List questions from uvw | MathQa

Inequality exercise (olympiad)

Published on 12 Jan 2025 - 18:13

Prove that $a^2+b^2+c^2+3\sqrt[3]{a^2b^2c^2} \geq 2(ab+bc+ca).$

Published on 12 Jan 2025 - 19:52

Prove that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$

Published on 12 Jan 2025 - 20:05

How to prove $|a^{2} + b^{2} + c^{2} - 2\left( ab+bc+ac\right)| \le \frac{1}{27}$

Published on 12 Jan 2025 - 19:58

Prove $\frac{x+y+z}{3}+\frac{3}{\frac1x+\frac1y+\frac1z}\geq5\sqrt[3]{\frac{xyz}{16}}$

Published on 12 Jan 2025 - 20:13

Prove $(a+b)^3(b+c)^3(c+d)^3(d+a)^3\ge 16a^2b^2c^2d^2$

Published on 12 Jan 2025 - 19:32

How prove this inequality: $\frac1{1-a}+\frac1{1-b}+\frac1{1-c}\ge \frac1{ab+bc+ac}+\frac1{2(a^2+b^2+c^2)}$ for $a+b+c=1$?

Published on 12 Jan 2025 - 20:28

An elegant but difficult inequality.

Published on 13 Jan 2025 - 7:40

Finding maximum with two constraints

Published on 12 Jan 2025 - 19:09

Prove $(\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x})^3+12\ge 13(x^3+y^3+z^3)$ for positives $xyz = 1$

Published on 12 Jan 2025 - 20:19