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ə̷̶̸͇̘̜́̍͗̂̄︣͟
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30 votes
Accepted

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

20 votes
Accepted

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

19 votes
Accepted

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

18 votes
Accepted

Is this a new characteristic function for the primes?

16 votes
Accepted

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

15 votes
Accepted

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

15 votes

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

15 votes

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

14 votes
Accepted

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

14 votes
Accepted

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 $n\ge1$?

13 votes

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

11 votes

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

11 votes
Accepted

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

11 votes

Can this determinant ever vanish?

11 votes

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

10 votes

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

10 votes
Accepted

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

10 votes

Relationship between Catalan's constant and $\pi$

10 votes
Accepted

Rotation of Hyperbola with any angle

10 votes
Accepted

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

9 votes
Accepted

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

9 votes

"Integral milking": working backward to construct nontrivial integrals

9 votes
Accepted

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

9 votes

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

9 votes

Using the fact that $\sqrt{n}$ is an irrational number whenever $n$ is not a perfect square, show $\sqrt{3} + \sqrt{7} + \sqrt{21}$ is irrational.

9 votes

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

9 votes
Accepted

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

9 votes
Accepted

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

9 votes
Accepted

Interesting integral $\int_0^\pi \frac{\sin(x)}{x} e^{x \cot(x)} \ \mathrm{d}x = \pi$

8 votes

Prove that $\int_0^\frac{\pi}{4}\frac{\cos (n-2)x}{\cos^nx}dx=\frac{1}{n-1}2^\frac{n-1}{2}\sin\frac{(n-1)\pi}{4}$

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