User Frank

24 votes

2 answers

502 views

Evaluating :$\int \frac{1}{x^{10} + x}dx$

asked Jul 17, 2012 at 20:49

13 votes

3 answers

525 views

Prove : $\frac{\cos(x_1) +\cos(x_2) +\cdots+\cos(x_{10})}{\sin(x_1) +\sin(x_2) +\cdots+\sin(x_{10})} \ge 3$

asked Aug 10, 2012 at 1:13

13 votes

4 answers

2k views

Finding all primes $p$ such that $\frac{(11^{p-1}-1)}{p}$ is a perfect square

asked Jul 12, 2012 at 13:34

12 votes

5 answers

604 views

proving :$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$.

asked Jul 7, 2012 at 14:14

12 votes

1 answer

997 views

Prove that there does not exist any positive integers pair $(m, n)$ satisfying: $n(n + 1)(n + 2)(n + 3) = m{(m + 1)^2}{(m + 2)^3}{(m + 3)^4}$

asked Jul 10, 2012 at 12:11

12 votes

1 answer

409 views

Proving that there exists $a_i \in \{a_1,\dots,a_k\}$ so that for any positive integer $n$ $F(n)$ is divisible by $a_i$

asked Jul 4, 2012 at 9:38

12 votes

4 answers

6k views

Integral:$ \int^\infty_{-\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}dx $

asked Jul 31, 2012 at 12:50

10 votes

2 answers

729 views

Finding all integers such that $a^2+4b^2 , 4a^2+b^2$ are both perfect squares

asked Aug 20, 2012 at 15:19

10 votes

1 answer

587 views

How to find all naturals $n$ such that $\sqrt{1 {\underbrace{4\cdots4}_{n\text{ times}}}}$ is an integer?

asked Jul 14, 2012 at 1:20

9 votes

2 answers

5k views

How many triangles in picture

asked Jun 26, 2013 at 9:10

8 votes

4 answers

518 views

Proving :$\frac{1}{2ab^2+1}+\frac{1}{2bc^2+1}+\frac{1}{2ca^2+1}\ge1$

asked Jul 9, 2012 at 16:54

8 votes

1 answer

858 views

Countably compact paracompact space is compact

asked Jul 15, 2012 at 16:56

7 votes

2 answers

528 views

proving :$\frac{(ab+b)(2b+1)}{(ab+a)(5b+1)}+\frac{(bc+c)(2c+1)}{(bc+b)(5c+1)}+\frac{(ca+a)(2a+1)}{(ca+c)(5a+1)}\ge\frac{3}{2}$

asked Jul 12, 2012 at 17:08

7 votes

1 answer

358 views

proving that $2^{m-1}$ has reminder $1$ when divided by $m$

asked Jul 3, 2012 at 9:55

7 votes

2 answers

783 views

Triples of positive real numbers $(a,b,c)$ such that $\lfloor a\rfloor bc=3,\; a\lfloor b\rfloor c=4,\;ab\lfloor c\rfloor=5$

asked Jun 22, 2012 at 23:31

7 votes

5 answers

8k views

Proving:$\tan(20^{\circ})\cdot \tan(30^{\circ}) \cdot \tan(40^{\circ})=\tan(10^{\circ})$

asked Jul 17, 2012 at 23:55

7 votes

1 answer

260 views

Proving XY is perpendicular on :CD

asked Aug 28, 2012 at 0:52

6 votes

2 answers

302 views

Finding the ratio

asked Feb 8, 2013 at 16:42

6 votes

1 answer

163 views

There are infinitely many $m$ such that $m^4 + 1$ has large prime factors

asked Jul 18, 2012 at 21:26

5 votes

2 answers

595 views

Proving $\frac{x}{\sqrt{y^2+yz+z^2}}+\frac{y}{\sqrt{z^2+zx+x^2}}+\frac{z}{\sqrt{x^2+xy+y^2}} \ge \sqrt{3}$

asked Jul 20, 2012 at 11:06

5 votes

1 answer

266 views

the integral :$ \int^\infty_{-\infty}e^{-x^{2}+x}dx $

asked Jul 22, 2012 at 12:27

5 votes

2 answers

679 views

congruent to mod p $1^{p-2}+2^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.$

asked Jun 30, 2012 at 8:15

5 votes

2 answers

341 views

Inequality $\frac{a^2+b^2+c^2}{a^5+b^5+c^5}+\cdots+\frac{d^2+a^2+b^2}{d ^5+a^5+b^5}\le\frac{a+b+c+d}{abcd}$

