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math110
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Like to share interesting and hard question and like solve math problem

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280
How does one prove the determinant inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?
Jul 8, 2014
113
Find all functions $f$ such that if $a+b$ is a square, then $f(a)+f(b)$ is a square
Apr 28, 2014
108
How prove this nice limit $\lim\limits_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$
Sep 9, 2013
98
Prove $\left(\frac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$
May 3, 2013
82
How to find this limit: $A=\lim_{n\to \infty}\sqrt{1+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{3}+\cdots+\sqrt{\frac{1}{n}}}}}$
Nov 21, 2013
60
How prove this inequality $\sin{\sin{\sin{\sin{x}}}}\le\frac{4}{5}\cos{\cos{\cos{\cos{x}}}}$
Dec 2, 2013
59
How to evaluate $\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)\ \mathrm dx$
Mar 14, 2013
53
Prove $\int\limits_{0}^{\pi/2}\frac{dx}{1+\sin^2{(\tan{x})}}=\frac{\pi}{2\sqrt{2}}\bigl(\frac{e^2+3-2\sqrt{2}}{e^2-3+2\sqrt{2}}\bigr)$
Mar 29, 2014
47
For $a$, $b$, $c$, $d$ the sides of a quadrilateral, show $ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$. (A generalization of IMO 1983 problem 6)
Aug 13, 2013
44
How to prove $k!+(2k)!+\cdots+(nk)!$ has a prime divisor greater than $k!$
Jun 12, 2014
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43
Proving $\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\frac{x_{i}}{x_{j}}\right\}\le \frac{9}{14}n^2$？
38
A family of polynomials whose zeros all lie on the unit circle
25
Is $\iiint_{[0, 1]^3} \lvert f(x)+f(y)+f(z)\rvert\, dx\, dy\, dz \ge \int_0^1 \lvert f(x)\rvert\, dx$?
24
This inequality why can't solve it by now (Only four variables inequality)?
24
show this nice and hard inequality with $ \prod_{i=1}^{n}|x_{i}-y_{i}|<e^{\frac{n}{2}}$