If $|p(z)| \leq 1$ for all $z$ with $|z| \leq 1$, then show that $p(z)=z^n, \forall z\in \mathbb{C}$
114812
Let $p(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+...+a_0$ be a complex polynomial satisfying $|p(z)| \leq 1$ for $|z|\leq 1$, then prove that $p(z)=z^n$
30568
Let $q \in \mathbb C$, $\|q\|=1$ and $q^n \neq 1, \forall n \in \mathbb N$. Show that $\{q^n: n \in \mathbb N\}$ is dense in $S^1$
79847
proving of inequality $(n!)^2 \leq n^n\cdot n!\leq (2n!)$, where $n\in \mathbb{N}$
46760
To show that $P(|X-Y| \leq 2) \leq 3P(|X-Y| \leq 1)$
140412
If $p$ and $q$ are distinct primes such that $p|c$ and $q|c$, prove that $pq|c$
96116
Show $P(z) $ is a polynomial of degree $n-1$ interpolating an Analytic function.
119436
Proving $2^n\leq 2^{n+1}-2^{n-1}-1$ for all $n\geq 1$ by induction
31436
Show that for all $a\in\mathbb{N}$, there exists $b\in\mathbb{N}$ and square-free integer $c$ such that $\sqrt{a}=b\sqrt{c}$.
153107
Does every connected component of $\{z : |P(z)|<1 \}$ contain a zero?
30630
Proving ${n \choose p} \equiv \Bigl[\frac{n}{p}\Bigr] \ (\text{mod} \ p)$
62755
The simplest way of proving that $|\mathcal{P}(\mathbb{N})| = |\mathbb{R}| = c$
110310
$|x_{n+1} - x_n| \le C |x_n - x_{n-1}|.$ Prove $(x_n)$ is cauchy
76049
Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for all $s, t \in \mathbb{Z}$
46147
Proving $(\forall a\in\mathbb{Z^+})(m\in\mathbb{Z^+}\to a^m\equiv a^{m-\phi(m)}\pmod{m})$
133228
Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.
80826
To prove $\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$ for some $M>0$
115726
For $p,q$ prime and $p \neq q$ show that $C_p \times C_q \cong C_{pq}$