Prove there exist $0\le c_1<c_2<\dots<c_n\le1$ where $\sum_{k=1}^n\frac1{f'(c_k)}=n$
82405
For f continuous on $[0,1]$, show that there exist points $\alpha_k$ such that $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $
15441
Show $\frac{1}{4}\leq \mu(A\cap B \cap C)$
61859
Prove $\sum \frac{a}{a+b^4+c^4} \le 1$
139453
Prove that $\frac{1}{f'(a)} = \frac{1}{2\pi i} \int_C \frac{1}{f(z)-f(a)} dz$
105164
If $abc=1$, prove that $\frac{2}{(a+1)^2+b^2+1} + \frac{2}{(b+1)^2+c^2+1} + \frac{2}{(c+1)^2+a^2+1} \le 1$.
73892
Prove $Σ_{cyc}\frac{1}{\left(a+1\right)^2}+\frac{1}{a+b+c+1}\ge 1$
145274
Prove by induction $\sum \frac {1}{2^n} < 1$
90199
Prove that there exists a constant $c>0$ such that the following happens
37871
Show that, for all $n > 1: \frac{1}{n + 1} < \log(1 + \frac1n) < \frac1n.$
137640
Proof that $\sum_{1}^{\infty} \frac{1}{n^2} <2$
349
Prove $| \sum_{i \leq n} \frac{\mu(i)}{i} | \leq 1$
134152
Prove $\sum_{n=1}^\infty(e-\sum_{k=0}^n\frac1{k!})=1$
54982
How to prove that $x_n=1+\frac{1}{2!}+\cdots+\frac{1}{n!}$ is a Cauchy sequence?
134812
Proving that if $\sum\|f_n-e_n\|^2< 1$, $\{f_n\}$ is a complete sequence
43885
Proving that $\sum\limits_{n = 0}^{2013} a_n z^n \neq 0$ if $a_0 > a_1 > \dots > a_{2013} > 0$ and $|z| \leq 1$
132541
Showing ${1\over n}\sum|S_i|=O(\sqrt n)$ for $S_i\subset [n]$, $|S_i\cap S_i|\le 1$ for $i\ne j$
59260
How to prove the convergence of $\sum_{n\geq 0}\frac{n!^d}{(dn)!}$ for $d\geq 2$?