# china math

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# 524 Questions

 70 How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$? 60 How to find this limit: $A=\lim_{n\to \infty}\sqrt{1+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{3}+\cdots+\sqrt{\frac{1}{n}}}}}$ 43 How prove this inequality $\sin{\sin{\sin{\sin{x}}}}\le\frac{4}{5}\cos{\cos{\cos{\cos{x}}}}$ 36 How to prove $k!+(2k)!+\cdots+(nk)!$ has a prime divisor greater than $k!$ 28 How prove this function $f(x)=x!-x^n$ is injective

# 6,898 Reputation

 +10 How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$? +5 How find the smallest $m$ such this $|A|=n,|B|=m,A\subseteq B$ +10 How find this limit $\lim_{x\to 0}\frac{1}{x^4}\left(\frac{1}{x}\left(\frac{1}{\tanh{x}}-\frac{1}{\tan{x}}\right)-\frac{2}{3}\right)=？$ +5 How find this inequality minimum $\sum_{cyc}\sqrt{a^2+b^2+ab-2a-b+1}$

 31 Solve $(4x+3)^2(2x+1)(x+1)=75$? 12 Find the maximum value of $\sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1}$ 7 Proving $\sqrt[3]{\sqrt[3]{2}-1}=\sqrt[3]{\frac19}-\sqrt[3]{\frac29}+\sqrt[3]{\frac49}$ 6 Hard Olympiad Inequality 5 Calculation of $\int_0^{\infty}\frac{\sin^2 x\,\cos^2 x}{x^2}dx=\frac{\pi}{4}$

# 93 Tags

 43 polynomials × 14 12 roots × 3 23 calculus × 64 7 number-theory × 58 21 inequality × 121 5 real-analysis × 18 15 algebra-precalculus × 11 5 improper-integrals × 9 12 radicals × 6 3 analysis × 57

# 1 Account

 Mathematics 6,898 rep 21580