User openletter.mousetail.nl

30 votes

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Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that $A^2$ is diagonal. Must $A$ be diagonal?

answered Apr 12, 2018 at 19:03

20 votes

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Why is $\sum\limits_{n=1}^{\infty}e^{-(n/10)^2}$ almost equal to $5\sqrt\pi-\frac12$ (agreeing up to $427$ digits)?

answered Nov 7, 2020 at 11:45

19 votes

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Does there exist a real function with domain $\Bbb{R}$ such that $f'(x)>0$ and $f''(x)+(f'(x))^2<0$ for all $x$?

answered Jul 3, 2021 at 10:10

18 votes

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Is this a new characteristic function for the primes?

answered Oct 12, 2019 at 13:31

16 votes

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Evaluating $\int_0^1\frac{3x^4+ 4x^3 + 3x^2}{(4x^3 + 3x^2 + 2x+ 1)^2}\, dx$

answered Jul 5, 2018 at 9:46

15 votes

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Maximum value of $ab+bc+ca$ given that $a+2b+c=4$

answered Sep 7, 2018 at 16:02

15 votes

Are there any "nonstandard" special angles for which trig functions yield radical expressions?

answered Mar 17, 2018 at 10:15

15 votes

Is every $N$th Fibonacci number where $N$ is divisible by $5$ itself divisible by $5$

answered Jan 29, 2018 at 9:45

14 votes

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What is the value of $\lim\limits_{n\to\infty}2^{n}x_{n}$ if $x_1\in(0,1)$ and $x_{n+1}=\frac{\sqrt{1+x_n}-\sqrt{1-x_n}}2$ for every $n\ge1$?

answered Jan 28 at 18:42

13 votes

How to prove :$\sqrt{1!\sqrt{2!\sqrt{3!\sqrt{\cdots\sqrt{n!}}}}} <3$

answered Jan 5, 2018 at 15:31

12 votes

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How to evaluate the limit of multifactorial $\lim_{n\to 0} \sqrt[n]{n!!!!\cdots !}$

answered Feb 17, 2021 at 11:05

12 votes

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Find $a$ and $b$ for which $\int_{0}^{1}( ax+b+\frac{1}{1+x^{2}} )^{2}\,dx$ takes its minimum possible value.

answered Apr 20, 2019 at 18:55

11 votes

Can this determinant ever vanish?

answered Jun 19, 2021 at 15:19

11 votes

Is there an elementary proof that $\int_0^{\infty}|\sin(x)|^{x}\ dx$ converges or diverges?

answered Jun 24, 2021 at 15:13

11 votes

How to map interval $[0, 100]$ to the interval $[100, 350]$?

answered Oct 13, 2018 at 13:12

10 votes

Show that $\int_a^b \sin\left(x+\frac{1}{x}\right) dx <3.$

answered Jul 1, 2021 at 15:05

10 votes

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Show $ \int_0^{\frac\pi2} \frac{(1+\sec^2t)\sqrt{\sec t}}{ (1+\sec t)^2-2}dt= \frac\pi{\sqrt2}$

answered Mar 21, 2021 at 11:01

10 votes

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Rotation of Hyperbola with any angle

answered Sep 18, 2018 at 19:29

10 votes

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Can we express $\pi$ in terms of $\sum_{n=1}^\infty\frac1{n^2}$?

answered May 29, 2018 at 12:59

9 votes

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Sum of reciprocal sine function $\sum\limits_{k=1}^{n-1} \frac{1}{\sin(\frac{k\pi}{n})}=?$

answered Apr 25, 2021 at 10:25

9 votes

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How to solve for $z$: $|z|=z+\bar{z}$

answered May 29, 2018 at 14:05

9 votes

Accepted

Minimum value of $\frac{b+1}{a+b-2}$

answered Jul 15, 2018 at 11:19

9 votes

Relationship between Catalan's constant and $\pi$

answered Dec 27, 2018 at 8:54

9 votes

Are there any other methods to apply to solving simultaneous equations?

