Question

Given numbers $a,b,c\geqq0$ and $-\frac{2}{11}\leqq k\leqq0$. Prove that $(k+1)^{6}(a+b+c)^{2}(\!ab+bc+ca\!)^{2}-81\prod\limits_{sym}(ka+b)\geqq0$ .

score 125 · Answer 1 · 2026-05-10 15:52:34

125

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Given numbers $a,b,c\geqq0$ and $-\frac{2}{11}\leqq k\leqq0$. Prove that $(k+1)^{6}(a+b+c)^{2}(\!ab+bc+ca\!)^{2}-81\prod\limits_{sym}(ka+b)\geqq0$ .

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120

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607

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199

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284

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66

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List Question

Given numbers $a,b,c\geqq0$ and $-\frac{2}{11}\leqq k\leqq0$. Prove that $(k+1)^{6}(a+b+c)^{2}(\!ab+bc+ca\!)^{2}-81\prod\limits_{sym}(ka+b)\geqq0$ .

Fransén-Robinson Constant and an intriguing series

Question on elementary set theory notation

The closed-form solution of the family $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(pn+m)}$?

Find $\int\ln^{n} x$ .

Complex numbers in Euler's rule

Prove that $\lim_{n \rightarrow \infty} \sum_{k=0}^{n} \frac{1}{k!} = e$

Prove that $e$ is irrational

Proving that $\sec\frac\pi{30}=\sqrt{2-\sqrt{5}+\sqrt{15-6\sqrt{5}}}$

Infinite product $\Gamma(\tfrac14)=\mathrm{A}^3e^{-\mathrm{G}/\pi}2^{1/6}\sqrt{\pi}\prod_{k\ge1}\left(1-\frac1{2k}\right)^{(-1)^k k}$

Given three positive numbers $x,y,z$ so that $xyz=1, xy+yz+zx=5$. Find the maximum value $x^{c}+ y^{c}+ z^{c}$ for $c\geqq -1$ .

Limits to infinity Finding Constant Number

If $\pi $ is a normal number, is $\tau $ one?

Statistics - scaling sets of data - mean & S.d

Calculus Derivative - Finding unknown constants

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