Let $n\in\mathbb{N}$ and $z\in\mathbb{C}$ with $|z|=1$ and $z^{2n}\neq-1$. Prove that $\frac{z^n}{1+z^{2n}}\in\mathbb{R}$.
54059
Showing that $\frac{z^n}{1+z^{2n}}$ is real for $|z|=1$.
13902
Show that $\frac{1}{x_1^2}+\frac{1}{4\cdot x_2^2}+\ldots+\frac{1}{n^2\cdot x_n^2}\leq\frac{3n-2}{2n-1}$.
108298
Prove $\sum_{i=1}^{i=n}\frac{a_i}{S-a_i}\geqslant \frac{n}{n-1}$ and $\sum_{i=1}^{i=n}\frac{S-a_i}{a_i}\geqslant n(n-1)$.
94580
Prove that, $\binom{3n}{r}=\binom{3n-1}{r} + \binom{3n-1}{r-1}$
12020
Prove $\sqrt{n+1}-\sqrt{n}\lt \frac{1}{2\sqrt{n}}$
48503
Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $
121874
Use Induction to prove that for all $n \in \mathbb{N}, (x^n + \frac{1}{x^n}) \in \mathbb{Z}$ if $x+\frac{1}{x}\in\mathbb{Z}$.
71299
Prove that $\frac{a}{b} < \frac{a+c}{b+d} <\frac{ c}{d}$
112907
Math Induction Proof: $(1+\frac1n)^n < n$
66145
>Prove that $\frac d {dx} x^n=nx^{n-1}$ for all $n \in \mathbb R$.
100375
Prove that ${\sqrt {n} }^{\sqrt {n+1}} > {\sqrt {n+1}}^{\sqrt {n}}$.
257
Prove that : $m_{a}m_{b}m_{c}\leq\frac{Rs^{2}}{2}$
15164
Proof of $\sum^{2N}_{n=1} \frac{(-1)^{n-1}}{n} = \sum^{N}_{n=1} \frac{1}{N+n}$
19896
Prove $\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\geq \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$
111847
For $a,b,c>0$, prove that $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c+\frac{4(a-b)^2}{a+b+c}$
80216
Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$
61186
For which $n$ does $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ imply $\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}$