Question

show this inequality with $xy+yz+zx=3$

score 328 · Answer 1 · 2026-03-25 06:12:16

328

Views

show this inequality with $xy+yz+zx=3$

Published on 25 Mar 2026 - 6:12

score 141 · Answer 2 · 2019-05-19 04:12:27

141

Views

Least value of $\alpha \in \mathbb R$ ($\forall x >0)$ for which $4 \alpha x^2 + \frac{1}{x} \geq 1$

Published on 19 May 2019 - 4:12

score 130 · Answer 3 · 2019-05-21 14:43:11

130

Views

Given $3$ positive reals $a$, $b$ and $c$ such that $a+b+c = 1$, show that $a^a b^b c^c + a^b b^c c^a + a^c b^a c^b \le1$.

Published on 21 May 2019 - 14:43

score 107 · Answer 4 · 2026-03-26 18:40:55

107

Views

Given three non-negative $a,b,c$ so that $c$ between $a$ and $b$. Prove that $2\sum\limits_{cyc}a^{3}+3abc\geqq3\sum\limits_{cyc}a^{2}b$ .

Published on 26 Mar 2026 - 18:40

score 78 · Answer 5 · 2019-05-25 02:01:03

78

Views

Proving an inequality given conditions.

Published on 25 May 2019 - 2:01

score 138 · Answer 6 · 2026-03-25 06:13:03

138

Views

Given three postive numbers $a,b,c$ so that $a\geqq b\geqq c$. Prove that $\sum\limits_{cyc}\frac{a+bW}{aW+b}\geqq 3$ .

Published on 25 Mar 2026 - 6:13

score 153 · Answer 7 · 2026-03-25 08:05:56

153

Views

Holder unsuccessful here!

Published on 25 Mar 2026 - 8:05

score 161 · Answer 8 · 2019-05-30 10:27:58

161

Views

Find the minimum value of $\sqrt{\frac{x^2}{y^2} + \frac{y^2}{x^2} + 2} + \frac{\sqrt{xy}}{x + y}$ where $x, y > 0$.

Published on 30 May 2019 - 10:27

score 104 · Answer 9 · 2019-05-30 13:51:53

104

Views

If $x + y + z = 2$, then show $\frac{x y z}{(1-x)(1-y)(1-z)} \geq 8$, added a second question(Problem 2).

Published on 30 May 2019 - 13:51

score 148 · Answer 10 · 2019-06-03 18:24:30

148

Views

Prove for all positive a,b,c that $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b} \geq6$

Published on 03 Jun 2019 - 18:24

score 104 · Answer 11 · 2026-03-25 09:29:13

104

Views

Calculate the minimum value of $\sum_\mathrm{cyc}\frac{a^2}{b + c}$ where $a, b, c > 0$ and $\sum_\mathrm{cyc}\sqrt{a^2 + b^2} = 1$.

Published on 25 Mar 2026 - 9:29

score 87 · Answer 12 · 2026-03-26 02:56:22

87

Views

Prove that $\sum_{cyc}\frac{1}{x^2 + 1} \ge \frac{2}{3}\bigl(\sum_{cyc}\frac{x}{\sqrt{x^2 + 1}}\bigr)^3$ where $x, y, z > 0$ and $xy + yz + zx = 1$.

Published on 26 Mar 2026 - 2:56

score 121 · Answer 13 · 2019-06-06 05:45:07

121

Views

Proving that a sequence converge to $\sqrt{2}$

Published on 06 Jun 2019 - 5:45

score 100 · Answer 14 · 2019-06-06 10:07:54

100

Views

Calculate the maximum value of $\frac{ab}{ab + a + b} + \frac{2ca}{ca + c + a} + \frac{3bc}{bc + b + c}$ where $3a + 4b + 5c = 12$

Published on 06 Jun 2019 - 10:07

score 1.3k · Answer 15 · 2019-06-06 12:54:42

1.3k

Views

New bound for Am-Gm of 2 variables

Published on 06 Jun 2019 - 12:54

List Question

show this inequality with $xy+yz+zx=3$

Least value of $\alpha \in \mathbb R$ ($\forall x >0)$ for which $4 \alpha x^2 + \frac{1}{x} \geq 1$

Given $3$ positive reals $a$, $b$ and $c$ such that $a+b+c = 1$, show that $a^a b^b c^c + a^b b^c c^a + a^c b^a c^b \le1$.

Given three non-negative $a,b,c$ so that $c$ between $a$ and $b$. Prove that $2\sum\limits_{cyc}a^{3}+3abc\geqq3\sum\limits_{cyc}a^{2}b$ .

Proving an inequality given conditions.

Given three postive numbers $a,b,c$ so that $a\geqq b\geqq c$. Prove that $\sum\limits_{cyc}\frac{a+bW}{aW+b}\geqq 3$ .

Holder unsuccessful here!

Find the minimum value of $\sqrt{\frac{x^2}{y^2} + \frac{y^2}{x^2} + 2} + \frac{\sqrt{xy}}{x + y}$ where $x, y > 0$.

If $x + y + z = 2$, then show $\frac{x y z}{(1-x)(1-y)(1-z)} \geq 8$, added a second question(Problem 2).

Prove for all positive a,b,c that $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b} \geq6$

Calculate the minimum value of $\sum_\mathrm{cyc}\frac{a^2}{b + c}$ where $a, b, c > 0$ and $\sum_\mathrm{cyc}\sqrt{a^2 + b^2} = 1$.

Prove that $\sum_{cyc}\frac{1}{x^2 + 1} \ge \frac{2}{3}\bigl(\sum_{cyc}\frac{x}{\sqrt{x^2 + 1}}\bigr)^3$ where $x, y, z > 0$ and $xy + yz + zx = 1$.

Proving that a sequence converge to $\sqrt{2}$

Calculate the maximum value of $\frac{ab}{ab + a + b} + \frac{2ca}{ca + c + a} + \frac{3bc}{bc + b + c}$ where $3a + 4b + 5c = 12$

New bound for Am-Gm of 2 variables

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