Question

Show this inequality $\frac{n}{a_1 - a_0} + \frac{n - 1}{a_2 - a_1} + \cdots + \frac{1}{a_n - a_{n-1}} \ge \sum_{k=1}^n \frac{k^2}{a_k}$

score 122 · Answer 1 · 2026-04-17 13:56:37

122

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Show this inequality $\frac{n}{a_1 - a_0} + \frac{n - 1}{a_2 - a_1} + \cdots + \frac{1}{a_n - a_{n-1}} \ge \sum_{k=1}^n \frac{k^2}{a_k}$

Published on 17 Apr 2026 - 13:56

score 126 · Answer 2 · 2026-04-16 22:41:30

Prove $\sum_{i=1}^n\frac{1}{\sqrt{i}}\le\frac{1}{\sqrt{n!}}\prod_{i=2}^n(\sqrt{i-1})+2\sum_{i=2}^n\frac{1}{\sqrt{i}}$ using Weierstrass inequality

score 101 · Answer 3 · 2026-04-16 22:07:17

101

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Values of $x$ satisfying $\lfloor{x^2\rfloor}=\left(\lfloor{x\rfloor}\right)^2$

Published on 16 Apr 2026 - 22:07

score 207 · Answer 4 · 2026-04-09 14:12:46

207

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Conditional Expectation Inequality for bounded moment

Published on 09 Apr 2026 - 14:12

score 108 · Answer 5 · 2026-03-25 09:28:56

108

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 25 Mar 2026 - 9:28

score 96 · Answer 6 · 2026-04-15 12:04:55

96

Views

How to prove by mathematical induction that $(y-x)x^n \leq \frac{y^{n+1}-x^{n+1}}{n+1} \leq (y-x)y^n$?

Published on 15 Apr 2026 - 12:04

score 318 · Answer 7 · 2026-04-12 13:37:57

318

Views

Challenging inequality with three variables

Published on 12 Apr 2026 - 13:37

score 54 · Answer 8 · 2026-03-25 03:35:31

54

Views

Approximating $\max \big\{\frac{x_\tau}{x}\big\}$ as $x$ and $x_\tau \rightarrow 0^+$

Published on 25 Mar 2026 - 3:35

score 53 · Answer 9 · 2026-04-13 17:35:06

53

Views

$\forall\varepsilon > 0,\exists\ a >0 : |f(x)|\,\le\, a\|f\|_2 + \varepsilon\|f'\|_2$ for $f\in H^1(0,1)$

Published on 13 Apr 2026 - 17:35

score 266 · Answer 10 · 2026-04-16 01:25:04

266

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Calculate the minimum value of $\left(\frac{n + p}{mx} + \frac{p + m}{ny} + \frac{m + n}{pz}\right) \cdot \sqrt{yz + zx + xy}$.

Published on 16 Apr 2026 - 1:25

score 88 · Answer 11 · 2026-04-10 05:03:17

88

Views

Reasoning about inequalities involving floor functions

Published on 10 Apr 2026 - 5:03

score 89 · Answer 12 · 2026-04-15 15:57:28

89

Views

Proof $|a-b| \lt \frac{|a|}{2} \rightarrow |b| \gt \frac{|a|}{2}$

Published on 15 Apr 2026 - 15:57

score 98 · Answer 13 · 2026-04-15 21:11:49

98

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Prove that $\sum_{cyc}\frac{a^2}{ca^2 + 2c^2} \ge 1$

Published on 15 Apr 2026 - 21:11

score 76 · Answer 14 · 2026-04-16 15:06:50

76

Views

Interesting Inequality With Exponents And Base > 1

Published on 16 Apr 2026 - 15:06

score 59 · Answer 15 · 2026-04-15 21:58:53

59

Views

Prove and interpret $|f(x)+g(x)|=|f(x)|+|g(x)| \implies f(x)g(x)\geq0$

Published on 15 Apr 2026 - 21:58

List Question

Show this inequality $\frac{n}{a_1 - a_0} + \frac{n - 1}{a_2 - a_1} + \cdots + \frac{1}{a_n - a_{n-1}} \ge \sum_{k=1}^n \frac{k^2}{a_k}$

Prove $\sum_{i=1}^n\frac{1}{\sqrt{i}}\le\frac{1}{\sqrt{n!}}\prod_{i=2}^n(\sqrt{i-1})+2\sum_{i=2}^n\frac{1}{\sqrt{i}}$ using Weierstrass inequality

Values of $x$ satisfying $\lfloor{x^2\rfloor}=\left(\lfloor{x\rfloor}\right)^2$

Conditional Expectation Inequality for bounded moment

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

How to prove by mathematical induction that $(y-x)x^n \leq \frac{y^{n+1}-x^{n+1}}{n+1} \leq (y-x)y^n$?

Challenging inequality with three variables

Approximating $\max \big\{\frac{x_\tau}{x}\big\}$ as $x$ and $x_\tau \rightarrow 0^+$

$\forall\varepsilon > 0,\exists\ a >0 : |f(x)|\,\le\, a\|f\|_2 + \varepsilon\|f'\|_2$ for $f\in H^1(0,1)$

Calculate the minimum value of $\left(\frac{n + p}{mx} + \frac{p + m}{ny} + \frac{m + n}{pz}\right) \cdot \sqrt{yz + zx + xy}$.

Reasoning about inequalities involving floor functions

Proof $|a-b| \lt \frac{|a|}{2} \rightarrow |b| \gt \frac{|a|}{2}$

Prove that $\sum_{cyc}\frac{a^2}{ca^2 + 2c^2} \ge 1$

Interesting Inequality With Exponents And Base > 1

Prove and interpret $|f(x)+g(x)|=|f(x)|+|g(x)| \implies f(x)g(x)\geq0$

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