Quintofron

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 6 Proving an equality involving compositions of an integer 3 Proving a statement about $k$-colouring of a graph 3 A property of a prime divisor of a number consisting of 1s 3 Expected value - removing digits from a number 3 A couple of asymptotics exercises

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 5 How to prove this inequality $\sqrt{\frac{ab+bc+cd+da+ac+bd}{6}}\geq \sqrt[3]{{\frac{abc+bcd+cda+dab}{4}}}$ 3 A question with the sequence $e_{n}=\left(1+\frac{1}{n}\right)^{n}$ 2 Prove $\frac{a^2}{b^2+c^2} + \frac{b^2}{c^2+a^2} + \frac{c^2}{a^2 + b^2} \geq \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}$ 2 I know that, $S_{2n}+4S_{n}=n(2n+1)^2$. Is there a way to find $S_{2n}$ or $S_{n}$ by some mathematical process with just this one expression? 1 Showing that $1<\sin\frac{\alpha}{2}+\sin\frac{\beta}{2}+\sin\frac{\gamma}{2}$

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 8 inequality × 4 3 sequences-and-series 8 contest-math × 2 2 number-theory × 2 5 radicals 2 recurrence-relations 5 substitution 1 trigonometry × 2 5 uvw 1 triangles

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