xz}{x^2+y^2+z^2}}$

Inequality with five variables

Prove $1<ab+bc+ca-abc<\frac{28}{27}$