answered Apr 9, 2019 at 8:34

9 votes

Evaluate $\int (1-x^{2008})^{\frac{1}{2007}} (1-x^{2007})^{\frac{1}{2008}} dx$

answered Nov 12, 2017 at 20:04

9 votes

Accepted

Finding $\sum_{n=1}^{\infty}\frac{(-1)^n (H_{2n}-H_{n})}{n2^n \binom{2n}{n}}$

answered Apr 9, 2022 at 11:00

8 votes

On $\int_0^{\pi} \operatorname{Si}^n(x) \ \mathrm{d}x $

answered Jun 17, 2021 at 15:34

8 votes

Accepted

Is there a way to analytically solve $x^\alpha + y^\alpha = \alpha(x + y)$ for $\alpha$, other than $\alpha = 1$?

answered Oct 30, 2021 at 10:03

8 votes

Accepted

Probability a random spherical triangle has area $> \pi$

answered Dec 3, 2021 at 10:35

8 votes

Accepted

Find the smallest $n$ for which there are real $a_{1}, a_{2}, \ldots,a_{n}$

answered Dec 29, 2021 at 20:32

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808 Answers

Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that $A^2$ is diagonal. Must $A$ be diagonal?

Why is $\sum\limits_{n=1}^{\infty}e^{-(n/10)^2}$ almost equal to $5\sqrt\pi-\frac12$ (agreeing up to $427$ digits)?

Does there exist a real function with domain $\Bbb{R}$ such that $f'(x)>0$ and $f''(x)+(f'(x))^2<0$ for all $x$?

Is this a new characteristic function for the primes?

Evaluating $\int_0^1\frac{3x^4+ 4x^3 + 3x^2}{(4x^3 + 3x^2 + 2x+ 1)^2}\, dx$

Maximum value of $ab+bc+ca$ given that $a+2b+c=4$

Are there any "nonstandard" special angles for which trig functions yield radical expressions?

Is every $N$th Fibonacci number where $N$ is divisible by $5$ itself divisible by $5$

What is the value of $\lim\limits_{n\to\infty}2^{n}x_{n}$ if $x_1\in(0,1)$ and $x_{n+1}=\frac{\sqrt{1+x_n}-\sqrt{1-x_n}}2$ for every $n\ge1$?

How to prove :$\sqrt{1!\sqrt{2!\sqrt{3!\sqrt{\cdots\sqrt{n!}}}}} <3$

How to evaluate the limit of multifactorial $\lim_{n\to 0} \sqrt[n]{n!!!!\cdots !}$

Find $a$ and $b$ for which $\int_{0}^{1}( ax+b+\frac{1}{1+x^{2}} )^{2}\,dx$ takes its minimum possible value.

Can this determinant ever vanish?

Is there an elementary proof that $\int_0^{\infty}|\sin(x)|^{x}\ dx$ converges or diverges?

How to map interval $[0, 100]$ to the interval $[100, 350]$?

Show that $\int_a^b \sin\left(x+\frac{1}{x}\right) dx <3.$

Show $ \int_0^{\frac\pi2} \frac{(1+\sec^2t)\sqrt{\sec t}}{ (1+\sec t)^2-2}dt= \frac\pi{\sqrt2}$

Rotation of Hyperbola with any angle

Can we express $\pi$ in terms of $\sum_{n=1}^\infty\frac1{n^2}$?

Sum of reciprocal sine function $\sum\limits_{k=1}^{n-1} \frac{1}{\sin(\frac{k\pi}{n})}=?$

How to solve for $z$: $|z|=z+\bar{z}$

Minimum value of $\frac{b+1}{a+b-2}$

Relationship between Catalan's constant and $\pi$

Are there any other methods to apply to solving simultaneous equations?

Evaluate $\int (1-x^{2008})^{\frac{1}{2007}} (1-x^{2007})^{\frac{1}{2008}} dx$

Finding $\sum_{n=1}^{\infty}\frac{(-1)^n (H_{2n}-H_{n})}{n2^n \binom{2n}{n}}$

On $\int_0^{\pi} \operatorname{Si}^n(x) \ \mathrm{d}x $

Is there a way to analytically solve $x^\alpha + y^\alpha = \alpha(x + y)$ for $\alpha$, other than $\alpha = 1$?

Probability a random spherical triangle has area $> \pi$

Find the smallest $n$ for which there are real $a_{1}, a_{2}, \ldots,a_{n}$