Question

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

score 89 · Answer 1 · 2019-06-12 14:14:48

89

Views

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

Published on 12 Jun 2019 - 14:14

score 115 · Answer 2 · 2026-03-25 06:13:28

115

Views

show this inequality $(x+y)^3+(y+z)^3+(z+w)^3+(w+x)^3\ge 8(x^2y+y^2z+z^2w+w^2x)$

Published on 25 Mar 2026 - 6:13

score 96 · Answer 3 · 2019-06-14 15:50:47

96

Views

Calculate the minimum value of $\frac{a}{\sqrt{a + 2b}} + \frac{b}{\sqrt{b + 2a}}$ where $a, b > 0$ and $\sqrt{a + 2b} = 2 + \sqrt{\frac{b}{3}}$.

Published on 14 Jun 2019 - 15:50

score 158 · Answer 4 · 2026-03-25 06:01:29

158

Views

Minimize value of the function $a^2+b^2+c^2+2\sqrt{3abc}$

Published on 25 Mar 2026 - 6:01

score 118 · Answer 5 · 2026-03-25 06:04:10

118

Views

How do you come up with great and ingenious questions

Published on 25 Mar 2026 - 6:04

score 31 · Answer 6 · 2019-06-20 10:05:06

31

Views

Substitution method for definite integrals

Published on 20 Jun 2019 - 10:05

score 69 · Answer 7 · 2019-06-20 10:05:24

69

Views

Differential equation of the form $x^2y''+(1-2p)xy'+(p^2-q^2)y=x^n$

Published on 20 Jun 2019 - 10:05

score 71 · Answer 8 · 2026-03-25 08:10:05

71

Views

Prove that inequality $27+7\left(ab+bc+ca\right)\le 8\left(\sum \sqrt{a+3b}\right)$

Published on 25 Mar 2026 - 8:10

score 266 · Answer 9 · 2026-03-25 11:04:48

266

Views

Given three positive numbers $a,\,b,\,c$ . Prove that $(\!abc+ a+ b+ c\!)^{3}\geqq 8\,abc(\!1+ a\!)(\!1+ b\!)(\!1+ c\!)$ .

Published on 25 Mar 2026 - 11:04

score 192 · Answer 10 · 2026-03-26 06:21:25

192

Views

Prove that $abc(a^2+b^2+c^2-2abc-2a-2b-2c)+ab(a+b-2)+bc(b+c-2)+ca(c+a-2)+a+b+c \le 2$

Published on 26 Mar 2026 - 6:21

score 418 · Answer 11 · 2019-06-25 10:36:50

418

Views

Prove that $\sum\limits_{cyc}\sqrt{a+11bc+6}\geq9\sqrt2.$

Published on 25 Jun 2019 - 10:36

score 211 · Answer 12 · 2026-03-25 09:28:46

211

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 Mar 2026 - 9:28

score 316 · Answer 13 · 2026-03-25 06:13:03

316

Views

Given two positive numbers $b,\,c$. Prove $\left ( \frac{3}{b}- 1 \right )(3- b)^{2}+ \left ( \frac{b}{c}- 1 \right )(b- c)^{2}+ (c- 1)^{3}\geqq 0$ .

Published on 25 Mar 2026 - 6:13

score 115 · Answer 14 · 2019-06-26 05:43:01

115

Views

Prove that $\sum \frac {xy}{\sqrt{ (x^2+z^2)(y^2+z^2)}}\leq \frac {3}{2} $

Published on 26 Jun 2019 - 5:43

score 47 · Answer 15 · 2019-06-26 23:57:46

47

Views

Evaluate the indefinite trigonometric double angle integral

Published on 26 Jun 2019 - 23:57

List Question

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

show this inequality $(x+y)^3+(y+z)^3+(z+w)^3+(w+x)^3\ge 8(x^2y+y^2z+z^2w+w^2x)$

Calculate the minimum value of $\frac{a}{\sqrt{a + 2b}} + \frac{b}{\sqrt{b + 2a}}$ where $a, b > 0$ and $\sqrt{a + 2b} = 2 + \sqrt{\frac{b}{3}}$.

Minimize value of the function $a^2+b^2+c^2+2\sqrt{3abc}$

How do you come up with great and ingenious questions

Substitution method for definite integrals

Differential equation of the form $x^2y''+(1-2p)xy'+(p^2-q^2)y=x^n$

Prove that inequality $27+7\left(ab+bc+ca\right)\le 8\left(\sum \sqrt{a+3b}\right)$

Given three positive numbers $a,\,b,\,c$ . Prove that $(\!abc+ a+ b+ c\!)^{3}\geqq 8\,abc(\!1+ a\!)(\!1+ b\!)(\!1+ c\!)$ .

Prove that $abc(a^2+b^2+c^2-2abc-2a-2b-2c)+ab(a+b-2)+bc(b+c-2)+ca(c+a-2)+a+b+c \le 2$

Prove that $\sum\limits_{cyc}\sqrt{a+11bc+6}\geq9\sqrt2.$

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

Given two positive numbers $b,\,c$. Prove $\left ( \frac{3}{b}- 1 \right )(3- b)^{2}+ \left ( \frac{b}{c}- 1 \right )(b- c)^{2}+ (c- 1)^{3}\geqq 0$ .

Prove that $\sum \frac {xy}{\sqrt{ (x^2+z^2)(y^2+z^2)}}\leq \frac {3}{2} $

Evaluate the indefinite trigonometric double angle integral

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