Question

Given three postive numbers $a,b,c$ so that $a+b+c=3$. Prove that $\sum\frac{1}{a}\geqq (\frac{1}{2}+\frac{5\sqrt{3}}{18})(\!a^{2}+b^{2}+c^{2}\!)$ .

score 203 · Answer 1 · 2026-03-25 06:06:55

203

Views

Given three postive numbers $a,b,c$ so that $a+b+c=3$. Prove that $\sum\frac{1}{a}\geqq (\frac{1}{2}+\frac{5\sqrt{3}}{18})(\!a^{2}+b^{2}+c^{2}\!)$ .

Published on 25 Mar 2026 - 6:06

score 142 · Answer 2 · 2026-03-25 09:32:00

142

Views

Find the minimum and maximum values of $P=\left( 6-a^2-b^2-c^2\right)\left(2-abc\right)$

Published on 25 Mar 2026 - 9:32

score 295 · Answer 3 · 2026-03-25 06:06:19

295

Views

Given three non-negatve numbers $a,b,c$. Prove that $1+a^{2}+b^{2}+c^{2}+4abc\geqq a+b+c+ab+bc+ca$ .

Published on 25 Mar 2026 - 6:06

score 177 · Answer 4 · 2026-03-24 23:46:38

177

Views

Given three non-negative numbers $a,b,c$ so that $a+b+c=3$. Prove that $\prod\limits_{cyc}(\!2+ a^{2}\!)+abc\geqq 28$ .

Published on 24 Mar 2026 - 23:46

score 122 · Answer 5 · 2026-03-25 06:06:58

122

Views

Prove/disprove $\sum_{cyc}a\sqrt{\frac{(ca + 1)(ab + 1)}{bc + 1}} \ge 2$ where $a$, $b$, $c > 0$ and $a^2 + b^2 + c^2 = 1$

Published on 25 Mar 2026 - 6:06

score 144 · Answer 6 · 2026-03-25 07:42:52

144

Views

Find maximum of function $A=\sum _{cyc}\frac{1}{a^2+2}$

Published on 25 Mar 2026 - 7:42

score 103 · Answer 7 · 2026-03-25 06:04:27

103

Views

show this $\sum_{cyc}\frac{x}{x^2-x+1}\le\frac{8}{3}$

Published on 25 Mar 2026 - 6:04

score 203 · Answer 8 · 2026-03-24 22:08:44

203

Views

Given three positive numbers $a,b,c$. Prove that $\prod\limits_{sym}(\!a+\frac{1}{a}\!)\geqq\frac{4}{3}\sum\limits_{sym}\frac{b+c}{a}$ .

Published on 24 Mar 2026 - 22:08

score 200 · Answer 9 · 2026-03-25 06:06:51

200

Views

Given three positive numbers $a,b,c$ so that $8abc\geqq a+b+c+5$. Prove that $\frac{1}{a+2b}+\frac{1}{b+ 2c}+\frac{1}{c+2a}\leqq1$ .

Published on 25 Mar 2026 - 6:06

score 164 · Answer 10 · 2026-05-10 13:51:31

164

Views

Find $f(x)$ so that $\sum\limits_{cyc}a^{2}-f(x)\left(\prod_{sym}a-\prod_{sym}(1-a)\right)\geqq3(\frac{x}{2})^{2}$ .

Published on 10 May 2026 - 13:51

score 297 · Answer 11 · 2026-03-25 06:07:04

297

Views

Given three positve numbers $a,b,c$. Prove that $\sum\limits_{cyc}\frac{a}{\sqrt{b(a+b)}}\geqq \sum\limits_{cyc}\frac{a}{\sqrt{b(c+a)}}$ .

Published on 25 Mar 2026 - 6:07

score 470 · Answer 12 · 2026-05-07 10:49:55

470

Views

For $a,b,c,d > 0$ and $abcd = 1$, show that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{12}{a + b + c + d} \geq 7$

Published on 07 May 2026 - 10:49

score 172 · Answer 13 · 2026-04-10 11:24:08

172

Views

Nice olympiad inequality

Published on 10 Apr 2026 - 11:24

score 158 · Answer 14 · 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 15 · 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

List Question

Given three postive numbers $a,b,c$ so that $a+b+c=3$. Prove that $\sum\frac{1}{a}\geqq (\frac{1}{2}+\frac{5\sqrt{3}}{18})(\!a^{2}+b^{2}+c^{2}\!)$ .

Find the minimum and maximum values of $P=\left( 6-a^2-b^2-c^2\right)\left(2-abc\right)$

Given three non-negatve numbers $a,b,c$. Prove that $1+a^{2}+b^{2}+c^{2}+4abc\geqq a+b+c+ab+bc+ca$ .

Given three non-negative numbers $a,b,c$ so that $a+b+c=3$. Prove that $\prod\limits_{cyc}(\!2+ a^{2}\!)+abc\geqq 28$ .

Prove/disprove $\sum_{cyc}a\sqrt{\frac{(ca + 1)(ab + 1)}{bc + 1}} \ge 2$ where $a$, $b$, $c > 0$ and $a^2 + b^2 + c^2 = 1$

Find maximum of function $A=\sum _{cyc}\frac{1}{a^2+2}$

show this $\sum_{cyc}\frac{x}{x^2-x+1}\le\frac{8}{3}$

Given three positive numbers $a,b,c$. Prove that $\prod\limits_{sym}(\!a+\frac{1}{a}\!)\geqq\frac{4}{3}\sum\limits_{sym}\frac{b+c}{a}$ .

Given three positive numbers $a,b,c$ so that $8abc\geqq a+b+c+5$. Prove that $\frac{1}{a+2b}+\frac{1}{b+ 2c}+\frac{1}{c+2a}\leqq1$ .

Find $f(x)$ so that $\sum\limits_{cyc}a^{2}-f(x)\left(\prod_{sym}a-\prod_{sym}(1-a)\right)\geqq3(\frac{x}{2})^{2}$ .

Given three positve numbers $a,b,c$. Prove that $\sum\limits_{cyc}\frac{a}{\sqrt{b(a+b)}}\geqq \sum\limits_{cyc}\frac{a}{\sqrt{b(c+a)}}$ .

For $a,b,c,d > 0$ and $abcd = 1$, show that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{12}{a + b + c + d} \geq 7$

Nice olympiad inequality

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

How do you come up with great and ingenious questions

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