Prove that for any integer $n > 3$ there exist positive integers $a_1 > \dots > a_n$ such that $1 = \frac{1}{a_1} + \dots + \frac{1}{a_n}$
91105
For every $n\ge3$ there exists $n$ different integers $a_1,...,a_n$ so that $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}=1$
34767
Proving $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$
20593
Show that if $a \neq b$ and a and b are positive then $\frac{a}{b}+\frac{b}{a}$ is never an integer
150565
Prove $(a+b)^\frac{1}{n} \le a^\frac{1}{n} + b^\frac{1}{n}$
59535
Prove that there exists such a set of $n$ positive integers
73191
Proving that $\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots + \frac{1}{n(n+1)} = \frac{n}{n+1}$
107448
proving that there is no solution for $n$ in positive integers except $n = 1$ for $2^n = nk + 1$
41784
Prove using mathematical induction that $n < 3 ^n$ for all positive integers n.
106868
Prove that $x+\frac1{4x} ≥ 1$ for $x>0$
139383
Prove that $\tau(2^n-1) \geq \tau(n)$ for all positive integers $n$.
75166
Find with proof all positive integers $n$ such that $2^n + 1 \equiv 0 \pmod{n}$
156542
Number of solutions to $a_1 + a_2 + \dots + a_k = n$ where $n \gt 0$ and $0 \lt a_1 \leq a_2 \leq \dots \leq a_k$ are integers.
82775
if $a>1$, Prove that $\lim a^{1\over n}=1$
140763
How to prove that $\sum_{n=1}^{2012} \frac{1}{x_n} = 1 $ has finitely many solutions for positive integer $x_i$?
151153
Prove $\frac{a(a^2+2)}{3}$ is an integer for all integer $a\geqslant 1$
88425
Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$
20691
prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer