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 186 · Answer 1 · 2026-03-25 04:42:43

186

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 25 Mar 2026 - 4:42

score 152 · Answer 2 · 2020-02-10 13:03:32

152

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 10 Feb 2020 - 13:03

score 18 · Answer 3 · 2020-02-24 16:49:55

18

Views

Number systems, squares and probability

Published on 24 Feb 2020 - 16:49

score 87 · Answer 4 · 2020-02-27 12:26:34

87

Views

Elementary proof for the inequality

Published on 27 Feb 2020 - 12:26

score 55 · Answer 5 · 2020-03-17 09:08:47

55

Views

Prove the following inequality using am gm inequality

Published on 17 Mar 2020 - 9:08

score 196 · Answer 6 · 2020-03-19 08:30:03

196

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 19 Mar 2020 - 8:30

score 171 · Answer 7 · 2020-03-20 00:45:46

171

Views

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

Published on 20 Mar 2020 - 0:45

score 323 · Answer 8 · 2020-03-21 13:28:09

323

Views

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

Published on 21 Mar 2020 - 13:28

score 152 · Answer 9 · 2026-03-25 04:43:51

152

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 25 Mar 2026 - 4:43

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 718 · Answer 11 · 2020-04-15 05:05:24

718

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 15 Apr 2020 - 5:05

score 70 · Answer 12 · 2020-04-18 02:44:25

70

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 18 Apr 2020 - 2:44

score 74 · Answer 13 · 2026-03-25 09:33:47

74

Views

Stuck when transforming and solving this

Published on 25 Mar 2026 - 9:33

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 62 · Answer 15 · 2020-05-04 12:35:17

62

Views

representation of sum of squares of a globally positive quadratic function

Published on 04 May 2020 - 12:35

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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