Question

Want less brutish proof: if $a+b+c=3abc$ then $\frac1a+\frac1b+\frac1c\geq 3$

score 334 · Answer 1 · 2026-03-18 21:59:46

334

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Want less brutish proof: if $a+b+c=3abc$ then $\frac1a+\frac1b+\frac1c\geq 3$

Published on 18 Mar 2026 - 21:59

score 102 · Answer 2 · 2023-03-16 11:36:00

102

Views

Write $7 \cdot 10^{100} + 7$ as a sum of four squares

Published on 16 Mar 2023 - 11:36

score 326 · Answer 3 · 2023-06-24 19:45:01

326

Views

Relationship between sum of squares and square of sums in a recursive way

Published on 24 Jun 2023 - 19:45

score 96 · Answer 4 · 2026-03-25 14:27:04

96

Views

Decomposition of nonnegative polynomial on interval into sum of squares

Published on 25 Mar 2026 - 14:27

score 152 · Answer 5 · 2026-02-23 08:27:02

152

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Prove: $\sum\limits_{cyc} \sqrt{9a^2+(a+b+c)^2} \ge \sqrt{12(a^2+b^2+c^2)+14(a+b+c)^2}$ with $a,b,c>0.$

Published on 23 Feb 2026 - 8:27

score 62 · Answer 6 · 2023-08-04 03:27:17

62

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Find minimum and maximum $P=\frac{bc}{a^2+2b^2+2c^2}+\frac{ca}{b^2+2a^2+2c^2}+\frac{ab}{c^2+2b^2+2a^2}.$

Published on 04 Aug 2023 - 3:27

score 63 · Answer 7 · 2023-08-04 07:43:05

63

Views

Prove $\sum_{cyc}xy\left(2\sqrt{z^2+1}-z\right)\ge\sum_{cyc}\left(\sqrt{z^2+1}-z\right)$ when $xy+yz+zx=1$

Published on 04 Aug 2023 - 7:43

score 37 · Answer 8 · 2023-08-05 15:54:25

37

Views

Find minimum $\sum\frac{a^2(b+c)}{b^2+bc+c^2}$ when $a^2+b^2+c^2=a+b+c$

Published on 05 Aug 2023 - 15:54

score 79 · Answer 9 · 2023-08-07 07:48:07

79

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Let $x+y+z=1$, $x,y,z> 0$. Show that $\sum_{cyc}\sqrt{1-2xy}\geq\sqrt7$ by Jensen's inequality?

Published on 07 Aug 2023 - 7:48

score 57 · Answer 10 · 2023-08-07 12:05:34

57

Views

Find minimum $\dfrac{bc}{b^2+c^2}+\dfrac{ca}{c^2+a^2}+\dfrac{ab}{a^2+b^2}+\frac{1}{a}+\frac{1}{b} +\frac{1}{c}$ when $a+b+c=4$

Published on 07 Aug 2023 - 12:05

score 67 · Answer 11 · 2026-03-26 02:57:56

67

Views

If $a^2+b^2+c^2=1$ prove $\frac{a^4-a^2}{bc-1}+\frac{b^4-b^2}{ca-1}+\frac{c^4-c^2}{ab-1}\le ab+bc+ca$

Published on 26 Mar 2026 - 2:57

score 93 · Answer 12 · 2023-10-08 10:00:49

93

Views

Solving the inequality $\sum_{cyc a, b, c}a(\sqrt[3]{b/a} -b) \leq 2/3$ with constraint $a+b+c = 1$

Published on 08 Oct 2023 - 10:00

score 104 · Answer 13 · 2023-11-13 10:55:05

104

Views

$a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right) \geq 3\left(a^3b+b^3c+c^3a\right)$

Published on 13 Nov 2023 - 10:55

score 70 · Answer 14 · 2023-11-29 00:38:36

70

Views

Prove $\frac{4}{(a+1)(b+1)(c+1)}+\frac{1}{4}\ge \frac{a}{(a+1)^2}+\frac{b}{(b+1)^2}+\frac{c}{(c+1)^2}.$

Published on 29 Nov 2023 - 0:38

score 176 · Answer 15 · 2024-01-27 10:50:02

176

Views

when could $5^m+5^n$ represented as sum of two squares

Published on 27 Jan 2024 - 10:50

List Question

Want less brutish proof: if $a+b+c=3abc$ then $\frac1a+\frac1b+\frac1c\geq 3$

Write $7 \cdot 10^{100} + 7$ as a sum of four squares

Relationship between sum of squares and square of sums in a recursive way

Decomposition of nonnegative polynomial on interval into sum of squares

Prove: $\sum\limits_{cyc} \sqrt{9a^2+(a+b+c)^2} \ge \sqrt{12(a^2+b^2+c^2)+14(a+b+c)^2}$ with $a,b,c>0.$

Find minimum and maximum $P=\frac{bc}{a^2+2b^2+2c^2}+\frac{ca}{b^2+2a^2+2c^2}+\frac{ab}{c^2+2b^2+2a^2}.$

Prove $\sum_{cyc}xy\left(2\sqrt{z^2+1}-z\right)\ge\sum_{cyc}\left(\sqrt{z^2+1}-z\right)$ when $xy+yz+zx=1$

Find minimum $\sum\frac{a^2(b+c)}{b^2+bc+c^2}$ when $a^2+b^2+c^2=a+b+c$

Let $x+y+z=1$, $x,y,z> 0$. Show that $\sum_{cyc}\sqrt{1-2xy}\geq\sqrt7$ by Jensen's inequality?

Find minimum $\dfrac{bc}{b^2+c^2}+\dfrac{ca}{c^2+a^2}+\dfrac{ab}{a^2+b^2}+\frac{1}{a}+\frac{1}{b} +\frac{1}{c}$ when $a+b+c=4$

If $a^2+b^2+c^2=1$ prove $\frac{a^4-a^2}{bc-1}+\frac{b^4-b^2}{ca-1}+\frac{c^4-c^2}{ab-1}\le ab+bc+ca$

Solving the inequality $\sum_{cyc a, b, c}a(\sqrt[3]{b/a} -b) \leq 2/3$ with constraint $a+b+c = 1$

$a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right) \geq 3\left(a^3b+b^3c+c^3a\right)$

Prove $\frac{4}{(a+1)(b+1)(c+1)}+\frac{1}{4}\ge \frac{a}{(a+1)^2}+\frac{b}{(b+1)^2}+\frac{c}{(c+1)^2}.$

when could $5^m+5^n$ represented as sum of two squares

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