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math110
  • Member for 11 years, 5 months
  • Last seen more than a week ago
43 votes

Collection of surprising identities and equations.

38 votes
Accepted

Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $a^2 + b^2 + c^2 \ge a + b + c$.

33 votes
Accepted

Solve $(4x+3)^2(2x+1)(x+1)=75$?

26 votes

Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$

25 votes
Accepted

Prove that there is a real number $a$ such that $\frac{1}{3} \leq \{ a^n \} \leq \frac{2}{3}$ for all $n=1,2,3,...$

24 votes
Accepted

How to evaluate the following integral $\int_0^{\pi/2}\sin{x}\cos{x}\ln{(\sin{x})}\ln{(\cos{x})}\,dx$?

24 votes
Accepted

Show that $\frac {a_1^2}{a_2}+\frac {a_2^2}{a_3}+...+\frac {a_n^2}{a_1}\geq a_1+a_2+...+a_n$ using AM-GM.

22 votes
Accepted

Prove that if $x_1,x_2,\ldots,x_n>0$, then $(1+x_1)(1+x_1+x_2)\ldots(1+x_1+x_2+\ldots+x_n) \ge \sqrt{(n+1)^{n+1}}\sqrt{x_1x_2\ldots x_n}$.

21 votes
Accepted

How can I prove $\sum_{n=1}^{\infty }\frac{1}{n^3(n+1)^3}=10-\pi ^2$

21 votes
Accepted

Computing $ \int_0^\infty \frac{\log x}{\exp x} \ dx $

21 votes
Accepted

Is there any explicit formula for $x_n$?

19 votes
Accepted

Prove $\sum_{n=1}^\infty \text{arccot }a_n^2=\frac{\pi}{12}$ where $a_n=\frac{\left(2+\sqrt{3}\right)^n-\left(2-\sqrt{3}\right)^n}{\sqrt{3}}$

19 votes

Estimate the integral of the absolute value of the Dirichlet kernel

18 votes
Accepted

Show that $\frac {a+b+c} 3\geq\sqrt[27]{\frac{a^3+b^3+c^3}3}$.

18 votes
Accepted

Evaluate $\int_0^\infty\!\!\int_0^\infty\!\!\int_0^\infty\!\frac{(xyz)^{-1/7}(yz)^{-1/7}z^{-1/7}}{(x+1)(y+1)(z+1)}dx\,dy\,dz$

17 votes

Evaluating the sum : $\;\frac{1}{3}+\frac{1}{4}.\frac{1}{2!}+\frac{1}{5}.\frac{1}{3!}+\ldots$

17 votes
Accepted

Find the maximum value of $ \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1} $

17 votes
Accepted

Gosper's unusual formula connecting $e$ and $\pi$

17 votes

How do I evaluate $\int_{1/3}^3 \frac{\arctan x}{x^2 - x + 1} \; dx$?

16 votes
Accepted

Recurrent sequence limit

15 votes
Accepted

Proving that $\lim_{x\to1^-}\left(\sqrt[a]{1-x}\cdot\sum_{n=0}^\infty~x^{n^a}\right)=\Gamma\left(1+\frac1a\right)$

15 votes
Accepted

For which $n$ does $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ imply $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}$

15 votes

Solving $2^{2x+1} - 2^{x+4} = 2^3 - 2^x$

15 votes
Accepted

Hard contest type trigonometry proof

13 votes

Limit of the sequence $\lim_{n\to\infty}(n!)^{1/n^2}$

13 votes
Accepted

Prove the following inequality without using induction: $\frac{1}{2^k-1}\leq \sin^{2k}\theta+\cos^{2k}\theta\leq 1$

12 votes
Accepted

Limit of this recursive sequence: $x_{n+1}=\bigl(1-\frac{1}{2n}\bigr)x_{n}+\frac{1}{2n}x_{n-1}.$

12 votes
Accepted

Prove that if $({x+\sqrt{x^2+1}})({y+\sqrt{y^2+1}})=1$ then $x+y=0$

12 votes

Maximum value of Product of Cosines

12 votes
Accepted

sum of polynoms of given property

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