Question

Prove that $1+b+(1+c)+1/c+1+a \ge 3$ if $a, b,$ and $c$ are positive real numbers.

score 201 · Answer 1 · 2019-02-05 00:58:40

201

Views

Prove that $1+b+(1+c)+1/c+1+a \ge 3$ if $a, b,$ and $c$ are positive real numbers.

Published on 05 Feb 2019 - 0:58

score 525 · Answer 2 · 2026-03-25 07:45:38

525

Views

Let $P$ be a polynomial with positive real coefficients. Prove that if $P(1/x) \geq 1/P(x)$ holds for $x = 1$, then it holds for every $x > 0$.

Published on 25 Mar 2026 - 7:45

score 120 · Answer 3 · 2019-02-06 08:18:29

120

Views

If positive $a$, $b$, $c$, $d$ satisfy ${1\over a+1}+{1\over b+1}+{1\over c+1}+{1\over d+1}=1$, then $abcd\geq 81$

Published on 06 Feb 2019 - 8:18

score 83 · Answer 4 · 2019-02-07 20:22:13

83

Views

Finding the upper bound of a 3-variable function.(inequality)

Published on 07 Feb 2019 - 20:22

score 78 · Answer 5 · 2026-03-26 01:23:30

78

Views

Prove $ \sum_{cyc}\frac{x}{\sqrt{x^2+8yz}} \ge 1, \forall x,y,z\gt 0 $

Published on 26 Mar 2026 - 1:23

score 127 · Answer 6 · 2019-02-11 13:31:04

127

Views

show this inequality $a_{1}+a_{2}+\cdots+a_{n}\ge\sqrt{3}(a_{1}a_{2}+a_{2}a_{3}+\cdots+a_{n}a_{1})$

Published on 11 Feb 2019 - 13:31

score 246 · Answer 7 · 2019-02-12 21:55:14

246

Views

prove that : $\dfrac{a^2}{2}+\dfrac{b^3}{3}+\dfrac{c^6}{6} \geq abc$ for $a ,b ,c \in \mathbb{R}^{>0}$

Published on 12 Feb 2019 - 21:55

score 1.5k · Answer 8 · 2019-02-15 07:52:30

1.5k

Views

In any triangle is $\sin A+\sin B+\sin C=\frac{3\sqrt3}{2}$ always

Published on 15 Feb 2019 - 7:52

score 204 · Answer 9 · 2019-02-16 10:21:51

204

Views

Proving an equality using given ones; no need for differentiation

Published on 16 Feb 2019 - 10:21

score 290 · Answer 10 · 2019-02-17 01:38:54

290

Views

Prove that $a^2+b^2+c^2\geqslant\frac{1}{3}$ given that $a\gt0, b\gt0, c\gt0$ and $a+b+c=1$, using existing AM GM inequality

Published on 17 Feb 2019 - 1:38

score 282 · Answer 11 · 2019-02-17 14:43:42

282

Views

Understanding a proof from the APMO 1998 on inequalities.

Published on 17 Feb 2019 - 14:43

score 268 · Answer 12 · 2019-02-17 19:33:00

268

Views

For three positive numbers verifying $xyz=2+x+y+z$, is there an upper bound on $x+y+z$? And, if so, which is that?

Published on 17 Feb 2019 - 19:33

score 818 · Answer 13 · 2026-03-24 23:41:50

818

Views

Find maximum value by using AM-GM inequality

Published on 24 Mar 2026 - 23:41

score 49 · Answer 14 · 2019-02-27 10:34:19

49

Views

Proving an inequality using am-gm inequality

Published on 27 Feb 2019 - 10:34

score 281 · Answer 15 · 2019-02-28 07:37:53

281

Views

Stronger than AM-GM and a conjecture

Published on 28 Feb 2019 - 7:37

List Question

Prove that $1+b+(1+c)+1/c+1+a \ge 3$ if $a, b,$ and $c$ are positive real numbers.

Let $P$ be a polynomial with positive real coefficients. Prove that if $P(1/x) \geq 1/P(x)$ holds for $x = 1$, then it holds for every $x > 0$.

If positive $a$, $b$, $c$, $d$ satisfy ${1\over a+1}+{1\over b+1}+{1\over c+1}+{1\over d+1}=1$, then $abcd\geq 81$

Finding the upper bound of a 3-variable function.(inequality)

Prove $ \sum_{cyc}\frac{x}{\sqrt{x^2+8yz}} \ge 1, \forall x,y,z\gt 0 $

show this inequality $a_{1}+a_{2}+\cdots+a_{n}\ge\sqrt{3}(a_{1}a_{2}+a_{2}a_{3}+\cdots+a_{n}a_{1})$

prove that : $\dfrac{a^2}{2}+\dfrac{b^3}{3}+\dfrac{c^6}{6} \geq abc$ for $a ,b ,c \in \mathbb{R}^{>0}$

In any triangle is $\sin A+\sin B+\sin C=\frac{3\sqrt3}{2}$ always

Proving an equality using given ones; no need for differentiation

Prove that $a^2+b^2+c^2\geqslant\frac{1}{3}$ given that $a\gt0, b\gt0, c\gt0$ and $a+b+c=1$, using existing AM GM inequality

Understanding a proof from the APMO 1998 on inequalities.

For three positive numbers verifying $xyz=2+x+y+z$, is there an upper bound on $x+y+z$? And, if so, which is that?

Find maximum value by using AM-GM inequality

Proving an inequality using am-gm inequality

Stronger than AM-GM and a conjecture

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