Question

Prove that $a\sqrt{b^3+1}+b\sqrt{c^3+1}+c\sqrt{a^3+1} \leq 5$

score 210 · Answer 1 · 2019-03-18 15:12:10

210

Views

Prove that $a\sqrt{b^3+1}+b\sqrt{c^3+1}+c\sqrt{a^3+1} \leq 5$

Published on 18 Mar 2019 - 15:12

score 161 · Answer 2 · 2019-03-20 15:15:34

161

Views

Prove that $\sum_{cyc}\frac{a}{a + b^4 + c^4} \le 1$ where $abc = 1$

Published on 20 Mar 2019 - 15:15

score 107 · Answer 3 · 2019-03-21 21:58:12

107

Views

If $a+b+c=1$ and a,b,c ＞0 prove $\dfrac{b^2}{a+b^2}+\dfrac{c^2}{b+c^2}+\dfrac{a^2}{c+a^2} \geqslant \dfrac{3}{4}$

Published on 21 Mar 2019 - 21:58

score 44 · Answer 4 · 2019-04-17 11:20:42

44

Views

$|\underset{i,j}{\sum}a_{ij}x_iy_j| \leq \underset{u,v \in \{-1,1\}^n }{sup} |\underset{i,j}{\sum} a_{ij}u_iv_j|$?

Published on 17 Apr 2019 - 11:20

score 50 · Answer 5 · 2019-05-14 20:12:56

50

Views

Prove that $A_1 \cdot B_1 + A_2 \cdot B_2 > A_1 \cdot B_2 + A_2 \cdot B_1$ for $0 < A_1 < A_2$ and $0 < B_1 < B_2$

Published on 14 May 2019 - 20:12

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 60 · Answer 7 · 2019-05-30 16:23:16

60

Views

Is there a problem in this statement: As $\sum_{n=1}^{\infty} \mu(n)/n = 0, \sum_{n=1}^{N} \mu(n)/n = -\sum_{n=N+1}^{\infty} \mu(n)/n$.

Published on 30 May 2019 - 16:23

score 103 · Answer 8 · 2019-06-04 10:18:40

103

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 04 Jun 2019 - 10:18

score 86 · Answer 9 · 2019-06-15 20:54:33

86

Views

How to prove this inequality for $a,b,c>0$?

Published on 15 Jun 2019 - 20:54

score 210 · Answer 10 · 2019-06-25 13:24:27

210

Views

Given three non-negative numbers $a, b, c$ so that $a+ b+ c= 3,\,a^{2}+ b^{2}+ c^{2}= 5$. Prove $a^{3}b+ b^{3}c+ c^{3}a\leqq 8$ .

Published on 25 Jun 2019 - 13:24

score 76 · Answer 11 · 2019-07-01 16:41:30

76

Views

Prove that $\sum_{cyc}\frac{m^2}{n + p^2} \ge \sum_{cyc}\frac{m}{n^2 + p}$ where $m, n, p \ge 0$ and $mnp = 1$.

Published on 01 Jul 2019 - 16:41

score 113 · Answer 12 · 2019-07-09 14:36:58

113

Views

minimum value of $\frac{x}{2-x}+\frac{y}{2-y}+\frac{z}{2-z}$ is

Published on 09 Jul 2019 - 14:36

score 819 · Answer 13 · 2019-07-18 05:36:42

819

Views

Calculate the maximum value of $x^3y + y^3z + z^3x$ where $x + y + z = 4$ and $x, y, z \ge 0$.

Published on 18 Jul 2019 - 5:36

score 164 · Answer 14 · 2019-08-11 17:54:47

164

Views

Inequalities: BMO 2009/10 Round 2

Published on 11 Aug 2019 - 17:54

score 248 · Answer 15 · 2019-08-15 18:51:08

248

Views

Maximizing a permutation of a quadratic form

Published on 15 Aug 2019 - 18:51

List Question

Prove that $a\sqrt{b^3+1}+b\sqrt{c^3+1}+c\sqrt{a^3+1} \leq 5$

Prove that $\sum_{cyc}\frac{a}{a + b^4 + c^4} \le 1$ where $abc = 1$

If $a+b+c=1$ and a,b,c ＞0 prove $\dfrac{b^2}{a+b^2}+\dfrac{c^2}{b+c^2}+\dfrac{a^2}{c+a^2} \geqslant \dfrac{3}{4}$

$|\underset{i,j}{\sum}a_{ij}x_iy_j| \leq \underset{u,v \in \{-1,1\}^n }{sup} |\underset{i,j}{\sum} a_{ij}u_iv_j|$?

Prove that $A_1 \cdot B_1 + A_2 \cdot B_2 > A_1 \cdot B_2 + A_2 \cdot B_1$ for $0 < A_1 < A_2$ and $0 < B_1 < B_2$

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

Is there a problem in this statement: As $\sum_{n=1}^{\infty} \mu(n)/n = 0, \sum_{n=1}^{N} \mu(n)/n = -\sum_{n=N+1}^{\infty} \mu(n)/n$.

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

How to prove this inequality for $a,b,c>0$?

Given three non-negative numbers $a, b, c$ so that $a+ b+ c= 3,\,a^{2}+ b^{2}+ c^{2}= 5$. Prove $a^{3}b+ b^{3}c+ c^{3}a\leqq 8$ .

Prove that $\sum_{cyc}\frac{m^2}{n + p^2} \ge \sum_{cyc}\frac{m}{n^2 + p}$ where $m, n, p \ge 0$ and $mnp = 1$.

minimum value of $\frac{x}{2-x}+\frac{y}{2-y}+\frac{z}{2-z}$ is

Calculate the maximum value of $x^3y + y^3z + z^3x$ where $x + y + z = 4$ and $x, y, z \ge 0$.

Inequalities: BMO 2009/10 Round 2

Maximizing a permutation of a quadratic form

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