# math110

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I Love math and inequality

# 653 Questions

 59 Prove $\left(\dfrac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$ 52 How prove this nice limit $\lim_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$ 38 How to evaluate $I=\displaystyle\int_0^{\pi/2}x^2\ln(\sin x)\ln(\cos x)\ \mathrm dx$ 35 A generalization of IMO 1983 problem 6 29 How to prove this inequality? $\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\ge 4+(x-y)^2$

# 15,178 Reputation

 +10 $x,y,z$ positive real numbers , $x+y+z=3$ $\implies x^4y^4z^4(x^3+y^3+z^3)≤3$ +10 The minimum value of $\frac{(x+\frac{1}{x})^6-(x^6+\frac{1}{x^6})-2}{(x+\frac{1}{x})^3+x^3+\frac{1}{x^3}}$ +5 How find the limit $\lim_{n\to+\infty}\sum_{i=2}^{n}\dfrac{\ln{i^2}}{i^2}$ +5 How can I find the example of $f(x)$ such that $\,\lim_{x\to\infty}f(x) \neq 0$?

 22 Surprising identities / equations 21 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. 20 Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $a^2 + b^2 + c^2 \ge a + b + c$. 16 Show that $\frac {a+b+c} 3\geq\sqrt[27]{\frac{a^3+b^3+c^3}3}$. 14 Prove $\sum_{n=1}^\infty \text{arccot }a_n^2=\frac{\pi}{12}$ where $a_n=\frac{\left(2+\sqrt{3}\right)^n-\left(2-\sqrt{3}\right)^n}{\sqrt{3}}$

# 124 Tags

 326 inequality × 306 57 real-analysis × 30 117 calculus × 103 42 limits × 68 71 trigonometry × 51 40 contest-math × 27 64 integration × 69 36 definite-integrals × 17 61 sequences-and-series × 97 23 polynomials × 27

# 3 Accounts

 Mathematics 15,178 rep 225142 MathOverflow 127 rep 8 Meta Stack Exchange 101 rep 1

 +50 How to find $f(x)$ when $f\left(\frac{x+y}{x-y}\right)\ge\frac{f(x)-f(y)}{f(x)+f(y)}$ +50 Find the number of ways giving name tags such that there exist a student who don't exit the table after 4 operations.