Question

How to prove this inequality $\frac{(\sum_i w_i x_i)}{(\sum_i w_i y_i^2)} \leq \sum_i w_i \frac{x_i}{y_i^2}$

score 94 · Answer 1 · 2023-12-31 13:40:40

94

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How to prove this inequality $\frac{(\sum_i w_i x_i)}{(\sum_i w_i y_i^2)} \leq \sum_i w_i \frac{x_i}{y_i^2}$

Published on 31 Dec 2023 - 13:40

score 88 · Answer 2 · 2024-01-04 12:55:17

88

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$ \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{a+c}=2 $ then $ \sum_{cyc}\sqrt{ab} \geq \frac{3}{2}$

Published on 04 Jan 2024 - 12:55

score 199 · Answer 3 · 2026-03-26 11:03:11

199

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Showing $\frac{1}{5}<(f(k,l,m,n))^3\leq\frac{64}{11}$, where $f(k,l,m,n)=\sum_{cyc}\frac{k^2}{\sqrt[3]{5k^6+6l^3m^3}}$ for positive $k$, $l$, $m$, $n$

Published on 26 Mar 2026 - 11:03

score 162 · Answer 4 · 2024-01-19 18:22:09

162

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Prove the inequality $\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right)$

Published on 19 Jan 2024 - 18:22

score 341 · Answer 5 · 2024-01-20 09:42:41

341

Views

A strengthening of a known inequality; looking for a neater solution

Published on 20 Jan 2024 - 9:42

score 276 · Answer 6 · 2024-02-03 12:25:43

276

Views

Finding all $x, y$ natural numbers

Published on 03 Feb 2024 - 12:25

score 102 · Answer 7 · 2024-02-03 15:34:00

102

Views

Enhanced Nesbitt's Inequality with Geometric Mean Term.

Published on 03 Feb 2024 - 15:34

score 152 · Answer 8 · 2024-02-05 10:26:56

152

Views

Variant Proof of the AM-GM Inequality

Published on 05 Feb 2024 - 10:26

score 94 · Answer 9 · 2024-02-06 05:22:16

94

Views

Regarding the specifity of the A.M.-G.M. inequality in finding maximum and minimum in Number Theory

Published on 06 Feb 2024 - 5:22

score 213 · Answer 10 · 2024-02-18 16:32:02

213

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Prove the inequality knowing $x,y,z \ge 0$ and $xyz=1$ $\frac{x^5}{x^2+1}+\frac{y^5}{y^2+1}+\frac{z^5}{z^2+1}\ge\frac{3}{2}$

Published on 18 Feb 2024 - 16:32

score 59 · Answer 11 · 2024-02-20 12:41:25

59

Views

Prove that $a^2+b^2+c^2 \geq ab(a+b+\sqrt{ab})+cb(c+b+\sqrt{cb})+ ac(a+c+\sqrt{ac} )$

Published on 20 Feb 2024 - 12:41

score 74 · Answer 12 · 2024-03-01 23:37:24

74

Views

If $x>0,y>0$ and $4xy=2^{x+y}$ then find the minimum and maximum values of $x+y$.

Published on 01 Mar 2024 - 23:37

score 78 · Answer 13 · 2024-03-06 12:37:18

78

Views

Prove $\frac{x^2}{a} + \frac{y^2}{b} \geq \frac{(x + y)^2}{a + b}$, with $x, y, a, b \in \mathbb{R}$ and $a, b > 0$

Published on 06 Mar 2024 - 12:37

score 90 · Answer 14 · 2024-03-07 19:33:32

90

Views

$\frac{3x^5+1}{x^4+x^3+1}+\frac{3y^5+1}{y^4+y^3+1}+\frac{3z^5+1}{z^4+z^3+1} \ge 4$

Published on 07 Mar 2024 - 19:33

score 41 · Answer 15 · 2024-03-07 20:38:39

41

Views

$\log_a 10 + \log_b 10 +\ log_c 10 \ge \sqrt{3 \log_a 10 * \log_b 10 * \log_c 10}$

Published on 07 Mar 2024 - 20:38

List Question

How to prove this inequality $\frac{(\sum_i w_i x_i)}{(\sum_i w_i y_i^2)} \leq \sum_i w_i \frac{x_i}{y_i^2}$

$ \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{a+c}=2 $ then $ \sum_{cyc}\sqrt{ab} \geq \frac{3}{2}$

Showing $\frac{1}{5}<(f(k,l,m,n))^3\leq\frac{64}{11}$, where $f(k,l,m,n)=\sum_{cyc}\frac{k^2}{\sqrt[3]{5k^6+6l^3m^3}}$ for positive $k$, $l$, $m$, $n$

Prove the inequality $\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right)$

A strengthening of a known inequality; looking for a neater solution

Finding all $x, y$ natural numbers

Enhanced Nesbitt's Inequality with Geometric Mean Term.

Variant Proof of the AM-GM Inequality

Regarding the specifity of the A.M.-G.M. inequality in finding maximum and minimum in Number Theory

Prove the inequality knowing $x,y,z \ge 0$ and $xyz=1$ $\frac{x^5}{x^2+1}+\frac{y^5}{y^2+1}+\frac{z^5}{z^2+1}\ge\frac{3}{2}$

Prove that $a^2+b^2+c^2 \geq ab(a+b+\sqrt{ab})+cb(c+b+\sqrt{cb})+ ac(a+c+\sqrt{ac} )$

If $x>0,y>0$ and $4xy=2^{x+y}$ then find the minimum and maximum values of $x+y$.

Prove $\frac{x^2}{a} + \frac{y^2}{b} \geq \frac{(x + y)^2}{a + b}$, with $x, y, a, b \in \mathbb{R}$ and $a, b > 0$

$\frac{3x^5+1}{x^4+x^3+1}+\frac{3y^5+1}{y^4+y^3+1}+\frac{3z^5+1}{z^4+z^3+1} \ge 4$

$\log_a 10 + \log_b 10 +\ log_c 10 \ge \sqrt{3 \log_a 10 * \log_b 10 * \log_c 10}$

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