asked Jul 11, 2012 at 13:40

5 votes

2 answers

278 views

finding all integer $n$ such that $ n\mid2^{n!}-1$

asked Jul 9, 2012 at 12:44

5 votes

1 answer

503 views

Finding the number in the fourth corner of a square

asked Feb 28, 2013 at 10:57

4 votes

0 answers

3k views

Finding the area of the shaded square inside a square created by connecting point-opposite midpoint [closed]

asked Mar 26, 2013 at 14:12

4 votes

1 answer

72 views

An increasing sequence $a_i \in \mathbb{N}$ such that $a_{j}-a_{i}\mid a_{j}-1$ for all $i<j$

asked Jul 7, 2012 at 2:54

4 votes

1 answer

138 views

Integers that are a sum of two $k$th powers in $n$ different ways

asked Jul 16, 2012 at 8:01

4 votes

1 answer

495 views

Finding $3$ distinct prime numbers $a| (bc+b+c)$, $b|(ac+a+c)$, $c|(ab+a+b)$

asked Jun 30, 2012 at 20:16

4 votes

2 answers

3k views

The arithmetic progression $ a, a+b, a+2b, a+3b,\dots $ has $k$ consecutive composites

asked Jun 29, 2012 at 10:13

Stack Exchange Network

80 Questions

Evaluating :$\int \frac{1}{x^{10} + x}dx$

Prove : $\frac{\cos(x_1) +\cos(x_2) +\cdots+\cos(x_{10})}{\sin(x_1) +\sin(x_2) +\cdots+\sin(x_{10})} \ge 3$

Finding all primes $p$ such that $\frac{(11^{p-1}-1)}{p}$ is a perfect square

proving :$\frac{ab}{a^2+3b^2}+\frac{cb}{b^2+3c^2}+\frac{ac}{c^2+3a^2}\le\frac{3}{4}$.

Prove that there does not exist any positive integers pair $(m, n)$ satisfying: $n(n + 1)(n + 2)(n + 3) = m{(m + 1)^2}{(m + 2)^3}{(m + 3)^4}$

Proving that there exists $a_i \in \{a_1,\dots,a_k\}$ so that for any positive integer $n$ $F(n)$ is divisible by $a_i$

Integral:$ \int^\infty_{-\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}dx $

Finding all integers such that $a^2+4b^2 , 4a^2+b^2$ are both perfect squares

How to find all naturals $n$ such that $\sqrt{1 {\underbrace{4\cdots4}_{n\text{ times}}}}$ is an integer?

How many triangles in picture

Proving :$\frac{1}{2ab^2+1}+\frac{1}{2bc^2+1}+\frac{1}{2ca^2+1}\ge1$

Countably compact paracompact space is compact

proving :$\frac{(ab+b)(2b+1)}{(ab+a)(5b+1)}+\frac{(bc+c)(2c+1)}{(bc+b)(5c+1)}+\frac{(ca+a)(2a+1)}{(ca+c)(5a+1)}\ge\frac{3}{2}$

proving that $2^{m-1}$ has reminder $1$ when divided by $m$

Triples of positive real numbers $(a,b,c)$ such that $\lfloor a\rfloor bc=3,\; a\lfloor b\rfloor c=4,\;ab\lfloor c\rfloor=5$

Proving:$\tan(20^{\circ})\cdot \tan(30^{\circ}) \cdot \tan(40^{\circ})=\tan(10^{\circ})$

Proving XY is perpendicular on :CD

Finding the ratio

There are infinitely many $m$ such that $m^4 + 1$ has large prime factors

Proving $\frac{x}{\sqrt{y^2+yz+z^2}}+\frac{y}{\sqrt{z^2+zx+x^2}}+\frac{z}{\sqrt{x^2+xy+y^2}} \ge \sqrt{3}$

the integral :$ \int^\infty_{-\infty}e^{-x^{2}+x}dx $

congruent to mod p $1^{p-2}+2^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.$

Inequality $\frac{a^2+b^2+c^2}{a^5+b^5+c^5}+\cdots+\frac{d^2+a^2+b^2}{d ^5+a^5+b^5}\le\frac{a+b+c+d}{abcd}$

finding all integer $n$ such that $ n\mid2^{n!}-1$

Finding the number in the fourth corner of a square

Finding the area of the shaded square inside a square created by connecting point-opposite midpoint [closed]

An increasing sequence $a_i \in \mathbb{N}$ such that $a_{j}-a_{i}\mid a_{j}-1$ for all $i<j$

Integers that are a sum of two $k$th powers in $n$ different ways

Finding $3$ distinct prime numbers $a| (bc+b+c)$, $b|(ac+a+c)$, $c|(ab+a+b)$

The arithmetic progression $ a, a+b, a+2b, a+3b,\dots $ has $k$ consecutive composites