User math110

43 votes

3 answers

2k views

Proving $\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\frac{x_{i}}{x_{j}}\right\}\le \frac{9}{14}n^2$？

asked Jan 3 at 15:25

38 votes

4 answers

4k views

A family of polynomials whose zeros all lie on the unit circle

asked May 3, 2018 at 9:20

24 votes

4 answers

1k views

show this nice and hard inequality with $ \prod_{i=1}^{n}|x_{i}-y_{i}|<e^{\frac{n}{2}}$

asked Oct 11, 2018 at 8:29

24 votes

2 answers

1k views

Is $\iiint_{[0, 1]^3} \lvert f(x)+f(y)+f(z)\rvert\, dx\, dy\, dz \ge \int_0^1 \lvert f(x)\rvert\, dx$?

asked Feb 5, 2021 at 6:28

24 votes

4 answers

2k views

This inequality why can't solve it by now (Only four variables inequality)?

asked Jan 12, 2015 at 17:02

20 votes

4 answers

2k views

When is $1^n+2^n+3^n+4^n+\cdots+n^n$ a square number?

asked Mar 5, 2018 at 13:28

18 votes

4 answers

2k views

An inequality concerning Lagrange's identity

asked May 19, 2016 at 2:40

16 votes

1 answer

1k views

Solve this Diophantine equation $(2^x-1)(3^y-1)=2z^2$

asked Apr 1, 2018 at 4:47

14 votes

4 answers

1k views

Find all solution $a,b,c$ with $(1-a^2)(1-b^2)(1-c^2)=8abc$

asked Jun 5, 2015 at 1:17

13 votes

3 answers

3k views

How do we show this matrix has full rank?

asked Dec 30, 2014 at 16:00

13 votes

0 answers

475 views

show that there exist $n$ such that $r|\binom{p^n}{q^n}$

asked Oct 20, 2017 at 11:24

13 votes

2 answers

731 views

Show that there exist $k\in\{1,2,\cdots,n\}$ such that $\frac{1}{n}\sum_{i=1}^{n}\left(\{kx_{i}\}-\frac{1}{2}\right)^2>\frac{1}{12}-\frac{1}{6n}$

asked Sep 13, 2018 at 7:46

13 votes

3 answers

2k views

I conjecture inequalities $\sum_{k=1}^{n}\{kx\}\le\frac{n}{2}x$

asked Nov 10, 2018 at 1:20

10 votes

2 answers

759 views

Find all $m$ such $2^m+1\mid5^m-1$

asked Dec 14, 2018 at 1:34

10 votes

2 answers

688 views

If $p_{n}$ is the largest prime factor of $p_{n-1}+p_{n-2}+m$, then $p_{n}$ is bounded

asked Mar 17, 2017 at 1:27

9 votes

2 answers

1k views

How to prove this inequality of Karamata type?

asked May 29, 2019 at 0:50

8 votes

2 answers

594 views

On the conjecture : $|m^2-n^3|>\frac{1}{5}\sqrt[6]{m^2+n^3}$

asked Apr 9, 2020 at 16:15

8 votes

1 answer

688 views

Prove an inequality related to sums of Legendre symbols

asked Apr 12, 2022 at 0:34

8 votes

2 answers

370 views

Find the maximum of the value $c(n)$ (similar to Hardy's inequality)

asked Nov 8, 2018 at 16:05

8 votes

2 answers

574 views

How to determine the maximums of certain cyclic sums?

asked Jan 27, 2018 at 4:43

8 votes

0 answers

666 views

How prove this polynomial inequality from a book

asked Dec 12, 2014 at 12:22

7 votes

1 answer

757 views

Find the maximum of $|a_{p}|$, if $a_0+a_1x+\dots+a_nx^n:[-1,1]\mapsto [-1,1]$

asked Dec 23, 2016 at 12:11

7 votes

2 answers

747 views

Conjecture: $a^n+b^n+c^n\ge x^n+y^n+z^n$

asked Apr 3, 2019 at 3:50

7 votes

3 answers

984 views

Prove $ n!$ is divisible by the number of its positive divisors

asked May 24, 2018 at 1:22

7 votes

1 answer

385 views

How prove this Webb inequality?

asked Jan 11, 2021 at 8:23

7 votes

1 answer

368 views

Does the limit of $x_n$, defined by $x_{n+1}=1/(m+1-nx_n)$ exist?

asked Sep 27, 2022 at 0:47

6 votes

1 answer

468 views

a problem in complex-variable inequality

asked Nov 9, 2022 at 4:39

6 votes

0 answers

233 views

Are there infinitely many positive integers $\ n\ $ satisfying $\ \varphi(n)\mid \sigma(n)$?

asked Mar 25, 2022 at 15:44

6 votes

3 answers

877 views

Is there a simple proof that $Ax^3+By^3=C$ has only finitely many integer solutions

asked Feb 6, 2021 at 1:25

6 votes

0 answers

664 views

Show this number always is composite number

asked May 8, 2018 at 11:15

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75 Questions

Proving $\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\frac{x_{i}}{x_{j}}\right\}\le \frac{9}{14}n^2$？

A family of polynomials whose zeros all lie on the unit circle

show this nice and hard inequality with $ \prod_{i=1}^{n}|x_{i}-y_{i}|<e^{\frac{n}{2}}$

Is $\iiint_{[0, 1]^3} \lvert f(x)+f(y)+f(z)\rvert\, dx\, dy\, dz \ge \int_0^1 \lvert f(x)\rvert\, dx$?

This inequality why can't solve it by now (Only four variables inequality)?

When is $1^n+2^n+3^n+4^n+\cdots+n^n$ a square number?

An inequality concerning Lagrange's identity

Solve this Diophantine equation $(2^x-1)(3^y-1)=2z^2$

Find all solution $a,b,c$ with $(1-a^2)(1-b^2)(1-c^2)=8abc$

How do we show this matrix has full rank?

show that there exist $n$ such that $r|\binom{p^n}{q^n}$

Show that there exist $k\in\{1,2,\cdots,n\}$ such that $\frac{1}{n}\sum_{i=1}^{n}\left(\{kx_{i}\}-\frac{1}{2}\right)^2>\frac{1}{12}-\frac{1}{6n}$

I conjecture inequalities $\sum_{k=1}^{n}\{kx\}\le\frac{n}{2}x$

Find all $m$ such $2^m+1\mid5^m-1$

If $p_{n}$ is the largest prime factor of $p_{n-1}+p_{n-2}+m$, then $p_{n}$ is bounded

How to prove this inequality of Karamata type?

On the conjecture : $|m^2-n^3|>\frac{1}{5}\sqrt[6]{m^2+n^3}$

Prove an inequality related to sums of Legendre symbols

Find the maximum of the value $c(n)$ (similar to Hardy's inequality)

How to determine the maximums of certain cyclic sums?

How prove this polynomial inequality from a book

Find the maximum of $|a_{p}|$, if $a_0+a_1x+\dots+a_nx^n:[-1,1]\mapsto [-1,1]$

Conjecture: $a^n+b^n+c^n\ge x^n+y^n+z^n$

Prove $ n!$ is divisible by the number of its positive divisors

How prove this Webb inequality?

Does the limit of $x_n$, defined by $x_{n+1}=1/(m+1-nx_n)$ exist?

a problem in complex-variable inequality

Are there infinitely many positive integers $\ n\ $ satisfying $\ \varphi(n)\mid \sigma(n)$?

Is there a simple proof that $Ax^3+By^3=C$ has only finitely many integer solutions

Show this number always is composite number