User mathlove

39 votes

3 answers

5k views

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

asked Mar 11, 2015 at 11:55

34 votes

4 answers

2k views

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

asked Oct 1, 2013 at 8:29

33 votes

3 answers

3k views

Can we simplify $\int_{0}^{\infty}\frac{{\sin}^px}{x^q}dx$?

asked Dec 1, 2013 at 14:57

23 votes

4 answers

2k views

Does this sequence always give an integer?

asked Sep 30, 2013 at 14:29

22 votes

3 answers

2k views

Letting $S(m)$ be the digit sum of $m$, then $\lim_{n\to\infty}S(3^n)=\infty$?

asked Sep 26, 2013 at 14:54

20 votes

1 answer

9k views

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem

asked Jul 31, 2013 at 14:35

20 votes

1 answer

770 views

Tetrahedra passing through a hole

asked Aug 7, 2013 at 6:29

17 votes

5 answers

1k views

Proving that every term of the sequence is an integer

asked May 29, 2013 at 7:06

17 votes

1 answer

917 views

If $\left(1^a+2^a+\cdots+n^{a}\right)^b=1^c+2^c+\cdots+n^c$ for some $n$, then $(a,b,c)=(1,2,3)$?

asked Oct 7, 2013 at 14:27

14 votes

3 answers

1k views

What is the max of $n$ such that $\sum_{i=1}^n\frac{1}{a_i}=1$ where $2\le a_1\lt a_2\lt \cdots\lt a_n\le m$?

asked Sep 16, 2013 at 14:41

13 votes

2 answers

2k views

Proving $\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}$

asked Sep 27, 2013 at 15:33

13 votes

0 answers

512 views

If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$?

asked Dec 7, 2013 at 15:58

13 votes

0 answers

626 views

Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$

asked Mar 21, 2015 at 9:29

12 votes

3 answers

1k views

A special tessellation

asked Nov 13, 2013 at 11:31

11 votes

0 answers

720 views

Making a convex polyhedron with two sheets of paper

asked Feb 24, 2014 at 11:31

10 votes

3 answers

2k views

On maximal regular polyhedra inscribed in a regular polyhedron

asked Jul 31, 2013 at 6:36

9 votes

1 answer

535 views

What is the shape of the $n$-gon which gives the maximum of a function?

asked Jul 14, 2013 at 14:29

9 votes

3 answers

412 views

About a solid which satisfies $\sum_{i=1}^{n}x_i=0, |x_i|\le1\ (i=1,2,\cdots,n)$

asked Nov 15, 2013 at 15:07

8 votes

0 answers

483 views

Proving a proposition which leads the irrationality of $\frac{\zeta(5)}{\zeta(2)\zeta(3)}$

asked Nov 7, 2013 at 15:07

7 votes

1 answer

449 views

Finding the min of $m$ such that $k = \pm 1^n \pm 2^n \pm 3^n \pm \cdots \pm m^n$ for a given pair $(n,k)$

asked Nov 22, 2013 at 11:48

7 votes

1 answer

2k views

The number of distinct prime factors of $n\in\mathbb N$

asked Nov 23, 2013 at 15:15

6 votes

0 answers

247 views

On simple normality to co-prime bases

asked Jun 23, 2014 at 11:19

5 votes

1 answer

408 views

A problem related with 'Postage stamp problem'

asked Nov 10, 2013 at 15:58

5 votes

3 answers

1k views

Finding an invisible circle by drawing another line

asked Aug 26, 2013 at 9:55

5 votes

2 answers

2k views

Solving $x^k+(x+1)^k+(x+2)^k+\cdots+(x+k-1)^k=(x+k)^k$ for $k\in\mathbb N$

asked Sep 12, 2013 at 15:11

5 votes

2 answers

1k views

About two 'negative' continued fractions whose sum equals $1$

asked Oct 16, 2013 at 15:15

5 votes

0 answers

750 views

Conjectures on perfect squares

asked Feb 15, 2015 at 17:34

5 votes

1 answer

494 views

Permutation polynomials mod $p$ of the form $(x+1)^n-x^n$

asked Mar 10, 2015 at 11:23

4 votes

0 answers

267 views

How many points does 'the-most-point-contained-circle' contain at least?

asked Jan 28, 2015 at 11:07

4 votes

1 answer

520 views

About the inscribed sphere and the exspheres of a $n$-dimensional simplex

asked Nov 24, 2013 at 15:32

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45 Questions

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

Can we simplify $\int_{0}^{\infty}\frac{{\sin}^px}{x^q}dx$?

Does this sequence always give an integer?

Letting $S(m)$ be the digit sum of $m$, then $\lim_{n\to\infty}S(3^n)=\infty$?

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem

Tetrahedra passing through a hole

Proving that every term of the sequence is an integer

If $\left(1^a+2^a+\cdots+n^{a}\right)^b=1^c+2^c+\cdots+n^c$ for some $n$, then $(a,b,c)=(1,2,3)$?

What is the max of $n$ such that $\sum_{i=1}^n\frac{1}{a_i}=1$ where $2\le a_1\lt a_2\lt \cdots\lt a_n\le m$?

Proving $\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}$

If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$?

Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$

A special tessellation

Making a convex polyhedron with two sheets of paper

On maximal regular polyhedra inscribed in a regular polyhedron

What is the shape of the $n$-gon which gives the maximum of a function?

About a solid which satisfies $\sum_{i=1}^{n}x_i=0, |x_i|\le1\ (i=1,2,\cdots,n)$

Proving a proposition which leads the irrationality of $\frac{\zeta(5)}{\zeta(2)\zeta(3)}$

Finding the min of $m$ such that $k = \pm 1^n \pm 2^n \pm 3^n \pm \cdots \pm m^n$ for a given pair $(n,k)$

The number of distinct prime factors of $n\in\mathbb N$

On simple normality to co-prime bases

A problem related with 'Postage stamp problem'

Finding an invisible circle by drawing another line

Solving $x^k+(x+1)^k+(x+2)^k+\cdots+(x+k-1)^k=(x+k)^k$ for $k\in\mathbb N$

About two 'negative' continued fractions whose sum equals $1$

Conjectures on perfect squares

Permutation polynomials mod $p$ of the form $(x+1)^n-x^n$

How many points does 'the-most-point-contained-circle' contain at least?

About the inscribed sphere and the exspheres of a $n$-dimensional simplex