Show that for any integer $n$ we have $1+2\left( abc \right)^{n}\ge a^{2n}+b^{2n}+c^{2n}$
51769
If $x,y,z\in[-1,1]$ and $1+2xyz\geq x^2+y^2+z^2$, then can we infer $1+2(xyz)^n\geq x^{2n}+y^{2n}+z^{2n}$?
83752
If $p + q = 1$ prove that for any natural $n, m$ following is true: $(1 - p^n)^m + (1 - q^m)^n \ge 1$
111098
Prove $2^{2n} = \sum_{k=0}^{n}\binom{2n+1}{k}$
77064
Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$
48081
Prove that for all $n \in \mathbb{N}$: $2^{n-1}(a^n + b^n) \ge (a + b)^n$
21612
How to prove this form of $n$?
61117
Prove by induction: $n! \ge 2^{(n-1)}$ for any $n \ge 1$
96327
Show that $a^x+b^x+c^x>(a+b+c)^x$ for all $a,b,c>0$ and $0<x<1$
87813
How to show that for all k, $k! \ge (k/2)^{k/2}$
89175
Proof of $(a+b)^{n+1}$
137864
Prove that $ \left(1+\frac a b \right) \left(1+\frac b c \right)\left(1+\frac c a \right) \geq 2\left(1+ \frac{a+b+c}{\sqrt[3]{abc}}\right)$.
92300
For positive $a$, $b$, $c$ with $abc=1$, show that $\sum_{cyc}\sqrt{a^2-a+1}\geq a+b+c$
84213
How to show $n\sum_{i=1}^n {x_i^2} \ge (\sum_{i=1}^n{x_i})^2$
136274
$n! \leq \left( \frac{n+1}{2} \right)^n$ via induction
5694
How to prove $a^2 + b^2 + c^2 \ge ab + bc + ca$?
6777
Showing $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$
35611
How can I show that $n! \leqslant \left(\frac{n+1}{2}\right)^n$?