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LHF
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24 votes
1 answer
932 views

Prove that $e^\pi+\frac{1}{\pi} < \pi^e+1$

4 votes
2 answers
172 views

Evaluate $\lim\limits_{n\to\infty}\left(\sqrt [n+1]{\frac {a_{n+1}}{b_{n+1}}}-\sqrt[n]{\frac {a_n}{b_n}}\ \right)$

3 votes
5 answers
694 views

Asymmetric inequality in three variables $\frac{3(a+b)^2(b+c)^2}{4ab^2c} \geq 7+\frac{5(a^2+2b^2+c^2)}{(a+b)(b+c)}$

3 votes
2 answers
133 views

Given a recurrence formula, evaluate $\lim\limits_{n\to \infty} n^2 x_n^3$

3 votes
2 answers
175 views

Evaluate definite integral $\int_0^{e^{\pi}} |\cos\ (\ln x)|dx$

2 votes
1 answer
104 views

Another definite integral $\int_{\frac{1}{n}}^1\,\cos\, \left(\{nx\} \cdot \pi\right)dx$

2 votes
2 answers
151 views

Maximize product $(a^3-a^2+2)(b^3-b^2+2)(2c^3+5c^2+9)$

1 vote
0 answers
157 views

Non-homogeneous cyclic $\frac{x+1}{\sqrt{x+y}}+\frac{y+1}{\sqrt{y+z}}+\frac{z+1}{\sqrt{z+x}} \geq 3\sqrt{2}$

1 vote
0 answers
90 views

Evaluate $\lim\limits_{n\to \infty} \left(\frac{a_{n+1}}{\sqrt[n+1]{(n+1)!}} - \frac{a_n}{\sqrt[n]{n!}} \right)$

0 votes
1 answer
137 views

Another asymmetric inequality $5+\frac{3(a^2+2b^2+c^2)}{(a+b)(b+c)} \geq \frac{9(a+b)(b+c)(c+a)}{(a+b+c)(ab+bc+ca)}$

-1 votes
1 answer
75 views

Evaluate $\lim\limits_{n\to\infty} \frac{1}{n^2}\sum_{2\leq i < j \leq n} \log_i j$