If $a_1a_2\cdots a_n =1$, then $\prod_{i=1}^n (1+a_i) \geq 2^n$
33903
Prove that $(1+a_1) \cdot (1+a_2) \cdot \dots \cdot (1+a_n) \geq 2^n$
139503
How to prove $|a+b|^k \leq 2^{k-1} (|a|^k+|b|^k)$?
119436
Proving $2^n\leq 2^{n+1}-2^{n-1}-1$ for all $n\geq 1$ by induction
120954
Prove that: $\sqrt[3]{a_1^3+ a_2^3 +\cdots+a_n^3} \le \sqrt{a_1^2 + a_2^2 +\cdots+a_n^2}$
31331
$|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
134007
Borel-Cantelli-related exercise: Show that $\sum_{n=1}^{\infty} p_n < 1 \implies \prod_{n=1}^{\infty} (1-p_n) \geq 1- S$.
68771
Assume that $ 1a_1+2a_2+\cdots+na_n=1$, where the $a_j$ are real numbers.
59479
For a positive integer $n\geq 2$ with divisors $1=d_1<d_2<\cdots<d_k=n$, prove that $d_1d_2+d_2d_3+\cdots+d_{k-1}d_k<n^2$
6539
Prove $\sum_{i=1}^n i! \cdot i = (n+1)! - 1$?
25258
Prove that if $ a,b,c > 0 $, then $ [(1 + a) (1 + b) (1 + c)]^{7} > 7^{7} (a^{4} b^{4} c^{4}) $.
151154
Prove the inequality $\sqrt\frac{a}{a+8} + \sqrt\frac{b}{b+8} +\sqrt\frac{c}{c+8} \geq 1$ with the constraint $abc=1$
142992
Prove that for every positive integer $n$, $1/1^2+1/2^2+1/3^2+\cdots+1/n^2\le2-1/n$
115715
Evaluate these infinite products $\prod_{n\geq 2}(1-\frac{1}{n^3})$ and $\prod_{n\geq 1}(1+\frac{1}{n^3})$
111515
Proving that $-a=(-1)\cdot a$
107494
prove $\frac {ab}{a+b} \geq \sum ^n_{i=1} \frac{a_ib_i}{a_i+b_i}$
18285
Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$
41792
If $abc=1$, then $\frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $