User Donald Splutterwit

81 votes

Can we calculate $ i\sqrt { i\sqrt { i\sqrt { \cdots } } }$?

answered Mar 1, 2018 at 21:19

55 votes

Prove that every positive integer divides a number such as $70, 700, 7770, 77000$.

answered Oct 8, 2017 at 20:06

35 votes

Accepted

Why do these paths all have the same length?

answered Jan 26, 2018 at 20:02

21 votes

If $u_1=1$ and $u_{n+1} = n+\sum_{k=1}^n u_k^2$, then $u_n$ is never a square.

answered May 14, 2019 at 0:18

18 votes

How to find $x$ given $\log_{9}\left(\frac{1}{\sqrt3}\right) =x$ without a calculator?

answered Nov 3, 2017 at 1:27

17 votes

Accepted

Sphere inside of a Sphere

answered Dec 9, 2019 at 23:20

16 votes

There is a number divisible by all integers from 1 to 200, except for two consecutive numbers. What are the two?

answered Nov 14, 2019 at 23:55

15 votes

How many pairs of numbers are there so they are the inverse of each other and they have the same decimal part?

answered Feb 25, 2017 at 20:05

14 votes

What's the expected number of times I have to roll two die until they both sum $7$?

answered Feb 2, 2018 at 20:19

13 votes

Accepted

Minimum of the given expression

answered Jul 28, 2017 at 13:47

13 votes

Solve the system of equations for $\sqrt{xy}$

answered Jan 3, 2021 at 17:40

12 votes

Accepted

Transforming a 8x8, 4x4 and 1x1 square into a 9x9 square

answered Oct 2, 2017 at 7:52

12 votes

Iterative calculation of mean and standard deviation

answered Feb 17, 2017 at 16:04

10 votes

How do you simplify an expression involving fourth and higher order trigonometric functions?

answered Oct 9, 2017 at 1:43

10 votes

Number of ways to color the edges of a tetrahedron with two colors?

answered Feb 17, 2018 at 18:47

10 votes

Accepted

Conjecture: $\sum\limits_{n\geq0}\left(\frac12\right)^n\prod\limits_{k=1}^{n}\frac{2n-2k+1}{2n-2k+2}=\sqrt2$

answered Jan 20, 2019 at 22:35

10 votes

How can I find a bound for $\frac{n}{1} +\cdots+ \frac{n}{n}$ given $n< 10^{18}$?

answered Jun 25, 2017 at 21:19

9 votes

Value of $(\alpha^2+1)(\beta^2+1)(\gamma^2+1)(\delta^2+1)$ if $z^4-2z^3+z^2+z-7=0$ for $z=\alpha$, $\beta$, $\gamma$, $\delta$

answered Oct 8, 2017 at 16:31

9 votes

Accepted

Prove by induction: $2^{n+2} +3^{2n+1}$ is divisible by $7$ for all $n \in \mathbb{N}$

answered Sep 13, 2017 at 10:23

9 votes

A direct proof for $\int_0^x \frac{- x \ln(1-u^2)}{u \sqrt{x^2-u^2}} \, \mathrm{d} u = \arcsin^2(x)$

answered Jun 28, 2018 at 23:34

9 votes

Accepted

Combinatorial proof of Negative Binomial Identity $(1+x)^{-n}= \sum_{k=0}^{\infty} (-1)^k \binom{n+k-1}{k} x^k$

answered Jul 18, 2018 at 19:37

9 votes

An expression in rational numbers $x, y,$ and $z$: Why is it a square of a rational number?

answered Sep 19, 2020 at 15:46

8 votes

Accepted

Binomial theorem relating proof

answered Oct 3, 2017 at 19:33

8 votes

Prove that $\left(1+\frac1 n\right)^n > 2$

answered Jul 17, 2018 at 0:07

8 votes

Need help figure out a Fibonacci related math trick

answered Sep 5, 2019 at 20:16

7 votes

Probabilities : A red die, a blue die, and a yellow die (all six-sided) are rolled.

answered Sep 27, 2017 at 20:15

7 votes

Number of ways of dividing 20 persons into 10 couples

answered Oct 20, 2017 at 13:12

7 votes

Accepted

Concerning this integral $\int_{0}^{1}\left({1\over \ln(x)}+{1\over 1-x} -{x^s\over 2}\right){\mathrm dx\over 1-x}$

answered Oct 31, 2017 at 21:51

7 votes

Accepted

How can we show that $\sum_{n=1}^{\infty}{\zeta(2n)\over a^{2n}n}=\ln\left({\pi\over a\sin({\pi\over a})}\right)$?

answered Aug 27, 2017 at 20:22

7 votes

Accepted

Derivative of binomial coefficients

answered Mar 2, 2018 at 21:19

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1,386 Answers

Can we calculate $ i\sqrt { i\sqrt { i\sqrt { \cdots } } }$?

Prove that every positive integer divides a number such as $70, 700, 7770, 77000$.

Why do these paths all have the same length?

If $u_1=1$ and $u_{n+1} = n+\sum_{k=1}^n u_k^2$, then $u_n$ is never a square.

How to find $x$ given $\log_{9}\left(\frac{1}{\sqrt3}\right) =x$ without a calculator?

Sphere inside of a Sphere

There is a number divisible by all integers from 1 to 200, except for two consecutive numbers. What are the two?

How many pairs of numbers are there so they are the inverse of each other and they have the same decimal part?

What's the expected number of times I have to roll two die until they both sum $7$?

Minimum of the given expression

Solve the system of equations for $\sqrt{xy}$

Transforming a 8x8, 4x4 and 1x1 square into a 9x9 square

Iterative calculation of mean and standard deviation

How do you simplify an expression involving fourth and higher order trigonometric functions?

Number of ways to color the edges of a tetrahedron with two colors?

Conjecture: $\sum\limits_{n\geq0}\left(\frac12\right)^n\prod\limits_{k=1}^{n}\frac{2n-2k+1}{2n-2k+2}=\sqrt2$

How can I find a bound for $\frac{n}{1} +\cdots+ \frac{n}{n}$ given $n< 10^{18}$?

Value of $(\alpha^2+1)(\beta^2+1)(\gamma^2+1)(\delta^2+1)$ if $z^4-2z^3+z^2+z-7=0$ for $z=\alpha$, $\beta$, $\gamma$, $\delta$

Prove by induction: $2^{n+2} +3^{2n+1}$ is divisible by $7$ for all $n \in \mathbb{N}$

A direct proof for $\int_0^x \frac{- x \ln(1-u^2)}{u \sqrt{x^2-u^2}} \, \mathrm{d} u = \arcsin^2(x)$

Combinatorial proof of Negative Binomial Identity $(1+x)^{-n}= \sum_{k=0}^{\infty} (-1)^k \binom{n+k-1}{k} x^k$

An expression in rational numbers $x, y,$ and $z$: Why is it a square of a rational number?

Binomial theorem relating proof

Prove that $\left(1+\frac1 n\right)^n > 2$

Need help figure out a Fibonacci related math trick

Probabilities : A red die, a blue die, and a yellow die (all six-sided) are rolled.

Number of ways of dividing 20 persons into 10 couples

Concerning this integral $\int_{0}^{1}\left({1\over \ln(x)}+{1\over 1-x} -{x^s\over 2}\right){\mathrm dx\over 1-x}$

How can we show that $\sum_{n=1}^{\infty}{\zeta(2n)\over a^{2n}n}=\ln\left({\pi\over a\sin({\pi\over a})}\right)$?

Derivative of binomial coefficients