Question

Given positives $a, b, c$, prove that $\frac{a}{(b + c)^2} + \frac{b}{(c + a)^2} + \frac{c}{(a + b)^2} \ge \frac{9}{4(a + b + c)}$.

score 187 · Answer 1 · 2026-05-10 16:35:58

187

Views

Given positives $a, b, c$, prove that $\frac{a}{(b + c)^2} + \frac{b}{(c + a)^2} + \frac{c}{(a + b)^2} \ge \frac{9}{4(a + b + c)}$.

Published on 10 May 2026 - 16:35

score 153 · Answer 2 · 2026-03-25 12:41:26

153

Views

Given positives $x, y , z$ such that $x + y + z = xyz$. Calculate the minimum value of $\frac{x - 1}{y^2} + \frac{y - 1}{z^2} + \frac{z - 1}{x^2}$.

Published on 25 Mar 2026 - 12:41

score 20 · Answer 3 · 2026-04-06 03:14:53

20

Views

Number systems, squares and probability

Published on 06 Apr 2026 - 3:14

score 89 · Answer 4 · 2026-04-06 00:11:13

89

Views

Elementary proof for the inequality

Published on 06 Apr 2026 - 0:11

score 56 · Answer 5 · 2026-03-25 12:38:44

56

Views

Prove the following inequality using am gm inequality

Published on 25 Mar 2026 - 12:38

score 197 · Answer 6 · 2026-03-25 12:41:22

197

Views

$\frac{a}{b}+ \frac{b}{c} + \frac{c}{a} \geq \frac{9(a^2+b^2+c^2)}{(a+b+c)^2}$

Published on 25 Mar 2026 - 12:41

score 172 · Answer 7 · 2026-03-25 12:40:01

172

Views

Proving $a^2 + b^2 + c^2 \geqslant ab + bc + ca$

Published on 25 Mar 2026 - 12:40

score 324 · Answer 8 · 2026-03-25 12:41:22

324

Views

Prove that $\sqrt{ab+c}+\sqrt{bc+a}+\sqrt{ac+b} \ge 1+\sqrt{ab}+\sqrt{bc}+\sqrt{ac}$

Published on 25 Mar 2026 - 12:41

score 153 · Answer 9 · 2026-05-10 16:35:49

153

Views

For $a,b,c\in\left[\frac{1}{\sqrt{6}}, 6\right]$: $\sum_{cyc}\frac{4}{a+3b}\geq \sum_{cyc}\frac{3}{a+2b}$

Published on 10 May 2026 - 16:35

score 130 · Answer 10 · 2026-03-25 06:06:56

130

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 25 Mar 2026 - 6:06

score 719 · Answer 11 · 2026-03-25 12:41:38

719

Views

If $a,b,c$ are the sides of a triangle, then $\dfrac{a}{b+c-a}+\dfrac{b}{c+a-b}+\dfrac{c}{a+b-c}$ is:

Published on 25 Mar 2026 - 12:41

score 71 · Answer 12 · 2026-03-25 12:41:22

71

Views

For $a,b,c\ge0$, show $a^2 + b^2 + c^2 = 3$ implies $(a^3 + b^3 + c^3)^2 \geq 3 + 2(a^4 + b^4 + c^4)$.

Published on 25 Mar 2026 - 12:41

score 75 · Answer 13 · 2026-05-10 16:08:24

75

Views

Stuck when transforming and solving this

Published on 10 May 2026 - 16:08

score 139 · Answer 14 · 2026-03-25 09:40:00

139

Views

Prove $\Big[\sum\limits_{cyc} a(a^2+2bc)\Big]^3 \geqq 3(ab+bc+ca)^2 . \sum\limits_{cyc} a(a^2+2bc)^2$

Published on 25 Mar 2026 - 9:40

score 63 · Answer 15 · 2026-03-25 12:41:22

63

Views

representation of sum of squares of a globally positive quadratic function

Published on 25 Mar 2026 - 12:41

List Question

Given positives $a, b, c$, prove that $\frac{a}{(b + c)^2} + \frac{b}{(c + a)^2} + \frac{c}{(a + b)^2} \ge \frac{9}{4(a + b + c)}$.

Given positives $x, y , z$ such that $x + y + z = xyz$. Calculate the minimum value of $\frac{x - 1}{y^2} + \frac{y - 1}{z^2} + \frac{z - 1}{x^2}$.

Number systems, squares and probability

Elementary proof for the inequality

Prove the following inequality using am gm inequality

$\frac{a}{b}+ \frac{b}{c} + \frac{c}{a} \geq \frac{9(a^2+b^2+c^2)}{(a+b+c)^2}$

Proving $a^2 + b^2 + c^2 \geqslant ab + bc + ca$

Prove that $\sqrt{ab+c}+\sqrt{bc+a}+\sqrt{ac+b} \ge 1+\sqrt{ab}+\sqrt{bc}+\sqrt{ac}$

For $a,b,c\in\left[\frac{1}{\sqrt{6}}, 6\right]$: $\sum_{cyc}\frac{4}{a+3b}\geq \sum_{cyc}\frac{3}{a+2b}$

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

If $a,b,c$ are the sides of a triangle, then $\dfrac{a}{b+c-a}+\dfrac{b}{c+a-b}+\dfrac{c}{a+b-c}$ is:

For $a,b,c\ge0$, show $a^2 + b^2 + c^2 = 3$ implies $(a^3 + b^3 + c^3)^2 \geq 3 + 2(a^4 + b^4 + c^4)$.

Stuck when transforming and solving this

Prove $\Big[\sum\limits_{cyc} a(a^2+2bc)\Big]^3 \geqq 3(ab+bc+ca)^2 . \sum\limits_{cyc} a(a^2+2bc)^2$

representation of sum of squares of a globally positive quadratic function

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