List questions from tangent-line-method

Published on 04 Apr 2020 - 14:48

129

Views

Given $a, b, c>0$, prove $\frac{a^4}{a+b}+\frac{b^4}{b+c}+\frac{c^4}{c+a}\geq \frac{1}{2}(a^{2}c+b^{2}a+c^{2}b)$

Published on 05 Apr 2020 - 12:17

133

Views

Prove that $ a^2+b^2+c^2 \le a^3 +b^3 +c^3 $

Published on 05 May 2020 - 7:36

110

Views

Prove this inequality with $xyz=1$

Published on 14 May 2020 - 16:06

177

Views

Question regarding Jensen Inequality

Published on 15 May 2020 - 14:16

92

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Inequality $\sum_{cyc}\frac{a}{2a^2+a+1}\leq \frac{3}{4}$

Published on 16 May 2020 - 2:18

186

Views

Maximize $\sum\limits_{k =1}^n x_k (1 - x_k)^2$

Published on 29 May 2020 - 2:32

100

Views

$\frac{3x+1}{x+1}+\frac{3y+1}{y+1}+\frac{3z+1}{z+1} \le \frac{9}{2}$

Published on 12 Jun 2020 - 18:13

85

Views

If $abc=1$, then how do you prove $\frac{b-1}{bc+1}+\frac{c-1}{ac+1}+\frac{a-1}{ab+1} \geq 0$?

Published on 14 Jun 2020 - 0:01

117

Views

Prove that $21(a^2+b^2+c^2)\ge 20 +9(a^3+b^3+c^3)$

Published on 21 Jun 2020 - 2:24

122

Views

Given positive real numbers $a$, $b$, $c$, $d$, $e$ with $\sum_{\text{cyc}}\,\frac{1}{4+a}=1$, prove that $\sum_{\text{cyc}}\,\frac{a}{4+a^2}\le1$.

Published on 07 Jul 2020 - 15:40

71

Views

Curve tangent line

Published on 24 Mar 2026 - 17:47

222

Views

Curve tangent line with unknown gradient

Published on 09 Jul 2020 - 17:12

236

Views

AM/GM inequalities

Published on 23 Mar 2026 - 19:29

179

Views

If $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\le 1$, prove that $(1+a^2)(1+b^2)(1+c^2)\ge 125$.

Published on 20 Jul 2020 - 16:24

List Question

Prove $\frac{1+a^2}{1-a^2}+\frac{1+b^2}{1-b^2}+\frac{1+c^2}{1-c^2}\ge \frac{15}{4}$

Given $a, b, c>0$, prove $\frac{a^4}{a+b}+\frac{b^4}{b+c}+\frac{c^4}{c+a}\geq \frac{1}{2}(a^{2}c+b^{2}a+c^{2}b)$

Prove that $ a^2+b^2+c^2 \le a^3 +b^3 +c^3 $

Prove this inequality with $xyz=1$

Question regarding Jensen Inequality

Inequality $\sum_{cyc}\frac{a}{2a^2+a+1}\leq \frac{3}{4}$

Maximize $\sum\limits_{k =1}^n x_k (1 - x_k)^2$

$\frac{3x+1}{x+1}+\frac{3y+1}{y+1}+\frac{3z+1}{z+1} \le \frac{9}{2}$

If $abc=1$, then how do you prove $\frac{b-1}{bc+1}+\frac{c-1}{ac+1}+\frac{a-1}{ab+1} \geq 0$?

Prove that $21(a^2+b^2+c^2)\ge 20 +9(a^3+b^3+c^3)$

Given positive real numbers $a$, $b$, $c$, $d$, $e$ with $\sum_{\text{cyc}}\,\frac{1}{4+a}=1$, prove that $\sum_{\text{cyc}}\,\frac{a}{4+a^2}\le1$.

Curve tangent line

Curve tangent line with unknown gradient

AM/GM inequalities

If $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\le 1$, prove that $(1+a^2)(1+b^2)(1+c^2)\ge 125$.

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